Integrand size = 22, antiderivative size = 454 \[ \int \frac {c+d x}{(e x)^{8/3} \left (a+b x^2\right )} \, dx=-\frac {3 c}{5 a e (e x)^{5/3}}-\frac {3 d}{2 a e^2 (e x)^{2/3}}-\frac {b^{5/6} c \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{a^{11/6} e^{8/3}}+\frac {\sqrt [3]{b} \left (\sqrt {b} c+\sqrt {3} \sqrt {a} d\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{11/6} e^{8/3}}-\frac {\sqrt [3]{b} \left (\sqrt {b} c-\sqrt {3} \sqrt {a} d\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{11/6} e^{8/3}}+\frac {\sqrt [3]{b} d \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 a^{4/3} e^{8/3}}+\frac {\sqrt [3]{b} \left (\sqrt {3} \sqrt {b} c-\sqrt {a} d\right ) \log \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{11/6} e^{8/3}}-\frac {\sqrt [3]{b} \left (\sqrt {3} \sqrt {b} c+\sqrt {a} d\right ) \log \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{11/6} e^{8/3}} \] Output:
-3/5*c/a/e/(e*x)^(5/3)-3/2*d/a/e^2/(e*x)^(2/3)-b^(5/6)*c*arctan(b^(1/6)*(e *x)^(1/3)/a^(1/6)/e^(1/3))/a^(11/6)/e^(8/3)-1/2*b^(1/3)*(b^(1/2)*c+3^(1/2) *a^(1/2)*d)*arctan(-3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(11/6 )/e^(8/3)-1/2*b^(1/3)*(b^(1/2)*c-3^(1/2)*a^(1/2)*d)*arctan(3^(1/2)+2*b^(1/ 6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(11/6)/e^(8/3)+1/2*b^(1/3)*d*ln(a^(1/3)* e^(2/3)+b^(1/3)*(e*x)^(2/3))/a^(4/3)/e^(8/3)+1/4*b^(1/3)*(3^(1/2)*b^(1/2)* c-a^(1/2)*d)*ln(a^(1/3)*e^(2/3)-3^(1/2)*a^(1/6)*b^(1/6)*e^(1/3)*(e*x)^(1/3 )+b^(1/3)*(e*x)^(2/3))/a^(11/6)/e^(8/3)-1/4*b^(1/3)*(3^(1/2)*b^(1/2)*c+a^( 1/2)*d)*ln(a^(1/3)*e^(2/3)+3^(1/2)*a^(1/6)*b^(1/6)*e^(1/3)*(e*x)^(1/3)+b^( 1/3)*(e*x)^(2/3))/a^(11/6)/e^(8/3)
Time = 0.60 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.78 \[ \int \frac {c+d x}{(e x)^{8/3} \left (a+b x^2\right )} \, dx=\frac {x \left (-6 a^{5/6} (2 c+5 d x)+10 \sqrt [3]{b} \left (\sqrt {b} c+\sqrt {3} \sqrt {a} d\right ) x^{5/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+10 \left (-b^{5/6} c+\sqrt {3} \sqrt {a} \sqrt [3]{b} d\right ) x^{5/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-20 b^{5/6} c x^{5/3} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+10 \sqrt {a} \sqrt [3]{b} d x^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}\right )+5 \left (\sqrt {3} b^{5/6} c-\sqrt {a} \sqrt [3]{b} d\right ) x^{5/3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )-5 \sqrt [3]{b} \left (\sqrt {3} \sqrt {b} c+\sqrt {a} d\right ) x^{5/3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )\right )}{20 a^{11/6} (e x)^{8/3}} \] Input:
Integrate[(c + d*x)/((e*x)^(8/3)*(a + b*x^2)),x]
Output:
(x*(-6*a^(5/6)*(2*c + 5*d*x) + 10*b^(1/3)*(Sqrt[b]*c + Sqrt[3]*Sqrt[a]*d)* x^(5/3)*ArcTan[Sqrt[3] - (2*b^(1/6)*x^(1/3))/a^(1/6)] + 10*(-(b^(5/6)*c) + Sqrt[3]*Sqrt[a]*b^(1/3)*d)*x^(5/3)*ArcTan[Sqrt[3] + (2*b^(1/6)*x^(1/3))/a ^(1/6)] - 20*b^(5/6)*c*x^(5/3)*ArcTan[(b^(1/6)*x^(1/3))/a^(1/6)] + 10*Sqrt [a]*b^(1/3)*d*x^(5/3)*Log[a^(1/3) + b^(1/3)*x^(2/3)] + 5*(Sqrt[3]*b^(5/6)* c - Sqrt[a]*b^(1/3)*d)*x^(5/3)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/ 3) + b^(1/3)*x^(2/3)] - 5*b^(1/3)*(Sqrt[3]*Sqrt[b]*c + Sqrt[a]*d)*x^(5/3)* Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)]))/(20*a^( 11/6)*(e*x)^(8/3))
Time = 1.48 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.17, 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, 753, 27, 218, 807, 821, 16, 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.
| \(\displaystyle \int \frac {c+d x}{(e x)^{8/3} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle -\frac {3 \int -\frac {5 (a d-b c x)}{3 (e x)^{5/3} \left (b x^2+a\right )}dx}{5 a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a d-b c x}{(e x)^{5/3} \left (b x^2+a\right )}dx}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {3 \int \frac {2 a b (c+d x)}{3 (e x)^{2/3} \left (b x^2+a\right )}dx}{2 a e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \int \frac {c+d x}{(e x)^{2/3} \left (b x^2+a\right )}dx}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle \frac {-\frac {b \left (c \int \frac {1}{(e x)^{2/3} \left (b x^2+a\right )}dx+\frac {d \int \frac {\sqrt [3]{e x}}{b x^2+a}dx}{e}\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \int \frac {1}{b x^2+a}d\sqrt [3]{e x}}{e}+\frac {3 d \int \frac {e^3 x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{e^2}\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \int \frac {1}{b x^2+a}d\sqrt [3]{e x}}{e}+3 d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \left (\frac {e^{2/3} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 a^{2/3}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 a^{5/6}}\right )}{e}+3 d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \left (\frac {e^{2/3} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 a^{2/3}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}\right )}{e}+3 d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+3 d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \int \frac {(e x)^{2/3}}{a e^2+b x e}d(e x)^{2/3}\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\int \frac {\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}\right )\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\int \frac {\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3 c \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\frac {3 \int \frac {1}{\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}-4}d\left (1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} d \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )+\frac {3 c \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {b \left (\frac {3}{2} d \left (\frac {\frac {\log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{2/3}\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )+\frac {3 c \left (\frac {\sqrt [3]{e} \left (-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}\right )}{e}-\frac {3 d}{2 e (e x)^{2/3}}}{a e}-\frac {3 c}{5 a e (e x)^{5/3}}\) |
Input:
Int[(c + d*x)/((e*x)^(8/3)*(a + b*x^2)),x]
Output:
(-3*c)/(5*a*e*(e*x)^(5/3)) + ((-3*d)/(2*e*(e*x)^(2/3)) - (b*((3*c*((e^(1/3 )*ArcTan[(b^(1/6)*(e*x)^(1/3))/(a^(1/6)*e^(1/3))])/(3*a^(5/6)*b^(1/6)) + ( 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^(5/6)) + ( 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^(5/6))))/e + (3*d*(-1/3*Log[a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3)]/(a^(1/3)*b^(2/3)*e^( 2/3)) + (-((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^(1/3)*b^(1/3)*e^(2/3))))/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_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ (r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a /b]
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_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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[((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 = 0.73 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(\frac {-x \left (\frac {\left (-\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 )+\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 )\right ) \sqrt {3}}{2}+\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 \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\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 )\right ) e b c \left (e x \right )^{\frac {2}{3}} \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+d x \left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} \left (\left (\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 )+\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 )\right ) \sqrt {3}+\ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )-\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 )}{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 )}{2}\right ) b \left (e x \right )^{\frac {2}{3}}-\frac {6 \left (\frac {5 d x}{2}+c \right ) e^{2} a}{5}}{2 \left (e x \right )^{\frac {2}{3}} a^{2} e^{4} x}\) | \(415\) |
| derivativedivides | \(-\frac {3 c}{5 a e \left (e x \right )^{\frac {5}{3}}}-\frac {3 d}{2 a \,e^{2} \left (e x \right )^{\frac {2}{3}}}-\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 {1}{6}} c}{12 a e}+\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 {2}{3}} d}{12 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) c}{6 a e}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{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}\, d}{6 a \,e^{2}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} d \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{6 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} c \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 a e}+\frac {b \left (\frac {a \,e^{2}}{b}\right )^{\frac {7}{6}} \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}\, c}{12 a^{2} e^{3}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} \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 ) d}{12 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) c}{6 a e}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{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}\, d}{6 a \,e^{2}}\right ) b}{a \,e^{2}}\) | \(521\) |
| default | \(-\frac {3 c}{5 a e \left (e x \right )^{\frac {5}{3}}}-\frac {3 d}{2 a \,e^{2} \left (e x \right )^{\frac {2}{3}}}-\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 {1}{6}} c}{12 a e}+\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 {2}{3}} d}{12 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) c}{6 a e}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{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}\, d}{6 a \,e^{2}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} d \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{6 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} c \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 a e}+\frac {b \left (\frac {a \,e^{2}}{b}\right )^{\frac {7}{6}} \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}\, c}{12 a^{2} e^{3}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} \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 ) d}{12 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) c}{6 a e}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{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}\, d}{6 a \,e^{2}}\right ) b}{a \,e^{2}}\) | \(521\) |
Input:
int((d*x+c)/(e*x)^(8/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2/(e*x)^(2/3)*(-x*(1/2*(-ln(3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)-(e*x)^(2 /3)-(a*e^2/b)^(1/3))+ln((e*x)^(2/3)+3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)+(a *e^2/b)^(1/3)))*3^(1/2)+arctan((3^(1/2)*(a*e^2/b)^(1/6)+2*(e*x)^(1/3))/(a* e^2/b)^(1/6))+2*arctan((e*x)^(1/3)/(a*e^2/b)^(1/6))-arctan((3^(1/2)*(a*e^2 /b)^(1/6)-2*(e*x)^(1/3))/(a*e^2/b)^(1/6)))*e*b*c*(e*x)^(2/3)*(a*e^2/b)^(1/ 6)+d*x*(a*e^2/b)^(2/3)*((arctan((3^(1/2)*(a*e^2/b)^(1/6)+2*(e*x)^(1/3))/(a *e^2/b)^(1/6))+arctan((3^(1/2)*(a*e^2/b)^(1/6)-2*(e*x)^(1/3))/(a*e^2/b)^(1 /6)))*3^(1/2)+ln((e*x)^(2/3)+(a*e^2/b)^(1/3))-1/2*ln(3^(1/2)*(a*e^2/b)^(1/ 6)*(e*x)^(1/3)-(e*x)^(2/3)-(a*e^2/b)^(1/3))-1/2*ln((e*x)^(2/3)+3^(1/2)*(a* e^2/b)^(1/6)*(e*x)^(1/3)+(a*e^2/b)^(1/3)))*b*(e*x)^(2/3)-6/5*(5/2*d*x+c)*e ^2*a)/a^2/e^4/x
Leaf count of result is larger than twice the leaf count of optimal. 1995 vs. \(2 (306) = 612\).
Time = 0.21 (sec) , antiderivative size = 1995, normalized size of antiderivative = 4.39 \[ \int \frac {c+d x}{(e x)^{8/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)/(e*x)^(8/3)/(b*x^2+a),x, algorithm="fricas")
Output:
1/20*(10*a*e^3*x^2*(-(a^5*e^8*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3 *c^2*d^4)/(a^11*e^16)) + 3*b^2*c^2*d - a*b*d^3)/(a^5*e^8))^(1/3)*log(-(b^4 *c^5 - 2*a*b^3*c^3*d^2 - 3*a^2*b^2*c*d^4)*(e*x)^(1/3) + (a^8*d*e^11*sqrt(- (b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^11*e^16)) + (a^2*b^3*c^ 4 - 3*a^3*b^2*c^2*d^2)*e^3)*(-(a^5*e^8*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^11*e^16)) + 3*b^2*c^2*d - a*b*d^3)/(a^5*e^8))^(1/3)) + 10*a*e^3*x^2*((a^5*e^8*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2 *d^4)/(a^11*e^16)) - 3*b^2*c^2*d + a*b*d^3)/(a^5*e^8))^(1/3)*log(-(b^4*c^5 - 2*a*b^3*c^3*d^2 - 3*a^2*b^2*c*d^4)*(e*x)^(1/3) - (a^8*d*e^11*sqrt(-(b^5 *c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^11*e^16)) - (a^2*b^3*c^4 - 3*a^3*b^2*c^2*d^2)*e^3)*((a^5*e^8*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2 *b^3*c^2*d^4)/(a^11*e^16)) - 3*b^2*c^2*d + a*b*d^3)/(a^5*e^8))^(1/3)) - 5* (sqrt(-3)*a*e^3*x^2 + a*e^3*x^2)*(-(a^5*e^8*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d ^2 + 9*a^2*b^3*c^2*d^4)/(a^11*e^16)) + 3*b^2*c^2*d - a*b*d^3)/(a^5*e^8))^( 1/3)*log(-(b^4*c^5 - 2*a*b^3*c^3*d^2 - 3*a^2*b^2*c*d^4)*(e*x)^(1/3) - 1/2* (sqrt(-3)*(a^2*b^3*c^4 - 3*a^3*b^2*c^2*d^2)*e^3 + (a^2*b^3*c^4 - 3*a^3*b^2 *c^2*d^2)*e^3 + (sqrt(-3)*a^8*d*e^11 + a^8*d*e^11)*sqrt(-(b^5*c^6 - 6*a*b^ 4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^11*e^16)))*(-(a^5*e^8*sqrt(-(b^5*c^6 - 6 *a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^11*e^16)) + 3*b^2*c^2*d - a*b*d^3)/ (a^5*e^8))^(1/3)) + 5*(sqrt(-3)*a*e^3*x^2 - a*e^3*x^2)*(-(a^5*e^8*sqrt(...
\[ \int \frac {c+d x}{(e x)^{8/3} \left (a+b x^2\right )} \, dx=\int \frac {c + d x}{\left (e x\right )^{\frac {8}{3}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)/(e*x)**(8/3)/(b*x**2+a),x)
Output:
Integral((c + d*x)/((e*x)**(8/3)*(a + b*x**2)), x)
Exception generated. \[ \int \frac {c+d x}{(e x)^{8/3} \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)/(e*x)^(8/3)/(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 = 0.13 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x}{(e x)^{8/3} \left (a+b x^2\right )} \, dx=\frac {\left (a b^{5} e^{2}\right )^{\frac {1}{6}} d {\left | b \right |} {\left | e \right |} \log \left (\left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2 \, \sqrt {a b} a b e^{4}} - \frac {\left (a b^{5} e^{2}\right )^{\frac {1}{6}} c \arctan \left (\frac {\left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a^{2} e^{3}} - \frac {3 \, {\left (5 \, d e x + 2 \, c e\right )}}{10 \, \left (e x\right )^{\frac {2}{3}} a e^{3} x} - \frac {{\left (\left (a b^{5} e^{2}\right )^{\frac {1}{6}} b^{3} c e - \sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {2}{3}} d\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} + 2 \, \left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, a^{2} b^{3} e^{4}} - \frac {{\left (\left (a b^{5} e^{2}\right )^{\frac {1}{6}} b^{3} c e + \sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {2}{3}} d\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} - 2 \, \left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, a^{2} b^{3} e^{4}} - \frac {{\left (\sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {1}{6}} b^{3} c e + \left (a b^{5} e^{2}\right )^{\frac {2}{3}} d\right )} \log \left (\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} \left (e x\right )^{\frac {1}{3}} + \left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, a^{2} b^{3} e^{4}} + \frac {{\left (\sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {1}{6}} b^{3} c e - \left (a b^{5} e^{2}\right )^{\frac {2}{3}} d\right )} \log \left (-\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} \left (e x\right )^{\frac {1}{3}} + \left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, a^{2} b^{3} e^{4}} \] Input:
integrate((d*x+c)/(e*x)^(8/3)/(b*x^2+a),x, algorithm="giac")
Output:
1/2*(a*b^5*e^2)^(1/6)*d*abs(b)*abs(e)*log((e*x)^(2/3) + (a*e^2/b)^(1/3))/( sqrt(a*b)*a*b*e^4) - (a*b^5*e^2)^(1/6)*c*arctan((e*x)^(1/3)/(a*e^2/b)^(1/6 ))/(a^2*e^3) - 3/10*(5*d*e*x + 2*c*e)/((e*x)^(2/3)*a*e^3*x) - 1/2*((a*b^5* e^2)^(1/6)*b^3*c*e - sqrt(3)*(a*b^5*e^2)^(2/3)*d)*arctan((sqrt(3)*(a*e^2/b )^(1/6) + 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))/(a^2*b^3*e^4) - 1/2*((a*b^5*e^2) ^(1/6)*b^3*c*e + sqrt(3)*(a*b^5*e^2)^(2/3)*d)*arctan(-(sqrt(3)*(a*e^2/b)^( 1/6) - 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))/(a^2*b^3*e^4) - 1/4*(sqrt(3)*(a*b^5 *e^2)^(1/6)*b^3*c*e + (a*b^5*e^2)^(2/3)*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^3*e^4) + 1/4*(sqrt(3)*(a* b^5*e^2)^(1/6)*b^3*c*e - (a*b^5*e^2)^(2/3)*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^3*e^4)
Time = 0.45 (sec) , antiderivative size = 1839, normalized size of antiderivative = 4.05 \[ \int \frac {c+d x}{(e x)^{8/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((c + d*x)/((e*x)^(8/3)*(a + b*x^2)),x)
Output:
log((486*a^5*b^9*c^4*e^14 - 486*a^7*b^7*d^4*e^14)*(e*x)^(1/3) + (2916*a^8* b^7*c*d^2*e^17 - 972*a^7*b^8*c^3*e^17 + 3888*a^11*b^6*d*e^22*(e*x)^(1/3)*( (a^7*b*d^3 + b*c^3*(-a^11*b^3)^(1/2) - 3*a^6*b^2*c^2*d - 3*a*c*d^2*(-a^11* b^3)^(1/2))/(8*a^11*e^8))^(2/3))*((a^7*b*d^3 + b*c^3*(-a^11*b^3)^(1/2) - 3 *a^6*b^2*c^2*d - 3*a*c*d^2*(-a^11*b^3)^(1/2))/(8*a^11*e^8))^(1/3))*((a^7*b *d^3 + b*c^3*(-a^11*b^3)^(1/2) - 3*a^6*b^2*c^2*d - 3*a*c*d^2*(-a^11*b^3)^( 1/2))/(8*a^11*e^8))^(1/3) - ((3*c)/(5*a*e) + (3*d*x)/(2*a*e))/(e*x)^(5/3) + log((486*a^5*b^9*c^4*e^14 - 486*a^7*b^7*d^4*e^14)*(e*x)^(1/3) + (2916*a^ 8*b^7*c*d^2*e^17 - 972*a^7*b^8*c^3*e^17 + 3888*a^11*b^6*d*e^22*(e*x)^(1/3) *((a^7*b*d^3 - b*c^3*(-a^11*b^3)^(1/2) - 3*a^6*b^2*c^2*d + 3*a*c*d^2*(-a^1 1*b^3)^(1/2))/(8*a^11*e^8))^(2/3))*((a^7*b*d^3 - b*c^3*(-a^11*b^3)^(1/2) - 3*a^6*b^2*c^2*d + 3*a*c*d^2*(-a^11*b^3)^(1/2))/(8*a^11*e^8))^(1/3))*((a^7 *b*d^3 - b*c^3*(-a^11*b^3)^(1/2) - 3*a^6*b^2*c^2*d + 3*a*c*d^2*(-a^11*b^3) ^(1/2))/(8*a^11*e^8))^(1/3) + log((486*a^5*b^9*c^4*e^14 - 486*a^7*b^7*d^4* e^14)*(e*x)^(1/3) + ((3^(1/2)*1i)/2 - 1/2)*(2916*a^8*b^7*c*d^2*e^17 - 972* a^7*b^8*c^3*e^17 + 3888*a^11*b^6*d*e^22*((3^(1/2)*1i)/2 - 1/2)^2*(e*x)^(1/ 3)*((a^7*b*d^3 + b*c^3*(-a^11*b^3)^(1/2) - 3*a^6*b^2*c^2*d - 3*a*c*d^2*(-a ^11*b^3)^(1/2))/(8*a^11*e^8))^(2/3))*((a^7*b*d^3 + b*c^3*(-a^11*b^3)^(1/2) - 3*a^6*b^2*c^2*d - 3*a*c*d^2*(-a^11*b^3)^(1/2))/(8*a^11*e^8))^(1/3))*((3 ^(1/2)*1i)/2 - 1/2)*((a^7*b*d^3 + b*c^3*(-a^11*b^3)^(1/2) - 3*a^6*b^2*c...
Time = 0.26 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.80 \[ \int \frac {c+d x}{(e x)^{8/3} \left (a+b x^2\right )} \, dx=\frac {10 x^{\frac {5}{3}} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) b c +10 x^{\frac {5}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) a b d -10 x^{\frac {5}{3}} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) b c +10 x^{\frac {5}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) a b d -20 x^{\frac {5}{3}} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right ) b c +5 x^{\frac {5}{3}} \sqrt {b}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) b c -5 x^{\frac {5}{3}} \sqrt {b}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) b c -12 b^{\frac {2}{3}} a^{\frac {4}{3}} c -30 b^{\frac {2}{3}} a^{\frac {4}{3}} d x +10 x^{\frac {5}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) a b d -5 x^{\frac {5}{3}} \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) a b d -5 x^{\frac {5}{3}} \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) a b d}{20 e^{\frac {8}{3}} x^{\frac {5}{3}} b^{\frac {2}{3}} a^{\frac {7}{3}}} \] Input:
int((d*x+c)/(e*x)^(8/3)/(b*x^2+a),x)
Output:
(e**(1/3)*(10*x**(2/3)*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)))*b*c*x + 10*x**(2/3)*sqrt(3)*atan( (b**(1/6)*a**(1/6)*sqrt(3) - 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a*b *d*x - 10*x**(2/3)*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)))*b*c*x + 10*x**(2/3)*sqrt(3)*atan((b** (1/6)*a**(1/6)*sqrt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a*b*d*x - 20*x**(2/3)*sqrt(b)*sqrt(a)*atan((x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6) ))*b*c*x + 5*x**(2/3)*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))*b*c*x - 5*x**(2/3)*sqrt(b)*s qrt(a)*sqrt(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 - 12*b**(2/3)*a**(1/3)*a*c - 30*b**(2/3)*a**(1/3)*a*d*x + 10*x**(2/3)*log(a**(1/3) + x**(2/3)*b**(1/3))*a*b*d*x - 5*x**(2/3)*log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*a*b*d *x - 5*x**(2/3)*log(x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/ 3)*b**(1/3))*a*b*d*x))/(20*x**(2/3)*b**(2/3)*a**(1/3)*a**2*e**3*x)