Integrand size = 24, antiderivative size = 735 \[ \int \frac {1}{(e x)^{7/3} (c+d x) \left (a+b x^2\right )} \, dx=-\frac {3}{4 a c e (e x)^{4/3}}+\frac {3 d}{a c^2 e^2 \sqrt [3]{e x}}+\frac {b^{7/6} d \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{a^{7/6} \left (b c^2+a d^2\right ) e^{7/3}}-\frac {b^{7/6} d \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{7/6} \left (b c^2+a d^2\right ) e^{7/3}}+\frac {b^{7/6} d \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{7/6} \left (b c^2+a d^2\right ) e^{7/3}}-\frac {\sqrt {3} d^{10/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e x}}{\sqrt [3]{c} \sqrt [3]{e}}}{\sqrt {3}}\right )}{c^{7/3} \left (b c^2+a d^2\right ) e^{7/3}}+\frac {\sqrt {3} b^{5/3} c \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{2 a^{5/3} \left (b c^2+a d^2\right ) e^{7/3}}-\frac {\sqrt {3} b^{7/6} d \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}\right )}{2 a^{7/6} \left (b c^2+a d^2\right ) e^{7/3}}-\frac {d^{10/3} \log \left (\sqrt [3]{c} \sqrt [3]{e}+\sqrt [3]{d} \sqrt [3]{e x}\right )}{c^{7/3} \left (b c^2+a d^2\right ) e^{7/3}}-\frac {b^{5/3} c \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 a^{5/3} \left (b c^2+a d^2\right ) e^{7/3}}+\frac {d^{10/3} \log \left (c^{2/3} e^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{e} \sqrt [3]{e x}+d^{2/3} (e x)^{2/3}\right )}{2 c^{7/3} \left (b c^2+a d^2\right ) e^{7/3}}+\frac {b^{5/3} c \log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{4/3}\right )}{4 a^{5/3} \left (b c^2+a d^2\right ) e^{7/3}} \] Output:
-3/4/a/c/e/(e*x)^(4/3)+3*d/a/c^2/e^2/(e*x)^(1/3)+b^(7/6)*d*arctan(b^(1/6)* (e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(7/6)/(a*d^2+b*c^2)/e^(7/3)+1/2*b^(7/6)*d*a rctan(-3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(7/6)/(a*d^2+b*c^2 )/e^(7/3)+1/2*b^(7/6)*d*arctan(3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/ 3))/a^(7/6)/(a*d^2+b*c^2)/e^(7/3)-3^(1/2)*d^(10/3)*arctan(1/3*(1-2*d^(1/3) *(e*x)^(1/3)/c^(1/3)/e^(1/3))*3^(1/2))/c^(7/3)/(a*d^2+b*c^2)/e^(7/3)+1/2*3 ^(1/2)*b^(5/3)*c*arctan(1/3*(1-2*b^(1/3)*(e*x)^(2/3)/a^(1/3)/e^(2/3))*3^(1 /2))/a^(5/3)/(a*d^2+b*c^2)/e^(7/3)-1/2*3^(1/2)*b^(7/6)*d*arctanh(3^(1/2)*a ^(1/6)*b^(1/6)*e^(1/3)*(e*x)^(1/3)/(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3)))/ a^(7/6)/(a*d^2+b*c^2)/e^(7/3)-d^(10/3)*ln(c^(1/3)*e^(1/3)+d^(1/3)*(e*x)^(1 /3))/c^(7/3)/(a*d^2+b*c^2)/e^(7/3)-1/2*b^(5/3)*c*ln(a^(1/3)*e^(2/3)+b^(1/3 )*(e*x)^(2/3))/a^(5/3)/(a*d^2+b*c^2)/e^(7/3)+1/2*d^(10/3)*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))/c^(7/3)/(a*d^2 +b*c^2)/e^(7/3)+1/4*b^(5/3)*c*ln(a^(2/3)*e^(4/3)-a^(1/3)*b^(1/3)*e^(2/3)*( e*x)^(2/3)+b^(2/3)*(e*x)^(4/3))/a^(5/3)/(a*d^2+b*c^2)/e^(7/3)
Time = 1.21 (sec) , antiderivative size = 673, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(e x)^{7/3} (c+d x) \left (a+b x^2\right )} \, dx=\frac {x \left (-3 a^{2/3} b c^{10/3}-3 a^{5/3} c^{4/3} d^2+12 a^{2/3} b c^{7/3} d x+12 a^{5/3} \sqrt [3]{c} d^3 x+2 b^{7/6} c^{7/3} \left (\sqrt {3} \sqrt {b} c-\sqrt {a} d\right ) x^{4/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+2 b^{7/6} c^{7/3} \left (\sqrt {3} \sqrt {b} c+\sqrt {a} d\right ) x^{4/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-4 \sqrt {3} a^{5/3} d^{10/3} x^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+4 \sqrt {a} b^{7/6} c^{7/3} d x^{4/3} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-4 a^{5/3} d^{10/3} x^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )-2 b^{5/3} c^{10/3} x^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}\right )+b^{5/3} c^{10/3} x^{4/3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+\sqrt {3} \sqrt {a} b^{7/6} c^{7/3} d x^{4/3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+b^{5/3} c^{10/3} x^{4/3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )-\sqrt {3} \sqrt {a} b^{7/6} c^{7/3} d x^{4/3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+2 a^{5/3} d^{10/3} x^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )\right )}{4 a^{5/3} c^{7/3} \left (b c^2+a d^2\right ) (e x)^{7/3}} \] Input:
Integrate[1/((e*x)^(7/3)*(c + d*x)*(a + b*x^2)),x]
Output:
(x*(-3*a^(2/3)*b*c^(10/3) - 3*a^(5/3)*c^(4/3)*d^2 + 12*a^(2/3)*b*c^(7/3)*d *x + 12*a^(5/3)*c^(1/3)*d^3*x + 2*b^(7/6)*c^(7/3)*(Sqrt[3]*Sqrt[b]*c - Sqr t[a]*d)*x^(4/3)*ArcTan[Sqrt[3] - (2*b^(1/6)*x^(1/3))/a^(1/6)] + 2*b^(7/6)* c^(7/3)*(Sqrt[3]*Sqrt[b]*c + Sqrt[a]*d)*x^(4/3)*ArcTan[Sqrt[3] + (2*b^(1/6 )*x^(1/3))/a^(1/6)] - 4*Sqrt[3]*a^(5/3)*d^(10/3)*x^(4/3)*ArcTan[(1 - (2*d^ (1/3)*x^(1/3))/c^(1/3))/Sqrt[3]] + 4*Sqrt[a]*b^(7/6)*c^(7/3)*d*x^(4/3)*Arc Tan[(b^(1/6)*x^(1/3))/a^(1/6)] - 4*a^(5/3)*d^(10/3)*x^(4/3)*Log[c^(1/3) + d^(1/3)*x^(1/3)] - 2*b^(5/3)*c^(10/3)*x^(4/3)*Log[a^(1/3) + b^(1/3)*x^(2/3 )] + b^(5/3)*c^(10/3)*x^(4/3)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3 ) + b^(1/3)*x^(2/3)] + Sqrt[3]*Sqrt[a]*b^(7/6)*c^(7/3)*d*x^(4/3)*Log[a^(1/ 3) - Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] + b^(5/3)*c^(10/3) *x^(4/3)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] - Sqrt[3]*Sqrt[a]*b^(7/6)*c^(7/3)*d*x^(4/3)*Log[a^(1/3) + Sqrt[3]*a^(1/6)* b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] + 2*a^(5/3)*d^(10/3)*x^(4/3)*Log[c^(2/3 ) - c^(1/3)*d^(1/3)*x^(1/3) + d^(2/3)*x^(2/3)]))/(4*a^(5/3)*c^(7/3)*(b*c^2 + a*d^2)*(e*x)^(7/3))
Time = 2.73 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.19, 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.
| \(\displaystyle \int \frac {1}{(e x)^{7/3} \left (a+b x^2\right ) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (\frac {b (c-d x)}{(e x)^{7/3} \left (a+b x^2\right ) \left (a d^2+b c^2\right )}+\frac {d^2}{(e x)^{7/3} (c+d x) \left (a d^2+b c^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e x}}{\sqrt [3]{c} \sqrt [3]{e}}}{\sqrt {3}}\right ) d^{10/3}}{c^{7/3} \left (b c^2+a d^2\right ) e^{7/3}}+\frac {\log (c+d x) d^{10/3}}{2 c^{7/3} \left (b c^2+a d^2\right ) e^{7/3}}-\frac {3 \log \left (\sqrt [3]{c} \sqrt [3]{e}+\sqrt [3]{d} \sqrt [3]{e x}\right ) d^{10/3}}{2 c^{7/3} \left (b c^2+a d^2\right ) e^{7/3}}+\frac {3 d^3}{c^2 \left (b c^2+a d^2\right ) e^2 \sqrt [3]{e x}}-\frac {3 d^2}{4 c \left (b c^2+a d^2\right ) e (e x)^{4/3}}+\frac {b^{7/6} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right ) d}{a^{7/6} \left (b c^2+a d^2\right ) e^{7/3}}-\frac {b^{7/6} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right ) d}{2 a^{7/6} \left (b c^2+a d^2\right ) e^{7/3}}+\frac {b^{7/6} \arctan \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}+\sqrt {3}\right ) d}{2 a^{7/6} \left (b c^2+a d^2\right ) e^{7/3}}+\frac {\sqrt {3} b^{7/6} \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 ) d}{4 a^{7/6} \left (b c^2+a d^2\right ) e^{7/3}}-\frac {\sqrt {3} b^{7/6} \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 ) d}{4 a^{7/6} \left (b c^2+a d^2\right ) e^{7/3}}+\frac {3 b d}{a \left (b c^2+a d^2\right ) e^2 \sqrt [3]{e x}}+\frac {\sqrt {3} b^{5/3} c \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{2 a^{5/3} \left (b c^2+a d^2\right ) e^{7/3}}-\frac {b^{5/3} c \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 a^{5/3} \left (b c^2+a d^2\right ) e^{7/3}}+\frac {b^{5/3} c \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 )}{4 a^{5/3} \left (b c^2+a d^2\right ) e^{7/3}}-\frac {3 b c}{4 a \left (b c^2+a d^2\right ) e (e x)^{4/3}}\) |
Input:
Int[1/((e*x)^(7/3)*(c + d*x)*(a + b*x^2)),x]
Output:
(-3*b*c)/(4*a*(b*c^2 + a*d^2)*e*(e*x)^(4/3)) - (3*d^2)/(4*c*(b*c^2 + a*d^2 )*e*(e*x)^(4/3)) + (3*b*d)/(a*(b*c^2 + a*d^2)*e^2*(e*x)^(1/3)) + (3*d^3)/( c^2*(b*c^2 + a*d^2)*e^2*(e*x)^(1/3)) + (b^(7/6)*d*ArcTan[(b^(1/6)*(e*x)^(1 /3))/(a^(1/6)*e^(1/3))])/(a^(7/6)*(b*c^2 + a*d^2)*e^(7/3)) - (b^(7/6)*d*Ar cTan[Sqrt[3] - (2*b^(1/6)*(e*x)^(1/3))/(a^(1/6)*e^(1/3))])/(2*a^(7/6)*(b*c ^2 + a*d^2)*e^(7/3)) + (b^(7/6)*d*ArcTan[Sqrt[3] + (2*b^(1/6)*(e*x)^(1/3)) /(a^(1/6)*e^(1/3))])/(2*a^(7/6)*(b*c^2 + a*d^2)*e^(7/3)) - (Sqrt[3]*d^(10/ 3)*ArcTan[(1 - (2*d^(1/3)*(e*x)^(1/3))/(c^(1/3)*e^(1/3)))/Sqrt[3]])/(c^(7/ 3)*(b*c^2 + a*d^2)*e^(7/3)) + (Sqrt[3]*b^(5/3)*c*ArcTan[(1 - (2*b^(1/3)*(e *x)^(2/3))/(a^(1/3)*e^(2/3)))/Sqrt[3]])/(2*a^(5/3)*(b*c^2 + a*d^2)*e^(7/3) ) + (d^(10/3)*Log[c + d*x])/(2*c^(7/3)*(b*c^2 + a*d^2)*e^(7/3)) - (3*d^(10 /3)*Log[c^(1/3)*e^(1/3) + d^(1/3)*(e*x)^(1/3)])/(2*c^(7/3)*(b*c^2 + a*d^2) *e^(7/3)) - (b^(5/3)*c*Log[a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3)])/(2*a^(5 /3)*(b*c^2 + a*d^2)*e^(7/3)) + (Sqrt[3]*b^(7/6)*d*Log[a^(1/3)*e^(2/3) - Sq rt[3]*a^(1/6)*b^(1/6)*e^(1/3)*(e*x)^(1/3) + b^(1/3)*(e*x)^(2/3)])/(4*a^(7/ 6)*(b*c^2 + a*d^2)*e^(7/3)) - (Sqrt[3]*b^(7/6)*d*Log[a^(1/3)*e^(2/3) + Sqr t[3]*a^(1/6)*b^(1/6)*e^(1/3)*(e*x)^(1/3) + b^(1/3)*(e*x)^(2/3)])/(4*a^(7/6 )*(b*c^2 + a*d^2)*e^(7/3)) + (b^(5/3)*c*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*a^(5/3)*(b*c^2 + a*d^2) *e^(7/3))
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 = 0.99 (sec) , antiderivative size = 565, normalized size of antiderivative = 0.77
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (-\frac {3 \left (\frac {c e}{d}\right )^{\frac {1}{3}} e \left (-4 d x +c \right ) \left (a \,d^{2}+b \,c^{2}\right )}{4}+d^{3} \left (e x \right )^{\frac {4}{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 ) a \right ) e a \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+\left (\frac {c e}{d}\right )^{\frac {1}{3}} b \left (e x \right )^{\frac {4}{3}} c^{2} \left (b \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 (\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}+\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 ) c \sqrt {\frac {a \,e^{2}}{b}}+d e a \left (\frac {\left (\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 )-\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )\right ) \sqrt {3}}{2}+\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )+2 \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )-\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}-2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )\right )\right )}{2 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {4}{3}} \left (\frac {c e}{d}\right )^{\frac {1}{3}} e^{3} c^{2} \left (a \,d^{2}+b \,c^{2}\right ) a^{2}}\) | \(565\) |
| risch | \(-\frac {3 \left (-4 d x +c \right )}{4 c^{2} a \left (e x \right )^{\frac {1}{3}} x \,e^{2}}+\frac {\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 ) a \,d^{4}}{a \,d^{2}+b \,c^{2}}+\frac {3 \left (\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{12 a \,e^{2}}+\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{12 a e}+\frac {d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a e}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{6 a e}+\frac {d \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{12 a \,e^{2}}+\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{12 a e}+\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, c}{6 a e}\right ) c^{2} b^{2}}{a \,d^{2}+b \,c^{2}}}{a \,c^{2} e^{2}}\) | \(655\) |
| derivativedivides | \(3 e^{2} \left (-\frac {\left (\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{6 a e}-\frac {d \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{12 a e}-\frac {d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a e}+\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{12 a e}-\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, c}{6 a e}\right ) b^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) e^{4} a}-\frac {1}{4 c \,e^{3} a \left (e x \right )^{\frac {4}{3}}}+\frac {d}{c^{2} e^{4} a \left (e x \right )^{\frac {1}{3}}}+\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 ) d^{4}}{c^{2} e^{4} \left (a \,d^{2}+b \,c^{2}\right )}\right )\) | \(663\) |
| default | \(3 e^{2} \left (-\frac {\left (\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{6 a e}-\frac {d \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{12 a e}-\frac {d \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \sqrt {3}\, c \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a e}+\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {5}{6}} d}{12 a \,e^{2}}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} c}{12 a e}-\frac {\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, c}{6 a e}\right ) b^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) e^{4} a}-\frac {1}{4 c \,e^{3} a \left (e x \right )^{\frac {4}{3}}}+\frac {d}{c^{2} e^{4} a \left (e x \right )^{\frac {1}{3}}}+\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 ) d^{4}}{c^{2} e^{4} \left (a \,d^{2}+b \,c^{2}\right )}\right )\) | \(663\) |
Input:
int(1/(e*x)^(7/3)/(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2/(a*e^2/b)^(1/6)/(e*x)^(4/3)*(2*(-3/4*(c*e/d)^(1/3)*e*(-4*d*x+c)*(a*d^2 +b*c^2)+d^3*(e*x)^(4/3)*(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*a*(a*e^2/b)^(1/6)+(c*e/d)^(1/ 3)*b*(e*x)^(4/3)*c^2*(b*((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((e*x)^(2/3)-3^(1/2)* (a*e^2/b)^(1/6)*(e*x)^(1/3)+(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)))*c*(a*e^2/b)^(1/2)+d*e*a*(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))-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)+arc tan((3^(1/2)*(a*e^2/b)^(1/6)+2*(e*x)^(1/3))/(a*e^2/b)^(1/6))+2*arctan((e*x )^(1/3)/(a*e^2/b)^(1/6))-arctan((3^(1/2)*(a*e^2/b)^(1/6)-2*(e*x)^(1/3))/(a *e^2/b)^(1/6)))))/(c*e/d)^(1/3)/e^3/c^2/(a*d^2+b*c^2)/a^2
Leaf count of result is larger than twice the leaf count of optimal. 5270 vs. \(2 (539) = 1078\).
Time = 46.09 (sec) , antiderivative size = 5270, normalized size of antiderivative = 7.17 \[ \int \frac {1}{(e x)^{7/3} (c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x)^(7/3)/(d*x+c)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{(e x)^{7/3} (c+d x) \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)**(7/3)/(d*x+c)/(b*x**2+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(e x)^{7/3} (c+d x) \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x)^(7/3)/(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 = 0.32 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(e x)^{7/3} (c+d x) \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/(e*x)^(7/3)/(d*x+c)/(b*x^2+a),x, algorithm="giac")
Output:
-d^4*(-c*e/d)^(2/3)*log(abs((e*x)^(1/3) - (-c*e/d)^(1/3)))/(b*c^5*e^3 + a* c^3*d^2*e^3) - 3*(-c*d^2*e)^(2/3)*d^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^5*e^3 + sqrt(3)*a*c^3*d^2*e^3 ) + 1/2*(-c*d^2*e)^(2/3)*d^2*log((e*x)^(2/3) + (e*x)^(1/3)*(-c*e/d)^(1/3) + (-c*e/d)^(2/3))/(b*c^5*e^3 + a*c^3*d^2*e^3) + (a*b^5*e^2)^(5/6)*d*arctan ((e*x)^(1/3)/(a*e^2/b)^(1/6))/(a^2*b^4*c^2*e^4 + a^3*b^3*d^2*e^4) - 1/2*(a *b^5*e^2)^(1/3)*c*log((e*x)^(2/3) + (a*e^2/b)^(1/3))/(a^2*b*c^2*e^3 + a^3* d^2*e^3) + 1/2*(sqrt(3)*(a*b^5*e^2)^(1/3)*b^3*c*e + (a*b^5*e^2)^(5/6)*d)*a rctan((sqrt(3)*(a*e^2/b)^(1/6) + 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))/(a^2*b^4* c^2*e^4 + a^3*b^3*d^2*e^4) - 1/2*(sqrt(3)*(a*b^5*e^2)^(1/3)*b^3*c*e - (a*b ^5*e^2)^(5/6)*d)*arctan(-(sqrt(3)*(a*e^2/b)^(1/6) - 2*(e*x)^(1/3))/(a*e^2/ b)^(1/6))/(a^2*b^4*c^2*e^4 + a^3*b^3*d^2*e^4) + 1/4*((a*b^5*e^2)^(1/3)*b^3 *c*e - sqrt(3)*(a*b^5*e^2)^(5/6)*d)*log(sqrt(3)*(a*e^2/b)^(1/6)*(e*x)^(1/3 ) + (e*x)^(2/3) + (a*e^2/b)^(1/3))/(a^2*b^4*c^2*e^4 + a^3*b^3*d^2*e^4) + 1 /4*((a*b^5*e^2)^(1/3)*b^3*c*e + sqrt(3)*(a*b^5*e^2)^(5/6)*d)*log(-sqrt(3)* (a*e^2/b)^(1/6)*(e*x)^(1/3) + (e*x)^(2/3) + (a*e^2/b)^(1/3))/(a^2*b^4*c^2* e^4 + a^3*b^3*d^2*e^4) + 3/4*(4*d*e*x - c*e)/((e*x)^(1/3)*a*c^2*e^3*x)
Time = 17.65 (sec) , antiderivative size = 9362, normalized size of antiderivative = 12.74 \[ \int \frac {1}{(e x)^{7/3} (c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int(1/((e*x)^(7/3)*(a + b*x^2)*(c + d*x)),x)
Output:
log(6561*a^10*b^14*c^33*d^6*e^35 - ((e*x)^(1/3)*(6561*a^10*b^15*c^37*d^3*e ^37 + 13122*a^11*b^14*c^35*d^5*e^37 + 6561*a^12*b^13*c^33*d^7*e^37 - 52488 *a^15*b^10*c^27*d^13*e^37 + 52488*a^16*b^9*c^25*d^15*e^37) + (-(a^5*b^5*c^ 3 + a*d^3*(-a^11*b^7)^(1/2) - 3*a^6*b^4*c*d^2 - 3*b*c^2*d*(-a^11*b^7)^(1/2 ))/(8*(a^13*d^6*e^7 + a^10*b^3*c^6*e^7 + 3*a^11*b^2*c^4*d^2*e^7 + 3*a^12*b *c^2*d^4*e^7)))^(2/3)*(((-(a^5*b^5*c^3 + a*d^3*(-a^11*b^7)^(1/2) - 3*a^6*b ^4*c*d^2 - 3*b*c^2*d*(-a^11*b^7)^(1/2))/(8*(a^13*d^6*e^7 + a^10*b^3*c^6*e^ 7 + 3*a^11*b^2*c^4*d^2*e^7 + 3*a^12*b*c^2*d^4*e^7)))^(2/3)*(419904*a^19*b^ 11*c^41*d^4*e^49 + 1259712*a^20*b^10*c^39*d^6*e^49 + 839808*a^21*b^9*c^37* d^8*e^49 - 839808*a^22*b^8*c^35*d^10*e^49 - 1259712*a^23*b^7*c^33*d^12*e^4 9 - 419904*a^24*b^6*c^31*d^14*e^49) - (e*x)^(1/3)*(209952*a^16*b^12*c^38*d ^5*e^44 - 52488*a^15*b^13*c^40*d^3*e^44 + 524880*a^17*b^11*c^36*d^7*e^44 + 209952*a^18*b^10*c^34*d^9*e^44 - 52488*a^19*b^9*c^32*d^11*e^44 + 419904*a ^20*b^8*c^30*d^13*e^44 + 839808*a^21*b^7*c^28*d^15*e^44 + 419904*a^22*b^6* c^26*d^17*e^44))*(-(a^5*b^5*c^3 + a*d^3*(-a^11*b^7)^(1/2) - 3*a^6*b^4*c*d^ 2 - 3*b*c^2*d*(-a^11*b^7)^(1/2))/(8*(a^13*d^6*e^7 + a^10*b^3*c^6*e^7 + 3*a ^11*b^2*c^4*d^2*e^7 + 3*a^12*b*c^2*d^4*e^7)))^(1/3) - 26244*a^14*b^13*c^38 *d^4*e^42 - 183708*a^15*b^12*c^36*d^6*e^42 + 131220*a^16*b^11*c^34*d^8*e^4 2 + 288684*a^17*b^10*c^32*d^10*e^42 + 419904*a^19*b^8*c^28*d^14*e^42 + 419 904*a^20*b^7*c^26*d^16*e^42))*(-(a^5*b^5*c^3 + a*d^3*(-a^11*b^7)^(1/2) ...
Time = 0.42 (sec) , antiderivative size = 539, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(e x)^{7/3} (c+d x) \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
int(1/(e*x)^(7/3)/(d*x+c)/(b*x^2+a),x)
Output:
( - 4*x**(1/3)*d**(1/3)*b**(1/3)*a**(2/3)*sqrt(3)*atan((c**(1/3) - 2*x**(1 /3)*d**(1/3))/(c**(1/3)*sqrt(3)))*a*d**3*x - 2*x**(1/3)*c**(1/3)*sqrt(b)*s qrt(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*d*x + 2*x**(1/3)*c**(1/3)*sqrt(3)*atan((b**(1/6)*a**(1/6)* sqrt(3) - 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*b**2*c**3*x + 2*x**(1/ 3)*c**(1/3)*sqrt(b)*sqrt(a)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*x**(1/3)*b **(1/3))/(b**(1/6)*a**(1/6)))*b*c**2*d*x + 2*x**(1/3)*c**(1/3)*sqrt(3)*ata n((b**(1/6)*a**(1/6)*sqrt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*b **2*c**3*x + 4*x**(1/3)*c**(1/3)*sqrt(b)*sqrt(a)*atan((x**(1/3)*b**(1/3))/ (b**(1/6)*a**(1/6)))*b*c**2*d*x + x**(1/3)*c**(1/3)*sqrt(b)*sqrt(a)*sqrt(3 )*log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3) )*b*c**2*d*x - x**(1/3)*c**(1/3)*sqrt(b)*sqrt(a)*sqrt(3)*log(x**(1/3)*b**( 1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*b*c**2*d*x - 3*c**(1 /3)*b**(1/3)*a**(2/3)*a*c*d**2 + 12*c**(1/3)*b**(1/3)*a**(2/3)*a*d**3*x - 3*c**(1/3)*b**(1/3)*a**(2/3)*b*c**3 + 12*c**(1/3)*b**(1/3)*a**(2/3)*b*c**2 *d*x + 2*x**(1/3)*d**(1/3)*b**(1/3)*a**(2/3)*log(c**(2/3) - x**(1/3)*d**(1 /3)*c**(1/3) + x**(2/3)*d**(2/3))*a*d**3*x - 4*x**(1/3)*d**(1/3)*b**(1/3)* a**(2/3)*log(c**(1/3) + x**(1/3)*d**(1/3))*a*d**3*x - 2*x**(1/3)*c**(1/3)* log(a**(1/3) + x**(2/3)*b**(1/3))*b**2*c**3*x + x**(1/3)*c**(1/3)*log( - x **(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*b**2*...