Integrand size = 24, antiderivative size = 494 \[ \int \frac {(c+d x)^2}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {3 d^2 \sqrt [3]{e x}}{b e}+\frac {\left (b c^2-a d^2\right ) \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{a^{5/6} b^{7/6} e^{2/3}}-\frac {\left (b c^2+2 \sqrt {3} \sqrt {a} \sqrt {b} c d-a d^2\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{5/6} b^{7/6} e^{2/3}}+\frac {\left (b c^2-2 \sqrt {3} \sqrt {a} \sqrt {b} c d-a d^2\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{5/6} b^{7/6} e^{2/3}}-\frac {c d \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{\sqrt [3]{a} b^{2/3} e^{2/3}}-\frac {\left (\sqrt {3} b c^2-2 \sqrt {a} \sqrt {b} c d-\sqrt {3} a d^2\right ) \log \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{5/6} b^{7/6} e^{2/3}}+\frac {\left (2 \sqrt {b} c d+\frac {\sqrt {3} \left (b c^2-a d^2\right )}{\sqrt {a}}\right ) \log \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 \sqrt [3]{a} b^{7/6} e^{2/3}} \] Output:
3*d^2*(e*x)^(1/3)/b/e+(-a*d^2+b*c^2)*arctan(b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^ (1/3))/a^(5/6)/b^(7/6)/e^(2/3)+1/2*(b*c^2+2*3^(1/2)*a^(1/2)*b^(1/2)*c*d-a* d^2)*arctan(-3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(5/6)/b^(7/6 )/e^(2/3)+1/2*(b*c^2-2*3^(1/2)*a^(1/2)*b^(1/2)*c*d-a*d^2)*arctan(3^(1/2)+2 *b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(5/6)/b^(7/6)/e^(2/3)-c*d*ln(a^(1/ 3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))/a^(1/3)/b^(2/3)/e^(2/3)-1/4*(3^(1/2)*b*c^2 -2*a^(1/2)*b^(1/2)*c*d-3^(1/2)*a*d^2)*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^(5/6)/b^(7/6)/e^(2/3)+1/ 4*(2*b^(1/2)*c*d+3^(1/2)*(-a*d^2+b*c^2)/a^(1/2))*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^(1/3)/b^(7/6) /e^(2/3)
Time = 0.61 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.77 \[ \int \frac {(c+d x)^2}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {x^{2/3} \left (12 a^{5/6} \sqrt [6]{b} d^2 \sqrt [3]{x}-2 \left (b c^2+2 \sqrt {3} \sqrt {a} \sqrt {b} c d-a d^2\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+2 \left (b c^2-2 \sqrt {3} \sqrt {a} \sqrt {b} c d-a d^2\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+4 \left (b c^2-a d^2\right ) \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-4 \sqrt {a} \sqrt {b} c d \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}\right )-\left (\sqrt {3} b c^2-2 \sqrt {a} \sqrt {b} c d-\sqrt {3} a d^2\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+\left (\sqrt {3} b c^2+2 \sqrt {a} \sqrt {b} c d-\sqrt {3} a d^2\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )\right )}{4 a^{5/6} b^{7/6} (e x)^{2/3}} \] Input:
Integrate[(c + d*x)^2/((e*x)^(2/3)*(a + b*x^2)),x]
Output:
(x^(2/3)*(12*a^(5/6)*b^(1/6)*d^2*x^(1/3) - 2*(b*c^2 + 2*Sqrt[3]*Sqrt[a]*Sq rt[b]*c*d - a*d^2)*ArcTan[Sqrt[3] - (2*b^(1/6)*x^(1/3))/a^(1/6)] + 2*(b*c^ 2 - 2*Sqrt[3]*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*ArcTan[Sqrt[3] + (2*b^(1/6)*x^( 1/3))/a^(1/6)] + 4*(b*c^2 - a*d^2)*ArcTan[(b^(1/6)*x^(1/3))/a^(1/6)] - 4*S qrt[a]*Sqrt[b]*c*d*Log[a^(1/3) + b^(1/3)*x^(2/3)] - (Sqrt[3]*b*c^2 - 2*Sqr t[a]*Sqrt[b]*c*d - Sqrt[3]*a*d^2)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x^ (1/3) + b^(1/3)*x^(2/3)] + (Sqrt[3]*b*c^2 + 2*Sqrt[a]*Sqrt[b]*c*d - Sqrt[3 ]*a*d^2)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)]) )/(4*a^(5/6)*b^(7/6)*(e*x)^(2/3))
Time = 1.48 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {559, 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.
| \(\displaystyle \int \frac {(c+d x)^2}{(e x)^{2/3} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 559 |
| \(\displaystyle \frac {3 \int \frac {b c^2+2 b d x c-a d^2}{3 (e x)^{2/3} \left (b x^2+a\right )}dx}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b c^2+2 b d x c-a d^2}{(e x)^{2/3} \left (b x^2+a\right )}dx}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle \frac {\left (b c^2-a d^2\right ) \int \frac {1}{(e x)^{2/3} \left (b x^2+a\right )}dx+\frac {2 b c d \int \frac {\sqrt [3]{e x}}{b x^2+a}dx}{e}}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \int \frac {1}{b x^2+a}d\sqrt [3]{e x}}{e}+\frac {6 b c d \int \frac {e^3 x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{e^2}}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \int \frac {1}{b x^2+a}d\sqrt [3]{e x}}{e}+6 b c d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \left (\frac {e^{2/3} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 a^{2/3}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 a^{5/6}}\right )}{e}+6 b c d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \left (\frac {e^{2/3} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 a^{2/3}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}\right )}{e}+6 b c d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+6 b c d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+3 b c d \int \frac {(e x)^{2/3}}{a e^2+b x e}d(e x)^{2/3}}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+3 b c d \left (\frac {\int \frac {\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}\right )}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+3 b c d \left (\frac {\int \frac {\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+3 b c d \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+3 b c d \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+3 b c d \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {3 \left (b c^2-a d^2\right ) \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+3 b c d \left (\frac {\frac {3 \int \frac {1}{\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}-4}d\left (1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 b c d \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )+\frac {3 \left (b c^2-a d^2\right ) \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 b c d \left (\frac {\frac {\log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{2/3}\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )+\frac {3 \left (b c^2-a d^2\right ) \left (\frac {\sqrt [3]{e} \left (-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}}{b}+\frac {3 d^2 \sqrt [3]{e x}}{b e}\) |
Input:
Int[(c + d*x)^2/((e*x)^(2/3)*(a + b*x^2)),x]
Output:
(3*d^2*(e*x)^(1/3))/(b*e) + ((3*(b*c^2 - a*d^2)*((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[Sq rt[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*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)) + (-((Sqr t[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))))/b
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 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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1)) Int[(e*x)^m*(a + b* x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 )*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && IGtQ[n, 1] && !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
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 = 0.77 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(\frac {e \left (a \,d^{2}-b \,c^{2}\right ) \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 ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+2 d \left (6 a d e \left (e x \right )^{\frac {1}{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 (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} c \right )}{4 a b \,e^{2}}\) | \(412\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(766\) |
| default | \(\text {Expression too large to display}\) | \(766\) |
Input:
int((d*x+c)^2/(e*x)^(2/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/4*(e*(a*d^2-b*c^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)-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)))*(a*e^2/b)^(1/6)+2*d*(6*a*d*e*(e *x)^(1/3)+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)))*(a*e^2/b)^(2/3)*c))/a/b/e^2
Leaf count of result is larger than twice the leaf count of optimal. 3637 vs. \(2 (352) = 704\).
Time = 1.32 (sec) , antiderivative size = 3637, normalized size of antiderivative = 7.36 \[ \int \frac {(c+d x)^2}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2/(e*x)^(2/3)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(c+d x)^2}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\int \frac {\left (c + d x\right )^{2}}{\left (e x\right )^{\frac {2}{3}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)**2/(e*x)**(2/3)/(b*x**2+a),x)
Output:
Integral((c + d*x)**2/((e*x)**(2/3)*(a + b*x**2)), x)
Exception generated. \[ \int \frac {(c+d x)^2}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^2/(e*x)^(2/3)/(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 = 538, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x)^2}{(e x)^{2/3} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^2/(e*x)^(2/3)/(b*x^2+a),x, algorithm="giac")
Output:
-(a*b^5*e^2)^(1/6)*c*d*abs(b)*abs(e)*log((e*x)^(2/3) + (a*e^2/b)^(1/3))/(s qrt(a*b)*b^2*e^2) + 3*(e*x)^(1/3)*d^2/(b*e) + ((a*b^5*e^2)^(1/6)*b*c^2 - ( a*b^5*e^2)^(1/6)*a*d^2)*arctan((e*x)^(1/3)/(a*e^2/b)^(1/6))/(a*b^2*e) + 1/ 2*((a*b^5*e^2)^(1/6)*b^3*c^2*e - (a*b^5*e^2)^(1/6)*a*b^2*d^2*e - 2*sqrt(3) *(a*b^5*e^2)^(2/3)*c*d)*arctan((sqrt(3)*(a*e^2/b)^(1/6) + 2*(e*x)^(1/3))/( a*e^2/b)^(1/6))/(a*b^4*e^2) + 1/2*((a*b^5*e^2)^(1/6)*b^3*c^2*e - (a*b^5*e^ 2)^(1/6)*a*b^2*d^2*e + 2*sqrt(3)*(a*b^5*e^2)^(2/3)*c*d)*arctan(-(sqrt(3)*( a*e^2/b)^(1/6) - 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))/(a*b^4*e^2) + 1/4*(sqrt(3 )*(a*b^5*e^2)^(1/6)*b^3*c^2*e - sqrt(3)*(a*b^5*e^2)^(1/6)*a*b^2*d^2*e + 2* (a*b^5*e^2)^(2/3)*c*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*b^4*e^2) - 1/4*(sqrt(3)*(a*b^5*e^2)^(1/6)*b^3*c^2 *e - sqrt(3)*(a*b^5*e^2)^(1/6)*a*b^2*d^2*e - 2*(a*b^5*e^2)^(2/3)*c*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*b ^4*e^2)
Time = 9.28 (sec) , antiderivative size = 3499, normalized size of antiderivative = 7.08 \[ \int \frac {(c+d x)^2}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((c + d*x)^2/((e*x)^(2/3)*(a + b*x^2)),x)
Output:
log((e*x)^(1/3)*(4860*a^2*b^3*c^4*d^4*e^4 - 486*a^4*b*d^8*e^4 - 486*b^5*c^ 8*e^4 + 1944*a^3*b^2*c^2*d^6*e^4 + 1944*a*b^4*c^6*d^2*e^4) + ((a^3*d^6*(-a ^5*b^7)^(1/2) - b^3*c^6*(-a^5*b^7)^(1/2) + 6*a^3*b^6*c^5*d + 6*a^5*b^4*c*d ^5 - 20*a^4*b^5*c^3*d^3 + 15*a*b^2*c^4*d^2*(-a^5*b^7)^(1/2) - 15*a^2*b*c^2 *d^4*(-a^5*b^7)^(1/2))/(8*a^5*b^7*e^2))^(1/3)*(972*a^4*b^2*d^6*e^5 - 972*a *b^5*c^6*e^5 + 14580*a^2*b^4*c^4*d^2*e^5 - 14580*a^3*b^3*c^2*d^4*e^5 + 777 6*a^3*b^5*c*d*e^6*(e*x)^(1/3)*((a^3*d^6*(-a^5*b^7)^(1/2) - b^3*c^6*(-a^5*b ^7)^(1/2) + 6*a^3*b^6*c^5*d + 6*a^5*b^4*c*d^5 - 20*a^4*b^5*c^3*d^3 + 15*a* b^2*c^4*d^2*(-a^5*b^7)^(1/2) - 15*a^2*b*c^2*d^4*(-a^5*b^7)^(1/2))/(8*a^5*b ^7*e^2))^(2/3)))*((a^3*d^6*(-a^5*b^7)^(1/2) - b^3*c^6*(-a^5*b^7)^(1/2) + 6 *a^3*b^6*c^5*d + 6*a^5*b^4*c*d^5 - 20*a^4*b^5*c^3*d^3 + 15*a*b^2*c^4*d^2*( -a^5*b^7)^(1/2) - 15*a^2*b*c^2*d^4*(-a^5*b^7)^(1/2))/(8*a^5*b^7*e^2))^(1/3 ) + log((e*x)^(1/3)*(4860*a^2*b^3*c^4*d^4*e^4 - 486*a^4*b*d^8*e^4 - 486*b^ 5*c^8*e^4 + 1944*a^3*b^2*c^2*d^6*e^4 + 1944*a*b^4*c^6*d^2*e^4) + ((b^3*c^6 *(-a^5*b^7)^(1/2) - a^3*d^6*(-a^5*b^7)^(1/2) + 6*a^3*b^6*c^5*d + 6*a^5*b^4 *c*d^5 - 20*a^4*b^5*c^3*d^3 - 15*a*b^2*c^4*d^2*(-a^5*b^7)^(1/2) + 15*a^2*b *c^2*d^4*(-a^5*b^7)^(1/2))/(8*a^5*b^7*e^2))^(1/3)*(972*a^4*b^2*d^6*e^5 - 9 72*a*b^5*c^6*e^5 + 14580*a^2*b^4*c^4*d^2*e^5 - 14580*a^3*b^3*c^2*d^4*e^5 + 7776*a^3*b^5*c*d*e^6*(e*x)^(1/3)*((b^3*c^6*(-a^5*b^7)^(1/2) - a^3*d^6*(-a ^5*b^7)^(1/2) + 6*a^3*b^6*c^5*d + 6*a^5*b^4*c*d^5 - 20*a^4*b^5*c^3*d^3 ...
Time = 0.21 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^2}{(e x)^{2/3} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^2/(e*x)^(2/3)/(b*x^2+a),x)
Output:
(2*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)))*a*d**2 - 2*sqrt(b)*sqrt(a)*atan((b**(1/6)*a**(1/6)*sq rt(3) - 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*b*c**2 - 4*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 *c*d - 2*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)))*a*d**2 + 2*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**2 - 4*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*c*d - 4*sqrt(b)*sqrt(a)*atan((x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6) ))*a*d**2 + 4*sqrt(b)*sqrt(a)*atan((x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)) )*b*c**2 + 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))*a*d**2 - 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**2 - 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))*a*d**2 + 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**2 + 12*x**(1/3) *b**(2/3)*a**(1/3)*a*d**2 - 4*log(a**(1/3) + x**(2/3)*b**(1/3))*a*b*c*d + 2*log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3) )*a*b*c*d + 2*log(x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3) *b**(1/3))*a*b*c*d)/(4*e**(2/3)*b**(2/3)*a**(1/3)*a*b)