Integrand size = 22, antiderivative size = 435 \[ \int \frac {c+d x}{(e x)^{5/3} \left (a+b x^2\right )} \, dx=-\frac {3 c}{2 a e (e x)^{2/3}}+\frac {d \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{a^{5/6} \sqrt [6]{b} e^{5/3}}+\frac {\left (\sqrt {3} \sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{4/3} \sqrt [6]{b} e^{5/3}}+\frac {\left (\sqrt {3} \sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{4/3} \sqrt [6]{b} e^{5/3}}+\frac {\sqrt [3]{b} c \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 a^{4/3} e^{5/3}}-\frac {\left (\sqrt {b} c+\sqrt {3} \sqrt {a} d\right ) \log \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{4/3} \sqrt [6]{b} e^{5/3}}-\frac {\left (\sqrt {b} c-\sqrt {3} \sqrt {a} d\right ) \log \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{4/3} \sqrt [6]{b} e^{5/3}} \] Output:
-3/2*c/a/e/(e*x)^(2/3)+d*arctan(b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(5/ 6)/b^(1/6)/e^(5/3)-1/2*(3^(1/2)*b^(1/2)*c-a^(1/2)*d)*arctan(-3^(1/2)+2*b^( 1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(4/3)/b^(1/6)/e^(5/3)+1/2*(3^(1/2)*b^( 1/2)*c+a^(1/2)*d)*arctan(3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^ (4/3)/b^(1/6)/e^(5/3)+1/2*b^(1/3)*c*ln(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3) )/a^(4/3)/e^(5/3)-1/4*(b^(1/2)*c+3^(1/2)*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^(4/3)/b^(1 /6)/e^(5/3)-1/4*(b^(1/2)*c-3^(1/2)*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^(4/3)/b^(1/6)/e^ (5/3)
Time = 0.42 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x}{(e x)^{5/3} \left (a+b x^2\right )} \, dx=-\frac {3 c x}{2 a (e x)^{5/3}}+\frac {\left (\sqrt {3} \sqrt {b} c-\sqrt {a} d\right ) x^{5/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )}{2 a^{4/3} \sqrt [6]{b} (e x)^{5/3}}+\frac {\left (\sqrt {3} \sqrt {b} c+\sqrt {a} d\right ) x^{5/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )}{2 a^{4/3} \sqrt [6]{b} (e x)^{5/3}}+\frac {d x^{5/3} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )}{a^{5/6} \sqrt [6]{b} (e x)^{5/3}}+\frac {\sqrt [3]{b} c x^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}\right )}{2 a^{4/3} (e x)^{5/3}}+\frac {\left (-\sqrt {b} c-\sqrt {3} \sqrt {a} d\right ) x^{5/3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )}{4 a^{4/3} \sqrt [6]{b} (e x)^{5/3}}+\frac {\left (-\sqrt {b} c+\sqrt {3} \sqrt {a} d\right ) x^{5/3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )}{4 a^{4/3} \sqrt [6]{b} (e x)^{5/3}} \] Input:
Integrate[(c + d*x)/((e*x)^(5/3)*(a + b*x^2)),x]
Output:
(-3*c*x)/(2*a*(e*x)^(5/3)) + ((Sqrt[3]*Sqrt[b]*c - Sqrt[a]*d)*x^(5/3)*ArcT an[Sqrt[3] - (2*b^(1/6)*x^(1/3))/a^(1/6)])/(2*a^(4/3)*b^(1/6)*(e*x)^(5/3)) + ((Sqrt[3]*Sqrt[b]*c + Sqrt[a]*d)*x^(5/3)*ArcTan[Sqrt[3] + (2*b^(1/6)*x^ (1/3))/a^(1/6)])/(2*a^(4/3)*b^(1/6)*(e*x)^(5/3)) + (d*x^(5/3)*ArcTan[(b^(1 /6)*x^(1/3))/a^(1/6)])/(a^(5/6)*b^(1/6)*(e*x)^(5/3)) + (b^(1/3)*c*x^(5/3)* Log[a^(1/3) + b^(1/3)*x^(2/3)])/(2*a^(4/3)*(e*x)^(5/3)) + ((-(Sqrt[b]*c) - Sqrt[3]*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)])/(4*a^(4/3)*b^(1/6)*(e*x)^(5/3)) + ((-(Sqrt[b]*c) + Sqr t[3]*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)])/(4*a^(4/3)*b^(1/6)*(e*x)^(5/3))
Time = 1.42 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.17, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {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.
| \(\displaystyle \int \frac {c+d x}{(e x)^{5/3} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle -\frac {3 \int -\frac {2 (a d-b c x)}{3 (e x)^{2/3} \left (b x^2+a\right )}dx}{2 a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a d-b c x}{(e x)^{2/3} \left (b x^2+a\right )}dx}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle \frac {a d \int \frac {1}{(e x)^{2/3} \left (b x^2+a\right )}dx-\frac {b c \int \frac {\sqrt [3]{e x}}{b x^2+a}dx}{e}}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {3 a d \int \frac {1}{b x^2+a}d\sqrt [3]{e x}}{e}-\frac {3 b c \int \frac {e^3 x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{e^2}}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 a d \int \frac {1}{b x^2+a}d\sqrt [3]{e x}}{e}-3 b c \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {e^{2/3} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 a^{2/3}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 a^{5/6}}\right )}{e}-3 b c \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {e^{2/3} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 a^{2/3}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}\right )}{e}-3 b c \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}-3 b c \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}-\frac {3}{2} b c \int \frac {(e x)^{2/3}}{a e^2+b x e}d(e x)^{2/3}}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}-\frac {3}{2} b c \left (\frac {\int \frac {\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}\right )}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}-\frac {3}{2} b c \left (\frac {\int \frac {\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}-\frac {3}{2} b c \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}-\frac {3}{2} b c \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}-\frac {3}{2} b c \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}-\frac {3}{2} b c \left (\frac {\frac {3 \int \frac {1}{\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}-4}d\left (1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}-\frac {3}{2} b c \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {3 a d \left (\frac {\sqrt [3]{e} \left (-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}-\frac {3}{2} b c \left (\frac {\frac {\log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{2/3}\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{a e}-\frac {3 c}{2 a e (e x)^{2/3}}\) |
Input:
Int[(c + d*x)/((e*x)^(5/3)*(a + b*x^2)),x]
Output:
(-3*c)/(2*a*e*(e*x)^(2/3)) + ((3*a*d*((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*b*c*(-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)/(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 = 0.71 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(-\frac {d \left (\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}-4 \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )+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 {\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 a \left (e x \right )^{\frac {2}{3}} \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+\left (\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} b \left (\left (-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 {\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 (\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 \ln \left (\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 ) \left (e x \right )^{\frac {2}{3}}+6 a \,e^{2}\right ) c}{4 \left (e x \right )^{\frac {2}{3}} e^{3} a^{2}}\) | \(408\) |
| derivativedivides | \(\frac {\frac {b \left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{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^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} d \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{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}\, d}{4 a \,e^{3}}-\frac {b \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 ) c}{4 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} 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 e}-\frac {b \left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{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^{2}}+\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}\, d}{4 a \,e^{3}}-\frac {b \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 ) c}{4 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 ) d}{2 e}+\frac {b \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}\, c}{2 a \,e^{2}}}{a e}-\frac {3 c}{2 a e \left (e x \right )^{\frac {2}{3}}}\) | \(497\) |
| default | \(\frac {\frac {b \left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{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^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} d \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{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}\, d}{4 a \,e^{3}}-\frac {b \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 ) c}{4 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} 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 e}-\frac {b \left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{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^{2}}+\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}\, d}{4 a \,e^{3}}-\frac {b \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 ) c}{4 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 ) d}{2 e}+\frac {b \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}\, c}{2 a \,e^{2}}}{a e}-\frac {3 c}{2 a e \left (e x \right )^{\frac {2}{3}}}\) | \(497\) |
Input:
int((d*x+c)/(e*x)^(5/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/4/(e*x)^(2/3)*(d*((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)-4*arctan((e*x)^(1/3)/(a*e^2/b)^(1/6))+2*arctan((3^(1/2) *(a*e^2/b)^(1/6)-2*(e*x)^(1/3))/(a*e^2/b)^(1/6))-2*arctan((3^(1/2)*(a*e^2/ b)^(1/6)+2*(e*x)^(1/3))/(a*e^2/b)^(1/6)))*e*a*(e*x)^(2/3)*(a*e^2/b)^(1/6)+ ((a*e^2/b)^(2/3)*b*((-2*arctan((3^(1/2)*(a*e^2/b)^(1/6)-2*(e*x)^(1/3))/(a* e^2/b)^(1/6))-2*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(3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)-(e*x)^(2/3)-(a*e^2/b )^(1/3))-2*ln((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)))*(e*x)^(2/3)+6*a*e^2)*c)/e^3/a^2
Leaf count of result is larger than twice the leaf count of optimal. 1840 vs. \(2 (291) = 582\).
Time = 0.14 (sec) , antiderivative size = 1840, normalized size of antiderivative = 4.23 \[ \int \frac {c+d x}{(e x)^{5/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)/(e*x)^(5/3)/(b*x^2+a),x, algorithm="fricas")
Output:
1/4*(2*a*e^2*x*((a^4*e^5*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/( a^7*b*e^10)) + b*c^3 - 3*a*c*d^2)/(a^4*e^5))^(1/3)*log(-(3*b^2*c^4*d + 2*a *b*c^2*d^3 - a^2*d^5)*(e*x)^(1/3) - (a^5*b*c*e^7*sqrt(-(9*b^2*c^4*d^2 - 6* a*b*c^2*d^4 + a^2*d^6)/(a^7*b*e^10)) + (3*a^2*b*c^2*d^2 - a^3*d^4)*e^2)*(( a^4*e^5*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^7*b*e^10)) + b* c^3 - 3*a*c*d^2)/(a^4*e^5))^(1/3)) + 2*a*e^2*x*(-(a^4*e^5*sqrt(-(9*b^2*c^4 *d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^7*b*e^10)) - b*c^3 + 3*a*c*d^2)/(a^4*e^ 5))^(1/3)*log(-(3*b^2*c^4*d + 2*a*b*c^2*d^3 - a^2*d^5)*(e*x)^(1/3) + (a^5* b*c*e^7*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^7*b*e^10)) - (3 *a^2*b*c^2*d^2 - a^3*d^4)*e^2)*(-(a^4*e^5*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2 *d^4 + a^2*d^6)/(a^7*b*e^10)) - b*c^3 + 3*a*c*d^2)/(a^4*e^5))^(1/3)) - (sq rt(-3)*a*e^2*x + a*e^2*x)*((a^4*e^5*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^7*b*e^10)) + b*c^3 - 3*a*c*d^2)/(a^4*e^5))^(1/3)*log(-(3*b^2* c^4*d + 2*a*b*c^2*d^3 - a^2*d^5)*(e*x)^(1/3) + 1/2*(sqrt(-3)*(3*a^2*b*c^2* d^2 - a^3*d^4)*e^2 + (3*a^2*b*c^2*d^2 - a^3*d^4)*e^2 + (sqrt(-3)*a^5*b*c*e ^7 + a^5*b*c*e^7)*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^7*b*e ^10)))*((a^4*e^5*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^7*b*e^ 10)) + b*c^3 - 3*a*c*d^2)/(a^4*e^5))^(1/3)) + (sqrt(-3)*a*e^2*x - a*e^2*x) *((a^4*e^5*sqrt(-(9*b^2*c^4*d^2 - 6*a*b*c^2*d^4 + a^2*d^6)/(a^7*b*e^10)) + b*c^3 - 3*a*c*d^2)/(a^4*e^5))^(1/3)*log(-(3*b^2*c^4*d + 2*a*b*c^2*d^3 ...
\[ \int \frac {c+d x}{(e x)^{5/3} \left (a+b x^2\right )} \, dx=\int \frac {c + d x}{\left (e x\right )^{\frac {5}{3}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)/(e*x)**(5/3)/(b*x**2+a),x)
Output:
Integral((c + d*x)/((e*x)**(5/3)*(a + b*x**2)), x)
Exception generated. \[ \int \frac {c+d x}{(e x)^{5/3} \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)/(e*x)^(5/3)/(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 = 0.14 (sec) , antiderivative size = 423, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x}{(e x)^{5/3} \left (a+b x^2\right )} \, dx=\frac {\frac {2 \, \left (a b^{5} e^{2}\right )^{\frac {1}{6}} c {\left | b \right |} {\left | e \right |} \log \left (\left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{\sqrt {a b} a b e^{2}} - \frac {6 \, c}{\left (e x\right )^{\frac {2}{3}} a} + \frac {4 \, \left (a b^{5} e^{2}\right )^{\frac {1}{6}} d \arctan \left (\frac {\left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a b e} + \frac {2 \, {\left (\left (a b^{5} e^{2}\right )^{\frac {1}{6}} a b^{2} d e + \sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {2}{3}} c\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} + 2 \, \left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a^{2} b^{3} e^{2}} + \frac {2 \, {\left (\left (a b^{5} e^{2}\right )^{\frac {1}{6}} a b^{2} d e - \sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {2}{3}} c\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} - 2 \, \left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a^{2} b^{3} e^{2}} + \frac {{\left (\sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {1}{6}} a b^{2} d e - \left (a b^{5} e^{2}\right )^{\frac {2}{3}} c\right )} \log \left (\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} \left (e x\right )^{\frac {1}{3}} + \left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{a^{2} b^{3} e^{2}} - \frac {{\left (\sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {1}{6}} a b^{2} d e + \left (a b^{5} e^{2}\right )^{\frac {2}{3}} c\right )} \log \left (-\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} \left (e x\right )^{\frac {1}{3}} + \left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{a^{2} b^{3} e^{2}}}{4 \, e} \] Input:
integrate((d*x+c)/(e*x)^(5/3)/(b*x^2+a),x, algorithm="giac")
Output:
1/4*(2*(a*b^5*e^2)^(1/6)*c*abs(b)*abs(e)*log((e*x)^(2/3) + (a*e^2/b)^(1/3) )/(sqrt(a*b)*a*b*e^2) - 6*c/((e*x)^(2/3)*a) + 4*(a*b^5*e^2)^(1/6)*d*arctan ((e*x)^(1/3)/(a*e^2/b)^(1/6))/(a*b*e) + 2*((a*b^5*e^2)^(1/6)*a*b^2*d*e + s qrt(3)*(a*b^5*e^2)^(2/3)*c)*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^2) + 2*((a*b^5*e^2)^(1/6)*a*b^2*d*e - sqrt( 3)*(a*b^5*e^2)^(2/3)*c)*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^2) + (sqrt(3)*(a*b^5*e^2)^(1/6)*a*b^2*d*e - (a *b^5*e^2)^(2/3)*c)*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^2) - (sqrt(3)*(a*b^5*e^2)^(1/6)*a*b^2*d*e + ( a*b^5*e^2)^(2/3)*c)*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^2))/e
Time = 0.41 (sec) , antiderivative size = 1787, normalized size of antiderivative = 4.11 \[ \int \frac {c+d x}{(e x)^{5/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((c + d*x)/((e*x)^(5/3)*(a + b*x^2)),x)
Output:
log((486*a^7*b^7*c^4*e^9 - 486*a^9*b^5*d^4*e^9)*(e*x)^(1/3) - (972*a^10*b^ 5*d^3*e^11 - 2916*a^9*b^6*c^2*d*e^11 + 3888*a^11*b^6*c*e^14*(e*x)^(1/3)*(( a*d^3*(-a^9*b)^(1/2) + a^4*b^2*c^3 - 3*a^5*b*c*d^2 - 3*b*c^2*d*(-a^9*b)^(1 /2))/(8*a^8*b*e^5))^(2/3))*((a*d^3*(-a^9*b)^(1/2) + a^4*b^2*c^3 - 3*a^5*b* c*d^2 - 3*b*c^2*d*(-a^9*b)^(1/2))/(8*a^8*b*e^5))^(1/3))*((a*d^3*(-a^9*b)^( 1/2) + a^4*b^2*c^3 - 3*a^5*b*c*d^2 - 3*b*c^2*d*(-a^9*b)^(1/2))/(8*a^8*b*e^ 5))^(1/3) + log((486*a^7*b^7*c^4*e^9 - 486*a^9*b^5*d^4*e^9)*(e*x)^(1/3) - (972*a^10*b^5*d^3*e^11 - 2916*a^9*b^6*c^2*d*e^11 + 3888*a^11*b^6*c*e^14*(e *x)^(1/3)*(-(a*d^3*(-a^9*b)^(1/2) - a^4*b^2*c^3 + 3*a^5*b*c*d^2 - 3*b*c^2* d*(-a^9*b)^(1/2))/(8*a^8*b*e^5))^(2/3))*(-(a*d^3*(-a^9*b)^(1/2) - a^4*b^2* c^3 + 3*a^5*b*c*d^2 - 3*b*c^2*d*(-a^9*b)^(1/2))/(8*a^8*b*e^5))^(1/3))*(-(a *d^3*(-a^9*b)^(1/2) - a^4*b^2*c^3 + 3*a^5*b*c*d^2 - 3*b*c^2*d*(-a^9*b)^(1/ 2))/(8*a^8*b*e^5))^(1/3) - log((486*a^7*b^7*c^4*e^9 - 486*a^9*b^5*d^4*e^9) *(e*x)^(1/3) + ((3^(1/2)*1i)/2 + 1/2)*(972*a^10*b^5*d^3*e^11 - 2916*a^9*b^ 6*c^2*d*e^11 + 3888*a^11*b^6*c*e^14*((3^(1/2)*1i)/2 + 1/2)^2*(e*x)^(1/3)*( (a*d^3*(-a^9*b)^(1/2) + a^4*b^2*c^3 - 3*a^5*b*c*d^2 - 3*b*c^2*d*(-a^9*b)^( 1/2))/(8*a^8*b*e^5))^(2/3))*((a*d^3*(-a^9*b)^(1/2) + a^4*b^2*c^3 - 3*a^5*b *c*d^2 - 3*b*c^2*d*(-a^9*b)^(1/2))/(8*a^8*b*e^5))^(1/3))*((3^(1/2)*1i)/2 + 1/2)*((a*d^3*(-a^9*b)^(1/2) + a^4*b^2*c^3 - 3*a^5*b*c*d^2 - 3*b*c^2*d*(-a ^9*b)^(1/2))/(8*a^8*b*e^5))^(1/3) + log((486*a^7*b^7*c^4*e^9 - 486*a^9*...
Time = 0.24 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.86 \[ \int \frac {c+d x}{(e x)^{5/3} \left (a+b x^2\right )} \, dx=\frac {-2 x^{\frac {2}{3}} b^{\frac {1}{6}} a^{\frac {7}{6}} \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 {2}{3}} b^{\frac {2}{3}} a^{\frac {2}{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 ) c +2 x^{\frac {2}{3}} b^{\frac {1}{6}} a^{\frac {7}{6}} \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 {2}{3}} b^{\frac {2}{3}} a^{\frac {2}{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 ) c +4 x^{\frac {2}{3}} b^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {x^{\frac {1}{3}} b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right ) d -x^{\frac {2}{3}} b^{\frac {1}{6}} a^{\frac {7}{6}} \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 {2}{3}} b^{\frac {1}{6}} a^{\frac {7}{6}} \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 +2 x^{\frac {2}{3}} b^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) c -x^{\frac {2}{3}} b^{\frac {2}{3}} a^{\frac {2}{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 ) c -x^{\frac {2}{3}} b^{\frac {2}{3}} a^{\frac {2}{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 ) c -6 b^{\frac {1}{3}} a c}{4 e^{\frac {5}{3}} x^{\frac {2}{3}} b^{\frac {1}{3}} a^{2}} \] Input:
int((d*x+c)/(e*x)^(5/3)/(b*x^2+a),x)
Output:
(e**(1/3)*( - 2*x**(2/3)*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a*d + 2*x**(2/3)*b**(2/3)*a** (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)))*c + 2*x**(2/3)*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*s qrt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a*d + 2*x**(2/3)*b**(2/ 3)*a**(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)))*c + 4*x**(2/3)*b**(1/6)*a**(1/6)*atan((x**(1/3)*b**( 1/3))/(b**(1/6)*a**(1/6)))*a*d - x**(2/3)*b**(1/6)*a**(1/6)*sqrt(3)*log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*a*d + x**(2/3)*b**(1/6)*a**(1/6)*sqrt(3)*log(x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*a*d + 2*x**(2/3)*b**(2/3)*a**(2/3)*log(a** (1/3) + x**(2/3)*b**(1/3))*c - x**(2/3)*b**(2/3)*a**(2/3)*log( - x**(1/3)* b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*c - x**(2/3)*b** (2/3)*a**(2/3)*log(x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3 )*b**(1/3))*c - 6*b**(1/3)*a*c))/(4*x**(2/3)*b**(1/3)*a**2*e**2)