Integrand size = 24, antiderivative size = 715 \[ \int \frac {(e x)^{8/3}}{(c+d x) \left (a+b x^2\right )} \, dx=\frac {3 e^2 (e x)^{2/3}}{2 b d}-\frac {a^{5/6} c e^{8/3} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{b^{5/6} \left (b c^2+a d^2\right )}+\frac {a^{5/6} c e^{8/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 b^{5/6} \left (b c^2+a d^2\right )}-\frac {a^{5/6} c e^{8/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 b^{5/6} \left (b c^2+a d^2\right )}+\frac {\sqrt {3} c^{8/3} e^{8/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e x}}{\sqrt [3]{c} \sqrt [3]{e}}}{\sqrt {3}}\right )}{d^{5/3} \left (b c^2+a d^2\right )}+\frac {\sqrt {3} a^{4/3} d e^{8/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{2 b^{4/3} \left (b c^2+a d^2\right )}+\frac {\sqrt {3} a^{5/6} c e^{8/3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}\right )}{2 b^{5/6} \left (b c^2+a d^2\right )}+\frac {c^{8/3} e^{8/3} \log \left (\sqrt [3]{c} \sqrt [3]{e}+\sqrt [3]{d} \sqrt [3]{e x}\right )}{d^{5/3} \left (b c^2+a d^2\right )}-\frac {a^{4/3} d e^{8/3} \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 b^{4/3} \left (b c^2+a d^2\right )}-\frac {c^{8/3} e^{8/3} \log \left (c^{2/3} e^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{e x}+d^{2/3} (e x)^{2/3}\right )}{2 d^{5/3} \left (b c^2+a d^2\right )}+\frac {a^{4/3} d e^{8/3} \log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{4/3}\right )}{4 b^{4/3} \left (b c^2+a d^2\right )} \] Output:
3/2*e^2*(e*x)^(2/3)/b/d-a^(5/6)*c*e^(8/3)*arctan(b^(1/6)*(e*x)^(1/3)/a^(1/ 6)/e^(1/3))/b^(5/6)/(a*d^2+b*c^2)-1/2*a^(5/6)*c*e^(8/3)*arctan(-3^(1/2)+2* b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/b^(5/6)/(a*d^2+b*c^2)-1/2*a^(5/6)*c*e ^(8/3)*arctan(3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/b^(5/6)/(a*d^ 2+b*c^2)+3^(1/2)*c^(8/3)*e^(8/3)*arctan(1/3*(1-2*d^(1/3)*(e*x)^(1/3)/c^(1/ 3)/e^(1/3))*3^(1/2))/d^(5/3)/(a*d^2+b*c^2)+1/2*3^(1/2)*a^(4/3)*d*e^(8/3)*a rctan(1/3*(1-2*b^(1/3)*(e*x)^(2/3)/a^(1/3)/e^(2/3))*3^(1/2))/b^(4/3)/(a*d^ 2+b*c^2)+1/2*3^(1/2)*a^(5/6)*c*e^(8/3)*arctanh(3^(1/2)*a^(1/6)*b^(1/6)*e^( 1/3)*(e*x)^(1/3)/(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3)))/b^(5/6)/(a*d^2+b*c ^2)+c^(8/3)*e^(8/3)*ln(c^(1/3)*e^(1/3)+d^(1/3)*(e*x)^(1/3))/d^(5/3)/(a*d^2 +b*c^2)-1/2*a^(4/3)*d*e^(8/3)*ln(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))/b^(4 /3)/(a*d^2+b*c^2)-1/2*c^(8/3)*e^(8/3)*ln(c^(2/3)*e^(2/3)-c^(1/3)*d^(1/3)*e ^(1/3)*(e*x)^(1/3)+d^(2/3)*(e*x)^(2/3))/d^(5/3)/(a*d^2+b*c^2)+1/4*a^(4/3)* d*e^(8/3)*ln(a^(2/3)*e^(4/3)-a^(1/3)*b^(1/3)*e^(2/3)*(e*x)^(2/3)+b^(2/3)*( e*x)^(4/3))/b^(4/3)/(a*d^2+b*c^2)
Time = 1.21 (sec) , antiderivative size = 601, normalized size of antiderivative = 0.84 \[ \int \frac {(e x)^{8/3}}{(c+d x) \left (a+b x^2\right )} \, dx=\frac {(e x)^{8/3} \left (6 b^{4/3} c^2 d^{2/3} x^{2/3}+6 a \sqrt [3]{b} d^{8/3} x^{2/3}+2 a^{5/6} d^{5/3} \left (\sqrt {b} c+\sqrt {3} \sqrt {a} d\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+2 a^{5/6} d^{5/3} \left (-\sqrt {b} c+\sqrt {3} \sqrt {a} d\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+4 \sqrt {3} b^{4/3} c^{8/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}}{\sqrt {3}}\right )-4 a^{5/6} \sqrt {b} c d^{5/3} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+4 b^{4/3} c^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )-2 a^{4/3} d^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}\right )-\sqrt {3} a^{5/6} \sqrt {b} c d^{5/3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+a^{4/3} d^{8/3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+\sqrt {3} a^{5/6} \sqrt {b} c d^{5/3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+a^{4/3} d^{8/3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )-2 b^{4/3} c^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )\right )}{4 b^{4/3} d^{5/3} \left (b c^2+a d^2\right ) x^{8/3}} \] Input:
Integrate[(e*x)^(8/3)/((c + d*x)*(a + b*x^2)),x]
Output:
((e*x)^(8/3)*(6*b^(4/3)*c^2*d^(2/3)*x^(2/3) + 6*a*b^(1/3)*d^(8/3)*x^(2/3) + 2*a^(5/6)*d^(5/3)*(Sqrt[b]*c + Sqrt[3]*Sqrt[a]*d)*ArcTan[Sqrt[3] - (2*b^ (1/6)*x^(1/3))/a^(1/6)] + 2*a^(5/6)*d^(5/3)*(-(Sqrt[b]*c) + Sqrt[3]*Sqrt[a ]*d)*ArcTan[Sqrt[3] + (2*b^(1/6)*x^(1/3))/a^(1/6)] + 4*Sqrt[3]*b^(4/3)*c^( 8/3)*ArcTan[(1 - (2*d^(1/3)*x^(1/3))/c^(1/3))/Sqrt[3]] - 4*a^(5/6)*Sqrt[b] *c*d^(5/3)*ArcTan[(b^(1/6)*x^(1/3))/a^(1/6)] + 4*b^(4/3)*c^(8/3)*Log[c^(1/ 3) + d^(1/3)*x^(1/3)] - 2*a^(4/3)*d^(8/3)*Log[a^(1/3) + b^(1/3)*x^(2/3)] - Sqrt[3]*a^(5/6)*Sqrt[b]*c*d^(5/3)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x ^(1/3) + b^(1/3)*x^(2/3)] + a^(4/3)*d^(8/3)*Log[a^(1/3) - Sqrt[3]*a^(1/6)* b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] + Sqrt[3]*a^(5/6)*Sqrt[b]*c*d^(5/3)*Log [a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] + a^(4/3)*d^ (8/3)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] - 2 *b^(4/3)*c^(8/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x^(1/3) + d^(2/3)*x^(2/3)]) )/(4*b^(4/3)*d^(5/3)*(b*c^2 + a*d^2)*x^(8/3))
Time = 3.11 (sec) , antiderivative size = 810, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {615, 2009}
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.
| \(\displaystyle \int \frac {(e x)^{8/3}}{\left (a+b x^2\right ) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (\frac {b (e x)^{8/3} (c-d x)}{\left (a+b x^2\right ) \left (a d^2+b c^2\right )}+\frac {d^2 (e x)^{8/3}}{(c+d x) \left (a d^2+b c^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{5/6} c \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right ) e^{8/3}}{b^{5/6} \left (b c^2+a d^2\right )}+\frac {a^{5/6} c \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right ) e^{8/3}}{2 b^{5/6} \left (b c^2+a d^2\right )}-\frac {a^{5/6} c \arctan \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}+\sqrt {3}\right ) e^{8/3}}{2 b^{5/6} \left (b c^2+a d^2\right )}+\frac {\sqrt {3} c^{8/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e x}}{\sqrt [3]{c} \sqrt [3]{e}}}{\sqrt {3}}\right ) e^{8/3}}{d^{5/3} \left (b c^2+a d^2\right )}+\frac {\sqrt {3} a^{4/3} d \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right ) e^{8/3}}{2 b^{4/3} \left (b c^2+a d^2\right )}-\frac {c^{8/3} \log (c+d x) e^{8/3}}{2 d^{5/3} \left (b c^2+a d^2\right )}+\frac {3 c^{8/3} \log \left (\sqrt [3]{c} \sqrt [3]{e}+\sqrt [3]{d} \sqrt [3]{e x}\right ) e^{8/3}}{2 d^{5/3} \left (b c^2+a d^2\right )}-\frac {a^{4/3} d \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right ) e^{8/3}}{2 b^{4/3} \left (b c^2+a d^2\right )}-\frac {\sqrt {3} a^{5/6} c \log \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right ) e^{8/3}}{4 b^{5/6} \left (b c^2+a d^2\right )}+\frac {\sqrt {3} a^{5/6} c \log \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right ) e^{8/3}}{4 b^{5/6} \left (b c^2+a d^2\right )}+\frac {a^{4/3} d \log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{4/3}\right ) e^{8/3}}{4 b^{4/3} \left (b c^2+a d^2\right )}+\frac {3 a d (e x)^{2/3} e^2}{2 b \left (b c^2+a d^2\right )}+\frac {3 c^2 (e x)^{2/3} e^2}{2 d \left (b c^2+a d^2\right )}\) |
Input:
Int[(e*x)^(8/3)/((c + d*x)*(a + b*x^2)),x]
Output:
(3*c^2*e^2*(e*x)^(2/3))/(2*d*(b*c^2 + a*d^2)) + (3*a*d*e^2*(e*x)^(2/3))/(2 *b*(b*c^2 + a*d^2)) - (a^(5/6)*c*e^(8/3)*ArcTan[(b^(1/6)*(e*x)^(1/3))/(a^( 1/6)*e^(1/3))])/(b^(5/6)*(b*c^2 + a*d^2)) + (a^(5/6)*c*e^(8/3)*ArcTan[Sqrt [3] - (2*b^(1/6)*(e*x)^(1/3))/(a^(1/6)*e^(1/3))])/(2*b^(5/6)*(b*c^2 + a*d^ 2)) - (a^(5/6)*c*e^(8/3)*ArcTan[Sqrt[3] + (2*b^(1/6)*(e*x)^(1/3))/(a^(1/6) *e^(1/3))])/(2*b^(5/6)*(b*c^2 + a*d^2)) + (Sqrt[3]*c^(8/3)*e^(8/3)*ArcTan[ (1 - (2*d^(1/3)*(e*x)^(1/3))/(c^(1/3)*e^(1/3)))/Sqrt[3]])/(d^(5/3)*(b*c^2 + a*d^2)) + (Sqrt[3]*a^(4/3)*d*e^(8/3)*ArcTan[(1 - (2*b^(1/3)*(e*x)^(2/3)) /(a^(1/3)*e^(2/3)))/Sqrt[3]])/(2*b^(4/3)*(b*c^2 + a*d^2)) - (c^(8/3)*e^(8/ 3)*Log[c + d*x])/(2*d^(5/3)*(b*c^2 + a*d^2)) + (3*c^(8/3)*e^(8/3)*Log[c^(1 /3)*e^(1/3) + d^(1/3)*(e*x)^(1/3)])/(2*d^(5/3)*(b*c^2 + a*d^2)) - (a^(4/3) *d*e^(8/3)*Log[a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3)])/(2*b^(4/3)*(b*c^2 + a*d^2)) - (Sqrt[3]*a^(5/6)*c*e^(8/3)*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)])/(4*b^(5/6)*(b*c^2 + a*d^2)) + (Sqrt[3]*a^(5/6)*c*e^(8/3)*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)])/(4*b^(5/6)*(b*c^2 + a *d^2)) + (a^(4/3)*d*e^(8/3)*Log[a^(2/3)*e^(4/3) - a^(1/3)*b^(1/3)*e^(2/3)* (e*x)^(2/3) + b^(2/3)*(e*x)^(4/3)])/(4*b^(4/3)*(b*c^2 + a*d^2))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Time = 0.82 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.78
| method | result | size |
| pseudoelliptic | \(-\frac {e^{2} \left (\left (-\frac {3 \left (e x \right )^{\frac {2}{3}} d \left (\frac {c e}{d}\right )^{\frac {1}{3}} \left (a \,d^{2}+b \,c^{2}\right )}{2}+e b \,c^{3} \left (\arctan \left (\frac {\sqrt {3}\, \left (2 \left (e x \right )^{\frac {1}{3}}-\left (\frac {c e}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {c e}{d}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\left (\frac {c e}{d}\right )^{\frac {1}{3}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {c e}{d}\right )^{\frac {2}{3}}\right )}{2}-\ln \left (\left (e x \right )^{\frac {1}{3}}+\left (\frac {c e}{d}\right )^{\frac {1}{3}}\right )\right )\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+\frac {d^{2} a \left (d \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 ) \sqrt {\frac {a \,e^{2}}{b}}+2 \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}}{4}+\arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )-\frac {\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}+\frac {\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}\right ) e c \right ) \left (\frac {c e}{d}\right )^{\frac {1}{3}}}{2}\right )}{\left (\frac {c e}{d}\right )^{\frac {1}{3}} \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} b \,d^{2} \left (a \,d^{2}+b \,c^{2}\right )}\) | \(555\) |
| derivativedivides | \(3 e^{2} \left (\frac {\left (e x \right )^{\frac {2}{3}}}{2 b d}-\frac {\left (-\frac {\ln \left (\left (e x \right )^{\frac {1}{3}}+\left (\frac {c e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c e}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\left (\frac {c e}{d}\right )^{\frac {1}{3}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {c e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c e}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {c e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c e}{d}\right )^{\frac {1}{3}}}\right ) c^{3} e}{d \left (a \,d^{2}+b \,c^{2}\right )}-\frac {\left (\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} d \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{6 e}+\frac {c \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {b \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} c}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} d}{12 e}+\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) c}{6 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d}{6 e}+\frac {b \ln \left (\left (e x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} \sqrt {3}\, c}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} d}{12 e}+\frac {c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 e}\right ) e a}{\left (a \,d^{2}+b \,c^{2}\right ) b}\right )\) | \(617\) |
| default | \(3 e^{2} \left (\frac {\left (e x \right )^{\frac {2}{3}}}{2 b d}-\frac {\left (-\frac {\ln \left (\left (e x \right )^{\frac {1}{3}}+\left (\frac {c e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c e}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\left (\frac {c e}{d}\right )^{\frac {1}{3}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {c e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c e}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {c e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c e}{d}\right )^{\frac {1}{3}}}\right ) c^{3} e}{d \left (a \,d^{2}+b \,c^{2}\right )}-\frac {\left (\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} d \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{6 e}+\frac {c \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {b \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} c}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} d}{12 e}+\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) c}{6 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d}{6 e}+\frac {b \ln \left (\left (e x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} \sqrt {3}\, c}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} d}{12 e}+\frac {c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 e}\right ) e a}{\left (a \,d^{2}+b \,c^{2}\right ) b}\right )\) | \(617\) |
| risch | \(\frac {3 x \,e^{3}}{2 d b \left (e x \right )^{\frac {1}{3}}}-\frac {\left (\frac {3 \left (-\frac {\ln \left (\left (e x \right )^{\frac {1}{3}}+\left (\frac {c e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c e}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\left (\frac {c e}{d}\right )^{\frac {1}{3}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {c e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c e}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {c e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c e}{d}\right )^{\frac {1}{3}}}\right ) b \,c^{3}}{a \,d^{2}+b \,c^{2}}+\frac {3 \left (\frac {b \ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} c}{12 a \,e^{2}}-\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} d}{12 e}+\frac {c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 e}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} d \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{6 e}+\frac {c \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {b \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} c}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} d}{12 e}+\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) c}{6 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d}{6 e}\right ) d a}{a \,d^{2}+b \,c^{2}}\right ) e^{3}}{b d}\) | \(628\) |
Input:
int((e*x)^(8/3)/(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-e^2*((-3/2*(e*x)^(2/3)*d*(c*e/d)^(1/3)*(a*d^2+b*c^2)+e*b*c^3*(arctan(1/3* 3^(1/2)*(2*(e*x)^(1/3)-(c*e/d)^(1/3))/(c*e/d)^(1/3))*3^(1/2)+1/2*ln((e*x)^ (2/3)-(c*e/d)^(1/3)*(e*x)^(1/3)+(c*e/d)^(2/3))-ln((e*x)^(1/3)+(c*e/d)^(1/3 ))))*(a*e^2/b)^(1/6)+1/2*d^2*a*(d*((-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)))*(a*e^2/b)^(1/2)+2*( 1/4*(ln(3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)-(e*x)^(2/3)-(a*e^2/b)^(1/3))-l n((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((e*x)^(1/3)/(a*e^2/b)^(1/6))-1/2*arctan((3^(1/2)*(a*e^2/b)^(1/6)- 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))+1/2*arctan((3^(1/2)*(a*e^2/b)^(1/6)+2*(e*x )^(1/3))/(a*e^2/b)^(1/6)))*e*c)*(c*e/d)^(1/3))/(c*e/d)^(1/3)/(a*e^2/b)^(1/ 6)/b/d^2/(a*d^2+b*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 5132 vs. \(2 (521) = 1042\).
Time = 9.46 (sec) , antiderivative size = 5132, normalized size of antiderivative = 7.18 \[ \int \frac {(e x)^{8/3}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(8/3)/(d*x+c)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e x)^{8/3}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(8/3)/(d*x+c)/(b*x**2+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {(e x)^{8/3}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(8/3)/(d*x+c)/(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 = 0.30 (sec) , antiderivative size = 660, normalized size of antiderivative = 0.92 \[ \int \frac {(e x)^{8/3}}{(c+d x) \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((e*x)^(8/3)/(d*x+c)/(b*x^2+a),x, algorithm="giac")
Output:
1/4*(4*c^3*(-c*e/d)^(2/3)*e*log(abs((e*x)^(1/3) - (-c*e/d)^(1/3)))/(b*c^3* d*e + a*c*d^3*e) + 12*(-c*d^2*e)^(2/3)*c^2*arctan(1/3*sqrt(3)*(2*(e*x)^(1/ 3) + (-c*e/d)^(1/3))/(-c*e/d)^(1/3))/(sqrt(3)*b*c^2*d^3 + sqrt(3)*a*d^5) - 2*(-c*d^2*e)^(2/3)*c^2*log((e*x)^(2/3) + (e*x)^(1/3)*(-c*e/d)^(1/3) + (-c *e/d)^(2/3))/(b*c^2*d^3 + a*d^5) - 2*(a*b^5*e^2)^(1/3)*a*d*log((e*x)^(2/3) + (a*e^2/b)^(1/3))/(b^4*c^2 + a*b^3*d^2) - 4*(a*b^5*e^2)^(5/6)*c*arctan(( e*x)^(1/3)/(a*e^2/b)^(1/6))/(b^6*c^2*e + a*b^5*d^2*e) + 2*(sqrt(3)*(a*b^5* e^2)^(1/3)*a*b^2*d*e - (a*b^5*e^2)^(5/6)*c)*arctan((sqrt(3)*(a*e^2/b)^(1/6 ) + 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))/(b^6*c^2*e + a*b^5*d^2*e) - 2*(sqrt(3) *(a*b^5*e^2)^(1/3)*a*b^2*d*e + (a*b^5*e^2)^(5/6)*c)*arctan(-(sqrt(3)*(a*e^ 2/b)^(1/6) - 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))/(b^6*c^2*e + a*b^5*d^2*e) + ( (a*b^5*e^2)^(1/3)*a*b^2*d*e + sqrt(3)*(a*b^5*e^2)^(5/6)*c)*log(sqrt(3)*(a* e^2/b)^(1/6)*(e*x)^(1/3) + (e*x)^(2/3) + (a*e^2/b)^(1/3))/(b^6*c^2*e + a*b ^5*d^2*e) + ((a*b^5*e^2)^(1/3)*a*b^2*d*e - sqrt(3)*(a*b^5*e^2)^(5/6)*c)*lo g(-sqrt(3)*(a*e^2/b)^(1/6)*(e*x)^(1/3) + (e*x)^(2/3) + (a*e^2/b)^(1/3))/(b ^6*c^2*e + a*b^5*d^2*e) + 6*(e*x)^(2/3)/(b*d))*e^2
Time = 47.45 (sec) , antiderivative size = 7091, normalized size of antiderivative = 9.92 \[ \int \frac {(e x)^{8/3}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((e*x)^(8/3)/((a + b*x^2)*(c + d*x)),x)
Output:
log(((((((((52488*a^5*b^4*c^2*d^3*e^24*(e*x)^(1/3)*(3*b^2*c^4 - a^2*d^4 + 2*a*b*c^2*d^2)^2 + 104976*a^4*b^6*c*d^4*e^19*(a*d^2 + b*c^2)^4*(a*d^2 - b* c^2)*(-(e^8*(a^4*b^4*d^3 + b*c^3*(-a^5*b^9)^(1/2) - 3*a^3*b^5*c^2*d - 3*a* c*d^2*(-a^5*b^9)^(1/2)))/(b^8*(a*d^2 + b*c^2)^3))^(2/3))*(-(e^8*(a^4*b^4*d ^3 + b*c^3*(-a^5*b^9)^(1/2) - 3*a^3*b^5*c^2*d - 3*a*c*d^2*(-a^5*b^9)^(1/2) ))/(b^8*(a*d^2 + b*c^2)^3))^(1/3))/2 + (26244*a^4*b^2*c*e^27*(a*d^2 + b*c^ 2)^2*(4*a^4*d^8 + 16*b^4*c^8 - 16*a*b^3*c^6*d^2 - 15*a^3*b*c^2*d^6 + 17*a^ 2*b^2*c^4*d^4))/d)*(-(e^8*(a^4*b^4*d^3 + b*c^3*(-a^5*b^9)^(1/2) - 3*a^3*b^ 5*c^2*d - 3*a*c*d^2*(-a^5*b^9)^(1/2)))/(b^8*(a*d^2 + b*c^2)^3))^(2/3))/4 + (6561*a^5*c^2*e^32*(e*x)^(1/3)*(a^5*d^10 - 8*b^5*c^10 + 56*a*b^4*c^8*d^2 + 2*a^4*b*c^2*d^8 - 64*a^2*b^3*c^6*d^4 + 17*a^3*b^2*c^4*d^6))/d^2)*(-(e^8* (a^4*b^4*d^3 + b*c^3*(-a^5*b^9)^(1/2) - 3*a^3*b^5*c^2*d - 3*a*c*d^2*(-a^5* b^9)^(1/2)))/(b^8*(a*d^2 + b*c^2)^3))^(1/3))/2 + (6561*a^6*c*e^35*(a^5*d^1 0 - 4*b^5*c^10 + a^4*b*c^2*d^8))/(b^2*d^2))*(-(e^8*(a^4*b^4*d^3 + b*c^3*(- a^5*b^9)^(1/2) - 3*a^3*b^5*c^2*d - 3*a*c*d^2*(-a^5*b^9)^(1/2)))/(b^8*(a*d^ 2 + b*c^2)^3))^(2/3))/4 + (6561*a^8*c^6*e^40*(e*x)^(1/3)*(a*d^2 - b*c^2))/ (b^2*d))*(-(a^4*b^4*d^3*e^8 + b*c^3*e^8*(-a^5*b^9)^(1/2) - 3*a^3*b^5*c^2*d *e^8 - 3*a*c*d^2*e^8*(-a^5*b^9)^(1/2))/(8*(b^11*c^6 + a^3*b^8*d^6 + 3*a*b^ 10*c^4*d^2 + 3*a^2*b^9*c^2*d^4)))^(1/3) + log(((((((((52488*a^5*b^4*c^2*d^ 3*e^24*(e*x)^(1/3)*(3*b^2*c^4 - a^2*d^4 + 2*a*b*c^2*d^2)^2 + 104976*a^4...
\[ \int \frac {(e x)^{8/3}}{(c+d x) \left (a+b x^2\right )} \, dx=\int \frac {\left (e x \right )^{\frac {8}{3}}}{\left (d x +c \right ) \left (b \,x^{2}+a \right )}d x \] Input:
int((e*x)^(8/3)/(d*x+c)/(b*x^2+a),x)
Output:
int((e*x)^(8/3)/(d*x+c)/(b*x^2+a),x)