Integrand size = 22, antiderivative size = 434 \[ \int \frac {c+d x}{(e x)^{7/3} \left (a+b x^2\right )} \, dx=-\frac {3 c}{4 a e (e x)^{4/3}}-\frac {3 d}{a e^2 \sqrt [3]{e x}}-\frac {\sqrt [6]{b} d \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{a^{7/6} e^{7/3}}+\frac {\sqrt [6]{b} d \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{7/6} e^{7/3}}-\frac {\sqrt [6]{b} d \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{7/6} e^{7/3}}+\frac {\sqrt {3} b^{2/3} c \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{2 a^{5/3} e^{7/3}}+\frac {\sqrt {3} \sqrt [6]{b} d \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}\right )}{2 a^{7/6} e^{7/3}}-\frac {b^{2/3} c \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 a^{5/3} e^{7/3}}+\frac {b^{2/3} c \log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{4/3}\right )}{4 a^{5/3} e^{7/3}} \] Output:
-3/4*c/a/e/(e*x)^(4/3)-3*d/a/e^2/(e*x)^(1/3)-b^(1/6)*d*arctan(b^(1/6)*(e*x )^(1/3)/a^(1/6)/e^(1/3))/a^(7/6)/e^(7/3)-1/2*b^(1/6)*d*arctan(-3^(1/2)+2*b ^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(7/6)/e^(7/3)-1/2*b^(1/6)*d*arctan(3 ^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(7/6)/e^(7/3)+1/2*3^(1/2)* b^(2/3)*c*arctan(1/3*(1-2*b^(1/3)*(e*x)^(2/3)/a^(1/3)/e^(2/3))*3^(1/2))/a^ (5/3)/e^(7/3)+1/2*3^(1/2)*b^(1/6)*d*arctanh(3^(1/2)*a^(1/6)*b^(1/6)*e^(1/3 )*(e*x)^(1/3)/(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3)))/a^(7/6)/e^(7/3)-1/2*b ^(2/3)*c*ln(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))/a^(5/3)/e^(7/3)+1/4*b^(2/ 3)*c*ln(a^(2/3)*e^(4/3)-a^(1/3)*b^(1/3)*e^(2/3)*(e*x)^(2/3)+b^(2/3)*(e*x)^ (4/3))/a^(5/3)/e^(7/3)
Time = 0.53 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.81 \[ \int \frac {c+d x}{(e x)^{7/3} \left (a+b x^2\right )} \, dx=\frac {x \left (-3 a^{2/3} (c+4 d x)+2 \sqrt [6]{b} \left (\sqrt {3} \sqrt {b} c+\sqrt {a} d\right ) x^{4/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+2 \sqrt [6]{b} \left (\sqrt {3} \sqrt {b} c-\sqrt {a} d\right ) x^{4/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-4 \sqrt {a} \sqrt [6]{b} d x^{4/3} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-2 b^{2/3} c x^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}\right )+\left (b^{2/3} c-\sqrt {3} \sqrt {a} \sqrt [6]{b} d\right ) x^{4/3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+\left (b^{2/3} c+\sqrt {3} \sqrt {a} \sqrt [6]{b} d\right ) x^{4/3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )\right )}{4 a^{5/3} (e x)^{7/3}} \] Input:
Integrate[(c + d*x)/((e*x)^(7/3)*(a + b*x^2)),x]
Output:
(x*(-3*a^(2/3)*(c + 4*d*x) + 2*b^(1/6)*(Sqrt[3]*Sqrt[b]*c + Sqrt[a]*d)*x^( 4/3)*ArcTan[Sqrt[3] - (2*b^(1/6)*x^(1/3))/a^(1/6)] + 2*b^(1/6)*(Sqrt[3]*Sq rt[b]*c - Sqrt[a]*d)*x^(4/3)*ArcTan[Sqrt[3] + (2*b^(1/6)*x^(1/3))/a^(1/6)] - 4*Sqrt[a]*b^(1/6)*d*x^(4/3)*ArcTan[(b^(1/6)*x^(1/3))/a^(1/6)] - 2*b^(2/ 3)*c*x^(4/3)*Log[a^(1/3) + b^(1/3)*x^(2/3)] + (b^(2/3)*c - Sqrt[3]*Sqrt[a] *b^(1/6)*d)*x^(4/3)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3 )*x^(2/3)] + (b^(2/3)*c + Sqrt[3]*Sqrt[a]*b^(1/6)*d)*x^(4/3)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)]))/(4*a^(5/3)*(e*x)^(7/ 3))
Time = 1.44 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.22, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {553, 27, 553, 27, 557, 266, 27, 807, 750, 16, 824, 27, 218, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {c+d x}{(e x)^{7/3} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle -\frac {3 \int -\frac {4 (a d-b c x)}{3 (e x)^{4/3} \left (b x^2+a\right )}dx}{4 a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a d-b c x}{(e x)^{4/3} \left (b x^2+a\right )}dx}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {3 \int \frac {a b (c+d x)}{3 \sqrt [3]{e x} \left (b x^2+a\right )}dx}{a e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \int \frac {c+d x}{\sqrt [3]{e x} \left (b x^2+a\right )}dx}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle \frac {-\frac {b \left (c \int \frac {1}{\sqrt [3]{e x} \left (b x^2+a\right )}dx+\frac {d \int \frac {(e x)^{2/3}}{b x^2+a}dx}{e}\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \int \frac {e^2 \sqrt [3]{e x}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{e}+\frac {3 d \int \frac {e^2 (e x)^{4/3}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{e^2}\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \left (3 c e \int \frac {\sqrt [3]{e x}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}+3 d \int \frac {(e x)^{4/3}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \int \frac {1}{a e^2+b x e}d(e x)^{2/3}+3 d \int \frac {(e x)^{4/3}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {\int \frac {2 \sqrt [3]{a} e^{2/3}-\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}\right )+3 d \int \frac {(e x)^{4/3}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {\int \frac {2 \sqrt [3]{a} e^{2/3}-\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \int \frac {(e x)^{4/3}}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {\int \frac {2 \sqrt [3]{a} e^{2/3}-\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (\frac {\int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 b^{2/3}}+\frac {\int -\frac {\sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\int -\frac {\sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}\right )\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {\int \frac {2 \sqrt [3]{a} e^{2/3}-\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (\frac {\int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 b^{2/3}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}\right )\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {\int \frac {2 \sqrt [3]{a} e^{2/3}-\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}+\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}+\frac {3 \int \frac {1}{\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}-4}d\left (1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} c e \left (\frac {-\frac {\log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{2/3}\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} e^{4/3}}+\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 a^{2/3} \sqrt [3]{b} e^{4/3}}\right )+3 d \left (-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}-\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{e}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{e}}\right )\right )}{e}-\frac {3 d}{e \sqrt [3]{e x}}}{a e}-\frac {3 c}{4 a e (e x)^{4/3}}\) |
Input:
Int[(c + d*x)/((e*x)^(7/3)*(a + b*x^2)),x]
Output:
(-3*c)/(4*a*e*(e*x)^(4/3)) + ((-3*d)/(e*(e*x)^(1/3)) - (b*(3*d*(ArcTan[(b^ (1/6)*(e*x)^(1/3))/(a^(1/6)*e^(1/3))]/(3*a^(1/6)*b^(5/6)*e^(1/3)) - (ArcTa n[Sqrt[3]*(1 - (2*b^(1/6)*(e*x)^(1/3))/(Sqrt[3]*a^(1/6)*e^(1/3)))]/b^(1/6) - (Sqrt[3]*Log[a^(1/3)*e^(2/3) - Sqrt[3]*a^(1/6)*b^(1/6)*e^(1/3)*(e*x)^(1 /3) + b^(1/3)*(e*x)^(2/3)])/(2*b^(1/6)))/(6*a^(1/6)*b^(2/3)*e^(1/3)) - (-( ArcTan[Sqrt[3]*(1 + (2*b^(1/6)*(e*x)^(1/3))/(Sqrt[3]*a^(1/6)*e^(1/3)))]/b^ (1/6)) + (Sqrt[3]*Log[a^(1/3)*e^(2/3) + Sqrt[3]*a^(1/6)*b^(1/6)*e^(1/3)*(e *x)^(1/3) + b^(1/3)*(e*x)^(2/3)])/(2*b^(1/6)))/(6*a^(1/6)*b^(2/3)*e^(1/3)) ) + (3*c*e*(Log[a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3)]/(3*a^(2/3)*b^(1/3)* e^(4/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(e*x)^(2/3))/(a^(1/3)*e^(2/3 )))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3)*e^(4/3) + b^(2/3)*(e*x)^(2/3) - a^(1/ 3)*b^(1/3)*e^(2/3)*(e*x)^(2/3)]/(2*b^(1/3)))/(3*a^(2/3)*e^(4/3))))/2))/e)/ (a*e)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp [1/(a*e*(m + 1)) Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 0.69 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (\frac {x \left (e x \right )^{\frac {1}{3}} \left (\sqrt {3}\, a d e -c \sqrt {\frac {a \,e^{2}}{b}}\, b \right ) \ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{3}-\frac {x \left (e x \right )^{\frac {1}{3}} \left (\sqrt {3}\, a d e +c \sqrt {\frac {a \,e^{2}}{b}}\, b \right ) \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{3}-\frac {2 x \left (e x \right )^{\frac {1}{3}} \left (\sqrt {3}\, \sqrt {\frac {a \,e^{2}}{b}}\, b c +a d e \right ) \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}-2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3}+\frac {2 x \left (e x \right )^{\frac {1}{3}} \left (-\sqrt {3}\, \sqrt {\frac {a \,e^{2}}{b}}\, b c +a d e \right ) \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3}+\frac {2 \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) b c x \left (e x \right )^{\frac {1}{3}} \sqrt {\frac {a \,e^{2}}{b}}}{3}+e \left (\frac {4 \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) d x \left (e x \right )^{\frac {1}{3}}}{3}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (4 d x +c \right )\right ) a \right )}{4 \left (e x \right )^{\frac {1}{3}} \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} a^{2} e^{3} x}\) | \(374\) |
| derivativedivides | \(-\frac {3 \left (\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{12 a \,e^{2}}-\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{12 a e}+\frac {d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a e}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{6 a e}+\frac {d \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{12 a e}+\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, c}{6 a e}\right ) b}{a \,e^{2}}-\frac {3 c}{4 a e \left (e x \right )^{\frac {4}{3}}}-\frac {3 d}{a \,e^{2} \left (e x \right )^{\frac {1}{3}}}\) | \(511\) |
| default | \(-\frac {3 \left (\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{12 a \,e^{2}}-\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{12 a e}+\frac {d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a e}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{6 a e}+\frac {d \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{12 a e}+\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, c}{6 a e}\right ) b}{a \,e^{2}}-\frac {3 c}{4 a e \left (e x \right )^{\frac {4}{3}}}-\frac {3 d}{a \,e^{2} \left (e x \right )^{\frac {1}{3}}}\) | \(511\) |
| risch | \(-\frac {3 \left (4 d x +c \right )}{4 a x \left (e x \right )^{\frac {1}{3}} e^{2}}-\frac {b \left (\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{4 a \,e^{2}}-\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{4 a e}+\frac {d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 a e}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2 a e}+\frac {d \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{4 a \,e^{2}}-\frac {b \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {4}{3}} c}{4 a^{2} e^{3}}+\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{2 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {b \left (\frac {a \,e^{2}}{b}\right )^{\frac {4}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, c}{2 a^{2} e^{3}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, c}{a e}\right )}{a \,e^{2}}\) | \(550\) |
Input:
int((d*x+c)/(e*x)^(7/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-3/4/(e*x)^(1/3)/(a*e^2/b)^(1/6)*(1/3*x*(e*x)^(1/3)*(3^(1/2)*a*d*e-c*(a*e^ 2/b)^(1/2)*b)*ln(3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)-(e*x)^(2/3)-(a*e^2/b) ^(1/3))-1/3*x*(e*x)^(1/3)*(3^(1/2)*a*d*e+c*(a*e^2/b)^(1/2)*b)*ln((e*x)^(2/ 3)+3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)+(a*e^2/b)^(1/3))-2/3*x*(e*x)^(1/3)* (3^(1/2)*(a*e^2/b)^(1/2)*b*c+a*d*e)*arctan((3^(1/2)*(a*e^2/b)^(1/6)-2*(e*x )^(1/3))/(a*e^2/b)^(1/6))+2/3*x*(e*x)^(1/3)*(-3^(1/2)*(a*e^2/b)^(1/2)*b*c+ a*d*e)*arctan((3^(1/2)*(a*e^2/b)^(1/6)+2*(e*x)^(1/3))/(a*e^2/b)^(1/6))+2/3 *ln((e*x)^(2/3)+(a*e^2/b)^(1/3))*b*c*x*(e*x)^(1/3)*(a*e^2/b)^(1/2)+e*(4/3* arctan((e*x)^(1/3)/(a*e^2/b)^(1/6))*d*x*(e*x)^(1/3)+(a*e^2/b)^(1/6)*(4*d*x +c))*a)/a^2/e^3/x
Leaf count of result is larger than twice the leaf count of optimal. 2164 vs. \(2 (293) = 586\).
Time = 0.28 (sec) , antiderivative size = 2164, normalized size of antiderivative = 4.99 \[ \int \frac {c+d x}{(e x)^{7/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)/(e*x)^(7/3)/(b*x^2+a),x, algorithm="fricas")
Output:
1/4*(2*a*e^3*x^2*(-(a^5*e^7*sqrt(-(9*b^3*c^4*d^2 - 6*a*b^2*c^2*d^4 + a^2*b *d^6)/(a^9*e^14)) + b^2*c^3 - 3*a*b*c*d^2)/(a^5*e^7))^(1/3)*log(-((a^8*b*c ^2 - a^9*d^2)*e^12*sqrt(-(9*b^3*c^4*d^2 - 6*a*b^2*c^2*d^4 + a^2*b*d^6)/(a^ 9*e^14)) + 2*(3*a^4*b^2*c^3*d^2 - a^5*b*c*d^4)*e^5)*(-(a^5*e^7*sqrt(-(9*b^ 3*c^4*d^2 - 6*a*b^2*c^2*d^4 + a^2*b*d^6)/(a^9*e^14)) + b^2*c^3 - 3*a*b*c*d ^2)/(a^5*e^7))^(2/3) - (3*b^4*c^6*d + 5*a*b^3*c^4*d^3 + a^2*b^2*c^2*d^5 - a^3*b*d^7)*(e*x)^(1/3)) + 2*a*e^3*x^2*((a^5*e^7*sqrt(-(9*b^3*c^4*d^2 - 6*a *b^2*c^2*d^4 + a^2*b*d^6)/(a^9*e^14)) - b^2*c^3 + 3*a*b*c*d^2)/(a^5*e^7))^ (1/3)*log(((a^8*b*c^2 - a^9*d^2)*e^12*sqrt(-(9*b^3*c^4*d^2 - 6*a*b^2*c^2*d ^4 + a^2*b*d^6)/(a^9*e^14)) - 2*(3*a^4*b^2*c^3*d^2 - a^5*b*c*d^4)*e^5)*((a ^5*e^7*sqrt(-(9*b^3*c^4*d^2 - 6*a*b^2*c^2*d^4 + a^2*b*d^6)/(a^9*e^14)) - b ^2*c^3 + 3*a*b*c*d^2)/(a^5*e^7))^(2/3) - (3*b^4*c^6*d + 5*a*b^3*c^4*d^3 + a^2*b^2*c^2*d^5 - a^3*b*d^7)*(e*x)^(1/3)) + (sqrt(-3)*a*e^3*x^2 - a*e^3*x^ 2)*(-(a^5*e^7*sqrt(-(9*b^3*c^4*d^2 - 6*a*b^2*c^2*d^4 + a^2*b*d^6)/(a^9*e^1 4)) + b^2*c^3 - 3*a*b*c*d^2)/(a^5*e^7))^(1/3)*log(1/2*(2*sqrt(-3)*(3*a^4*b ^2*c^3*d^2 - a^5*b*c*d^4)*e^5 + 2*(3*a^4*b^2*c^3*d^2 - a^5*b*c*d^4)*e^5 + (sqrt(-3)*(a^8*b*c^2 - a^9*d^2)*e^12 + (a^8*b*c^2 - a^9*d^2)*e^12)*sqrt(-( 9*b^3*c^4*d^2 - 6*a*b^2*c^2*d^4 + a^2*b*d^6)/(a^9*e^14)))*(-(a^5*e^7*sqrt( -(9*b^3*c^4*d^2 - 6*a*b^2*c^2*d^4 + a^2*b*d^6)/(a^9*e^14)) + b^2*c^3 - 3*a *b*c*d^2)/(a^5*e^7))^(2/3) - (3*b^4*c^6*d + 5*a*b^3*c^4*d^3 + a^2*b^2*c...
\[ \int \frac {c+d x}{(e x)^{7/3} \left (a+b x^2\right )} \, dx=\int \frac {c + d x}{\left (e x\right )^{\frac {7}{3}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)/(e*x)**(7/3)/(b*x**2+a),x)
Output:
Integral((c + d*x)/((e*x)**(7/3)*(a + b*x**2)), x)
Exception generated. \[ \int \frac {c+d x}{(e x)^{7/3} \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)/(e*x)^(7/3)/(b*x^2+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.47 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x}{(e x)^{7/3} \left (a+b x^2\right )} \, dx=-\frac {\left (\frac {a e^{2}}{b}\right )^{\frac {5}{6}} b d \arctan \left (\frac {\left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a^{2} e^{4}} - \frac {\left (a b^{5} e^{2}\right )^{\frac {1}{3}} c \log \left (\left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2 \, a^{2} b e^{3}} - \frac {3 \, {\left (4 \, d e x + c e\right )}}{4 \, \left (e x\right )^{\frac {1}{3}} a e^{3} x} + \frac {{\left (\sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {1}{3}} b^{3} c e - \left (a b^{5} e^{2}\right )^{\frac {5}{6}} d\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} + 2 \, \left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, a^{2} b^{4} e^{4}} - \frac {{\left (\sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {1}{3}} b^{3} c e + \left (a b^{5} e^{2}\right )^{\frac {5}{6}} d\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} - 2 \, \left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, a^{2} b^{4} e^{4}} + \frac {{\left (\left (a b^{5} e^{2}\right )^{\frac {1}{3}} b^{3} c e + \sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {5}{6}} d\right )} \log \left (\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} \left (e x\right )^{\frac {1}{3}} + \left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, a^{2} b^{4} e^{4}} + \frac {{\left (\left (a b^{5} e^{2}\right )^{\frac {1}{3}} b^{3} c e - \sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {5}{6}} d\right )} \log \left (-\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} \left (e x\right )^{\frac {1}{3}} + \left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, a^{2} b^{4} e^{4}} \] Input:
integrate((d*x+c)/(e*x)^(7/3)/(b*x^2+a),x, algorithm="giac")
Output:
-(a*e^2/b)^(5/6)*b*d*arctan((e*x)^(1/3)/(a*e^2/b)^(1/6))/(a^2*e^4) - 1/2*( a*b^5*e^2)^(1/3)*c*log((e*x)^(2/3) + (a*e^2/b)^(1/3))/(a^2*b*e^3) - 3/4*(4 *d*e*x + c*e)/((e*x)^(1/3)*a*e^3*x) + 1/2*(sqrt(3)*(a*b^5*e^2)^(1/3)*b^3*c *e - (a*b^5*e^2)^(5/6)*d)*arctan((sqrt(3)*(a*e^2/b)^(1/6) + 2*(e*x)^(1/3)) /(a*e^2/b)^(1/6))/(a^2*b^4*e^4) - 1/2*(sqrt(3)*(a*b^5*e^2)^(1/3)*b^3*c*e + (a*b^5*e^2)^(5/6)*d)*arctan(-(sqrt(3)*(a*e^2/b)^(1/6) - 2*(e*x)^(1/3))/(a *e^2/b)^(1/6))/(a^2*b^4*e^4) + 1/4*((a*b^5*e^2)^(1/3)*b^3*c*e + sqrt(3)*(a *b^5*e^2)^(5/6)*d)*log(sqrt(3)*(a*e^2/b)^(1/6)*(e*x)^(1/3) + (e*x)^(2/3) + (a*e^2/b)^(1/3))/(a^2*b^4*e^4) + 1/4*((a*b^5*e^2)^(1/3)*b^3*c*e - sqrt(3) *(a*b^5*e^2)^(5/6)*d)*log(-sqrt(3)*(a*e^2/b)^(1/6)*(e*x)^(1/3) + (e*x)^(2/ 3) + (a*e^2/b)^(1/3))/(a^2*b^4*e^4)
Time = 0.41 (sec) , antiderivative size = 1971, normalized size of antiderivative = 4.54 \[ \int \frac {c+d x}{(e x)^{7/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((c + d*x)/((e*x)^(7/3)*(a + b*x^2)),x)
Output:
log(- ((1944*a^9*b^7*c^2*e^20 - 1944*a^10*b^6*d^2*e^20)*(e*x)^(1/3)*(-(a*d ^3*(-a^11*b)^(1/2) + a^5*b^2*c^3 - 3*a^6*b*c*d^2 - 3*b*c^2*d*(-a^11*b)^(1/ 2))/(8*a^10*e^7))^(1/3) - 972*a^9*b^6*d^3*e^18 + 2916*a^8*b^7*c^2*d*e^18)* (-(a*d^3*(-a^11*b)^(1/2) + a^5*b^2*c^3 - 3*a^6*b*c*d^2 - 3*b*c^2*d*(-a^11* b)^(1/2))/(8*a^10*e^7))^(2/3) - (e*x)^(1/3)*(243*a^4*b^9*c^5*e^13 + 486*a^ 5*b^8*c^3*d^2*e^13 + 243*a^6*b^7*c*d^4*e^13))*(-(a*d^3*(-a^11*b)^(1/2) + a ^5*b^2*c^3 - 3*a^6*b*c*d^2 - 3*b*c^2*d*(-a^11*b)^(1/2))/(8*a^10*e^7))^(1/3 ) + log(- ((1944*a^9*b^7*c^2*e^20 - 1944*a^10*b^6*d^2*e^20)*(e*x)^(1/3)*(( a*d^3*(-a^11*b)^(1/2) - a^5*b^2*c^3 + 3*a^6*b*c*d^2 - 3*b*c^2*d*(-a^11*b)^ (1/2))/(8*a^10*e^7))^(1/3) - 972*a^9*b^6*d^3*e^18 + 2916*a^8*b^7*c^2*d*e^1 8)*((a*d^3*(-a^11*b)^(1/2) - a^5*b^2*c^3 + 3*a^6*b*c*d^2 - 3*b*c^2*d*(-a^1 1*b)^(1/2))/(8*a^10*e^7))^(2/3) - (e*x)^(1/3)*(243*a^4*b^9*c^5*e^13 + 486* a^5*b^8*c^3*d^2*e^13 + 243*a^6*b^7*c*d^4*e^13))*((a*d^3*(-a^11*b)^(1/2) - a^5*b^2*c^3 + 3*a^6*b*c*d^2 - 3*b*c^2*d*(-a^11*b)^(1/2))/(8*a^10*e^7))^(1/ 3) - ((3*c)/(4*a*e) + (3*d*x)/(a*e))/(e*x)^(4/3) - log(((3^(1/2)*1i)/2 + 1 /2)^2*(((3^(1/2)*1i)/2 + 1/2)*(1944*a^9*b^7*c^2*e^20 - 1944*a^10*b^6*d^2*e ^20)*(e*x)^(1/3)*(-(a*d^3*(-a^11*b)^(1/2) + a^5*b^2*c^3 - 3*a^6*b*c*d^2 - 3*b*c^2*d*(-a^11*b)^(1/2))/(8*a^10*e^7))^(1/3) + 972*a^9*b^6*d^3*e^18 - 29 16*a^8*b^7*c^2*d*e^18)*(-(a*d^3*(-a^11*b)^(1/2) + a^5*b^2*c^3 - 3*a^6*b*c* d^2 - 3*b*c^2*d*(-a^11*b)^(1/2))/(8*a^10*e^7))^(2/3) - (e*x)^(1/3)*(243...
Time = 0.23 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.80 \[ \int \frac {c+d x}{(e x)^{7/3} \left (a+b x^2\right )} \, dx=\frac {2 x^{\frac {4}{3}} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d +2 x^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) b c -2 x^{\frac {4}{3}} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d +2 x^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) b c -4 x^{\frac {4}{3}} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right ) d -x^{\frac {4}{3}} \sqrt {b}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) d +x^{\frac {4}{3}} \sqrt {b}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) d -3 b^{\frac {1}{3}} a^{\frac {2}{3}} c -12 b^{\frac {1}{3}} a^{\frac {2}{3}} d x -2 x^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) b c +x^{\frac {4}{3}} \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) b c +x^{\frac {4}{3}} \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) b c}{4 x^{\frac {4}{3}} e^{\frac {7}{3}} b^{\frac {1}{3}} a^{\frac {5}{3}}} \] Input:
int((d*x+c)/(e*x)^(7/3)/(b*x^2+a),x)
Output:
(2*x**(1/3)*sqrt(b)*sqrt(a)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*x**(1/3)*b **(1/3))/(b**(1/6)*a**(1/6)))*d*x + 2*x**(1/3)*sqrt(3)*atan((b**(1/6)*a**( 1/6)*sqrt(3) - 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*b*c*x - 2*x**(1/3 )*sqrt(b)*sqrt(a)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*x**(1/3)*b**(1/3))/( b**(1/6)*a**(1/6)))*d*x + 2*x**(1/3)*sqrt(3)*atan((b**(1/6)*a**(1/6)*sqrt( 3) + 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*b*c*x - 4*x**(1/3)*sqrt(b)* sqrt(a)*atan((x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*d*x - x**(1/3)*sqrt( b)*sqrt(a)*sqrt(3)*log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*d*x + x**(1/3)*sqrt(b)*sqrt(a)*sqrt(3)*log(x**(1/3)*b** (1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*d*x - 3*b**(1/3)*a* *(2/3)*c - 12*b**(1/3)*a**(2/3)*d*x - 2*x**(1/3)*log(a**(1/3) + x**(2/3)*b **(1/3))*b*c*x + x**(1/3)*log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**( 1/3) + x**(2/3)*b**(1/3))*b*c*x + x**(1/3)*log(x**(1/3)*b**(1/6)*a**(1/6)* sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*b*c*x)/(4*x**(1/3)*e**(1/3)*b**(1/ 3)*a**(2/3)*a*e**2*x)