\(\int \frac {1}{(e x)^{2/3} (c+d x)^2 (a+b x^2)} \, dx\) [965]

Optimal result

Integrand size = 24, antiderivative size = 837 \[ \int \frac {1}{(e x)^{2/3} (c+d x)^2 \left (a+b x^2\right )} \, dx=\frac {d^2 \sqrt [3]{e x}}{c \left (b c^2+a d^2\right ) e (c+d x)}+\frac {b^{5/6} \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} \left (b c^2+a d^2\right )^2 e^{2/3}}-\frac {b^{5/6} \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} \left (b c^2+a d^2\right )^2 e^{2/3}}+\frac {b^{5/6} \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} \left (b c^2+a d^2\right )^2 e^{2/3}}-\frac {2 d^{5/3} \left (4 b c^2+a d^2\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e x}}{\sqrt [3]{c} \sqrt [3]{e}}}{\sqrt {3}}\right )}{\sqrt {3} c^{5/3} \left (b c^2+a d^2\right )^2 e^{2/3}}+\frac {2 d^{5/3} \left (4 b c^2+a d^2\right ) \log \left (\sqrt [3]{c} \sqrt [3]{e}+\sqrt [3]{d} \sqrt [3]{e x}\right )}{3 c^{5/3} \left (b c^2+a d^2\right )^2 e^{2/3}}+\frac {b^{4/3} c d \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{\sqrt [3]{a} \left (b c^2+a d^2\right )^2 e^{2/3}}-\frac {b^{5/6} \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} \left (b c^2+a d^2\right )^2 e^{2/3}}+\frac {b^{5/6} \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} \left (b c^2+a d^2\right )^2 e^{2/3}}-\frac {d^{5/3} \left (4 b c^2+a d^2\right ) \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 )}{3 c^{5/3} \left (b c^2+a d^2\right )^2 e^{2/3}} \] Output:

d^2*(e*x)^(1/3)/c/(a*d^2+b*c^2)/e/(d*x+c)+b^(5/6)*(-a*d^2+b*c^2)*arctan(b^ 
(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(5/6)/(a*d^2+b*c^2)^2/e^(2/3)+1/2*b^( 
5/6)*(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)/(a*d^2+b*c^2)^2/e^(2/3)+1/2*b^(5/6)* 
(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)/(a*d^2+b*c^2)^2/e^(2/3)-2/3*d^(5/3)*(a*d^2 
+4*b*c^2)*arctan(1/3*(1-2*d^(1/3)*(e*x)^(1/3)/c^(1/3)/e^(1/3))*3^(1/2))*3^ 
(1/2)/c^(5/3)/(a*d^2+b*c^2)^2/e^(2/3)+2/3*d^(5/3)*(a*d^2+4*b*c^2)*ln(c^(1/ 
3)*e^(1/3)+d^(1/3)*(e*x)^(1/3))/c^(5/3)/(a*d^2+b*c^2)^2/e^(2/3)+b^(4/3)*c* 
d*ln(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))/a^(1/3)/(a*d^2+b*c^2)^2/e^(2/3)- 
1/4*b^(5/6)*(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)/(a*d^2+b*c^2)^2/e^(2/3)+1/4*b^(5/6)*(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)/(a*d^2+b*c^2)^2/e^(2/3)-1/3*d^(5/3 
)*(a*d^2+4*b*c^2)*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^(5/3)/(a*d^2+b*c^2)^2/e^(2/3)
 

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 599, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(e x)^{2/3} (c+d x)^2 \left (a+b x^2\right )} \, dx=\frac {x^{2/3} \left (\frac {12 d^2 \left (b c^2+a d^2\right ) \sqrt [3]{x}}{c (c+d x)}+\frac {6 b^{5/6} \left (-b c^2+2 \sqrt {3} \sqrt {a} \sqrt {b} c d+a d^2\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )}{a^{5/6}}+\frac {6 b^{5/6} \left (b c^2+2 \sqrt {3} \sqrt {a} \sqrt {b} c d-a d^2\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )}{a^{5/6}}-\frac {8 \sqrt {3} d^{5/3} \left (4 b c^2+a d^2\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{5/3}}+\frac {12 b^{5/6} \left (b c^2-a d^2\right ) \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )}{a^{5/6}}+\frac {8 d^{5/3} \left (4 b c^2+a d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{c^{5/3}}+\frac {12 b^{4/3} c d \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}\right )}{\sqrt [3]{a}}+\frac {3 b^{5/6} \left (-\sqrt {3} b c^2-2 \sqrt {a} \sqrt {b} c d+\sqrt {3} a d^2\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )}{a^{5/6}}-\frac {3 b^{5/6} \left (-\sqrt {3} b c^2+2 \sqrt {a} \sqrt {b} c d+\sqrt {3} a d^2\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )}{a^{5/6}}-\frac {4 d^{5/3} \left (4 b c^2+a d^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{c^{5/3}}\right )}{12 \left (b c^2+a d^2\right )^2 (e x)^{2/3}} \] Input:

Integrate[1/((e*x)^(2/3)*(c + d*x)^2*(a + b*x^2)),x]
 

Output:

(x^(2/3)*((12*d^2*(b*c^2 + a*d^2)*x^(1/3))/(c*(c + d*x)) + (6*b^(5/6)*(-(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)])/a^(5/6) + (6*b^(5/6)*(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)])/a^(5/6) - (8 
*Sqrt[3]*d^(5/3)*(4*b*c^2 + a*d^2)*ArcTan[(1 - (2*d^(1/3)*x^(1/3))/c^(1/3) 
)/Sqrt[3]])/c^(5/3) + (12*b^(5/6)*(b*c^2 - a*d^2)*ArcTan[(b^(1/6)*x^(1/3)) 
/a^(1/6)])/a^(5/6) + (8*d^(5/3)*(4*b*c^2 + a*d^2)*Log[c^(1/3) + d^(1/3)*x^ 
(1/3)])/c^(5/3) + (12*b^(4/3)*c*d*Log[a^(1/3) + b^(1/3)*x^(2/3)])/a^(1/3) 
+ (3*b^(5/6)*(-(Sqrt[3]*b*c^2) - 2*Sqrt[a]*Sqrt[b]*c*d + Sqrt[3]*a*d^2)*Lo 
g[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)])/a^(5/6) - 
(3*b^(5/6)*(-(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)])/a^(5/6) - (4 
*d^(5/3)*(4*b*c^2 + a*d^2)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x^(1/3) + d^(2/3) 
*x^(2/3)])/c^(5/3)))/(12*(b*c^2 + a*d^2)^2*(e*x)^(2/3))
 

Rubi [A] (verified)

Time = 2.70 (sec) , antiderivative size = 994, 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.

\(\displaystyle \int \frac {1}{(e x)^{2/3} \left (a+b x^2\right ) (c+d x)^2} \, dx\)

\(\Big \downarrow \) 615

\(\displaystyle \int \left (\frac {b \left (-a d^2+b c^2-2 b c d x\right )}{(e x)^{2/3} \left (a+b x^2\right ) \left (a d^2+b c^2\right )^2}+\frac {2 b c d^2}{(e x)^{2/3} (c+d x) \left (a d^2+b c^2\right )^2}+\frac {d^2}{(e x)^{2/3} (c+d x)^2 \left (a d^2+b c^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt [3]{e x} d^2}{c \left (b c^2+a d^2\right ) e (c+d x)}-\frac {2 \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}}{\sqrt {3} c^{5/3} \left (b c^2+a d^2\right ) e^{2/3}}-\frac {2 \sqrt {3} b \sqrt [3]{c} \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 )^2 e^{2/3}}-\frac {\log (c+d x) d^{5/3}}{3 c^{5/3} \left (b c^2+a d^2\right ) e^{2/3}}-\frac {b \sqrt [3]{c} \log (c+d x) d^{5/3}}{\left (b c^2+a d^2\right )^2 e^{2/3}}+\frac {\log \left (\sqrt [3]{c} \sqrt [3]{e}+\sqrt [3]{d} \sqrt [3]{e x}\right ) d^{5/3}}{c^{5/3} \left (b c^2+a d^2\right ) e^{2/3}}+\frac {3 b \sqrt [3]{c} \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 )^2 e^{2/3}}+\frac {\sqrt {3} b^{4/3} c \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right ) d}{\sqrt [3]{a} \left (b c^2+a d^2\right )^2 e^{2/3}}+\frac {b^{4/3} c \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right ) d}{\sqrt [3]{a} \left (b c^2+a d^2\right )^2 e^{2/3}}-\frac {b^{4/3} 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 ) d}{2 \sqrt [3]{a} \left (b c^2+a d^2\right )^2 e^{2/3}}+\frac {b^{5/6} \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} \left (b c^2+a d^2\right )^2 e^{2/3}}-\frac {b^{5/6} \left (b c^2-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} \left (b c^2+a d^2\right )^2 e^{2/3}}+\frac {b^{5/6} \left (b c^2-a d^2\right ) \arctan \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}+\sqrt {3}\right )}{2 a^{5/6} \left (b c^2+a d^2\right )^2 e^{2/3}}-\frac {\sqrt {3} b^{5/6} \left (b c^2-a d^2\right ) \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 )}{4 a^{5/6} \left (b c^2+a d^2\right )^2 e^{2/3}}+\frac {\sqrt {3} b^{5/6} \left (b c^2-a d^2\right ) \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 )}{4 a^{5/6} \left (b c^2+a d^2\right )^2 e^{2/3}}\)

Input:

Int[1/((e*x)^(2/3)*(c + d*x)^2*(a + b*x^2)),x]
 

Output:

(d^2*(e*x)^(1/3))/(c*(b*c^2 + a*d^2)*e*(c + d*x)) + (b^(5/6)*(b*c^2 - a*d^ 
2)*ArcTan[(b^(1/6)*(e*x)^(1/3))/(a^(1/6)*e^(1/3))])/(a^(5/6)*(b*c^2 + a*d^ 
2)^2*e^(2/3)) - (b^(5/6)*(b*c^2 - a*d^2)*ArcTan[Sqrt[3] - (2*b^(1/6)*(e*x) 
^(1/3))/(a^(1/6)*e^(1/3))])/(2*a^(5/6)*(b*c^2 + a*d^2)^2*e^(2/3)) + (b^(5/ 
6)*(b*c^2 - a*d^2)*ArcTan[Sqrt[3] + (2*b^(1/6)*(e*x)^(1/3))/(a^(1/6)*e^(1/ 
3))])/(2*a^(5/6)*(b*c^2 + a*d^2)^2*e^(2/3)) - (2*Sqrt[3]*b*c^(1/3)*d^(5/3) 
*ArcTan[(1 - (2*d^(1/3)*(e*x)^(1/3))/(c^(1/3)*e^(1/3)))/Sqrt[3]])/((b*c^2 
+ a*d^2)^2*e^(2/3)) - (2*d^(5/3)*ArcTan[(1 - (2*d^(1/3)*(e*x)^(1/3))/(c^(1 
/3)*e^(1/3)))/Sqrt[3]])/(Sqrt[3]*c^(5/3)*(b*c^2 + a*d^2)*e^(2/3)) + (Sqrt[ 
3]*b^(4/3)*c*d*ArcTan[(1 - (2*b^(1/3)*(e*x)^(2/3))/(a^(1/3)*e^(2/3)))/Sqrt 
[3]])/(a^(1/3)*(b*c^2 + a*d^2)^2*e^(2/3)) - (b*c^(1/3)*d^(5/3)*Log[c + d*x 
])/((b*c^2 + a*d^2)^2*e^(2/3)) - (d^(5/3)*Log[c + d*x])/(3*c^(5/3)*(b*c^2 
+ a*d^2)*e^(2/3)) + (3*b*c^(1/3)*d^(5/3)*Log[c^(1/3)*e^(1/3) + d^(1/3)*(e* 
x)^(1/3)])/((b*c^2 + a*d^2)^2*e^(2/3)) + (d^(5/3)*Log[c^(1/3)*e^(1/3) + d^ 
(1/3)*(e*x)^(1/3)])/(c^(5/3)*(b*c^2 + a*d^2)*e^(2/3)) + (b^(4/3)*c*d*Log[a 
^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3)])/(a^(1/3)*(b*c^2 + a*d^2)^2*e^(2/3)) 
 - (Sqrt[3]*b^(5/6)*(b*c^2 - a*d^2)*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*a^(5/6)*(b*c^2 + a* 
d^2)^2*e^(2/3)) + (Sqrt[3]*b^(5/6)*(b*c^2 - a*d^2)*Log[a^(1/3)*e^(2/3) + S 
qrt[3]*a^(1/6)*b^(1/6)*e^(1/3)*(e*x)^(1/3) + b^(1/3)*(e*x)^(2/3)])/(4*a...
 

Defintions of rubi rules used

rule 615

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 544, normalized size of antiderivative = 0.65

method	result	size
pseudoelliptic	\(-\frac {2 d \left (\frac {c e}{d}\right )^{\frac {2}{3}} b^{2} \left (d x +c \right ) \left (\left (\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )-\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\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 ) c^{2} \left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}}+\left (\left (\frac {\left (-\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )+\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )\right ) \sqrt {3}}{2}+\arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )+\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 \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )\right ) \left (\frac {c e}{d}\right )^{\frac {2}{3}} b \left (d x +c \right ) \left (a \,d^{2}-b \,c^{2}\right ) c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+\frac {2 d a \left (-3 \left (\frac {c e}{d}\right )^{\frac {2}{3}} d \left (e x \right )^{\frac {1}{3}} \left (a \,d^{2}+b \,c^{2}\right )+\left (a \,d^{2}+4 b \,c^{2}\right ) e \left (d x +c \right ) \left (-2 \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 ) \sqrt {3}+\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 )}{3}\right ) e}{2 \left (\frac {c e}{d}\right )^{\frac {2}{3}} e^{2} c \left (d x +c \right ) \left (a \,d^{2}+b \,c^{2}\right )^{2} a}\)	\(544\)
derivativedivides	\(\text {Expression too large to display}\)	\(938\)
default	\(\text {Expression too large to display}\)	\(938\)

Input:

int(1/(e*x)^(2/3)/(d*x+c)^2/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-1/2/(c*e/d)^(2/3)*(2*d*(c*e/d)^(2/3)*b^2*(d*x+c)*((arctan(2*(e*x)^(1/3)/( 
a*e^2/b)^(1/6)-3^(1/2))-arctan(2*(e*x)^(1/3)/(a*e^2/b)^(1/6)+3^(1/2)))*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)))*c^2*(a*e^2/b)^(2/3)+((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))+ln((e*x)^(2/3)+3^( 
1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)+(a*e^2/b)^(1/3)))*3^(1/2)+arctan(2*(e*x)^ 
(1/3)/(a*e^2/b)^(1/6)-3^(1/2))+arctan(2*(e*x)^(1/3)/(a*e^2/b)^(1/6)+3^(1/2 
))+2*arctan((e*x)^(1/3)/(a*e^2/b)^(1/6)))*(c*e/d)^(2/3)*b*(d*x+c)*(a*d^2-b 
*c^2)*c*(a*e^2/b)^(1/6)+2/3*d*a*(-3*(c*e/d)^(2/3)*d*(e*x)^(1/3)*(a*d^2+b*c 
^2)+(a*d^2+4*b*c^2)*e*(d*x+c)*(-2*arctan(1/3*3^(1/2)*(2/(c*e/d)^(1/3)*(e*x 
)^(1/3)-1))*3^(1/2)+ln((e*x)^(2/3)-(c*e/d)^(1/3)*(e*x)^(1/3)+(c*e/d)^(2/3) 
)-2*ln((e*x)^(1/3)+(c*e/d)^(1/3)))))*e)/e^2/c/(d*x+c)/(a*d^2+b*c^2)^2/a
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{2/3} (c+d x)^2 \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)^(2/3)/(d*x+c)^2/(b*x^2+a),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{2/3} (c+d x)^2 \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)**(2/3)/(d*x+c)**2/(b*x**2+a),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(e x)^{2/3} (c+d x)^2 \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x)^(2/3)/(d*x+c)^2/(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
 

Giac [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 1040, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(e x)^{2/3} (c+d x)^2 \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x)^(2/3)/(d*x+c)^2/(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))/(sq 
rt(a*b)*b^2*c^4*e^2 + 2*sqrt(a*b)*a*b*c^2*d^2*e^2 + sqrt(a*b)*a^2*d^4*e^2) 
 - 2/3*(4*b*c^2*d^2 + a*d^4)*(-c*e/d)^(1/3)*log(abs((e*x)^(1/3) - (-c*e/d) 
^(1/3)))/(b^2*c^6*e + 2*a*b*c^4*d^2*e + a^2*c^2*d^4*e) + 2*(4*(-c*d^2*e)^( 
1/3)*b*c^2*d + (-c*d^2*e)^(1/3)*a*d^3)*arctan(1/3*sqrt(3)*(2*(e*x)^(1/3) + 
 (-c*e/d)^(1/3))/(-c*e/d)^(1/3))/(sqrt(3)*b^2*c^6*e + 2*sqrt(3)*a*b*c^4*d^ 
2*e + sqrt(3)*a^2*c^2*d^4*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*c^4*e^2 + 2*a^2*b^ 
3*c^2*d^2*e^2 + a^3*b^2*d^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*c^4*e^2 + 2*a 
^2*b^3*c^2*d^2*e^2 + a^3*b^2*d^4*e^2) + ((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*c^4*e + 2*a^2 
*b*c^2*d^2*e + a^3*d^4*e) + 1/4*(sqrt(3)*(a*b^5*e^2)^(1/6)*b^3*c^2*e - sqr 
t(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*c^4*e^ 
2 + 2*a^2*b^3*c^2*d^2*e^2 + a^3*b^2*d^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/...
 

Mupad [B] (verification not implemented)

Time = 79.64 (sec) , antiderivative size = 11877, normalized size of antiderivative = 14.19 \[ \int \frac {1}{(e x)^{2/3} (c+d x)^2 \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

int(1/((e*x)^(2/3)*(a + b*x^2)*(c + d*x)^2),x)
 

Output:

log((((((((((139968*a^3*b^6*d^4*e^18*(e*x)^(1/3)*(a*d^2 + b*c^2)^3*(2*a^2* 
d^4 - 3*b^2*c^4 + 6*a*b*c^2*d^2))/c + 209952*a^4*b^6*c*d^4*e^19*(a*d^2 + b 
*c^2)^4*(a*d^2 - b*c^2)*(-(a^3*d^6*(-a^5*b^5)^(1/2) - b^3*c^6*(-a^5*b^5)^( 
1/2) + 6*a^3*b^5*c^5*d + 6*a^5*b^3*c*d^5 - 20*a^4*b^4*c^3*d^3 + 15*a*b^2*c 
^4*d^2*(-a^5*b^5)^(1/2) - 15*a^2*b*c^2*d^4*(-a^5*b^5)^(1/2))/(a^5*e^2*(a*d 
^2 + b*c^2)^6))^(1/3))*(-(a^3*d^6*(-a^5*b^5)^(1/2) - b^3*c^6*(-a^5*b^5)^(1 
/2) + 6*a^3*b^5*c^5*d + 6*a^5*b^3*c*d^5 - 20*a^4*b^4*c^3*d^3 + 15*a*b^2*c^ 
4*d^2*(-a^5*b^5)^(1/2) - 15*a^2*b*c^2*d^4*(-a^5*b^5)^(1/2))/(a^5*e^2*(a*d^ 
2 + b*c^2)^6))^(2/3))/4 + (972*a*b^7*d^3*e^17*(128*a^5*d^10 - 27*b^5*c^10 
+ 135*a*b^4*c^8*d^2 + 1408*a^4*b*c^2*d^8 + 783*a^2*b^3*c^6*d^4 + 5357*a^3* 
b^2*c^4*d^6))/(b*c^4 + a*c^2*d^2))*(-(a^3*d^6*(-a^5*b^5)^(1/2) - b^3*c^6*( 
-a^5*b^5)^(1/2) + 6*a^3*b^5*c^5*d + 6*a^5*b^3*c*d^5 - 20*a^4*b^4*c^3*d^3 + 
 15*a*b^2*c^4*d^2*(-a^5*b^5)^(1/2) - 15*a^2*b*c^2*d^4*(-a^5*b^5)^(1/2))/(a 
^5*e^2*(a*d^2 + b*c^2)^6))^(1/3))/2 - (486*b^8*d^3*e^16*(e*x)^(1/3)*(256*a 
^7*d^14 + 27*b^7*c^14 + 81*a*b^6*c^12*d^2 + 1952*a^6*b*c^2*d^12 + 54*a^2*b 
^5*c^10*d^4 - 19894*a^3*b^4*c^8*d^6 + 14671*a^4*b^3*c^6*d^8 + 8613*a^5*b^2 
*c^4*d^10))/(c^2*(a*d^2 + b*c^2)^4))*(-(a^3*d^6*(-a^5*b^5)^(1/2) - b^3*c^6 
*(-a^5*b^5)^(1/2) + 6*a^3*b^5*c^5*d + 6*a^5*b^3*c*d^5 - 20*a^4*b^4*c^3*d^3 
 + 15*a*b^2*c^4*d^2*(-a^5*b^5)^(1/2) - 15*a^2*b*c^2*d^4*(-a^5*b^5)^(1/2))/ 
(a^5*e^2*(a*d^2 + b*c^2)^6))^(2/3))/4 + (243*b^9*d^6*e^15*(32*a^6*d^12 ...
 

Reduce [F]

\[ \int \frac {1}{(e x)^{2/3} (c+d x)^2 \left (a+b x^2\right )} \, dx=\int \frac {1}{\left (e x \right )^{\frac {2}{3}} \left (d x +c \right )^{2} \left (b \,x^{2}+a \right )}d x \] Input:

int(1/(e*x)^(2/3)/(d*x+c)^2/(b*x^2+a),x)
 

Output:

int(1/(e*x)^(2/3)/(d*x+c)^2/(b*x^2+a),x)