Optimal. Leaf size=139 \[ -\frac{e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}+\frac{e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac{1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}-\frac{e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac{e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0984611, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 44} \[ -\frac{e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}+\frac{e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac{1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}-\frac{e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac{e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac{1}{(a e+c d x)^4 (d+e x)} \, dx\\ &=\int \left (\frac{c d}{\left (c d^2-a e^2\right ) (a e+c d x)^4}-\frac{c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)^3}+\frac{c d e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac{c d e^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac{e^4}{\left (c d^2-a e^2\right )^4 (d+e x)}\right ) \, dx\\ &=-\frac{1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}+\frac{e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac{e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac{e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac{e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}\\ \end{align*}
Mathematica [A] time = 0.100535, size = 117, normalized size = 0.84 \[ -\frac{\frac{\left (c d^2-a e^2\right ) \left (11 a^2 e^4+a c d e^2 (15 e x-7 d)+c^2 d^2 \left (2 d^2-3 d e x+6 e^2 x^2\right )\right )}{(a e+c d x)^3}+6 e^3 \log (a e+c d x)-6 e^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.052, size = 135, normalized size = 1. \begin{align*}{\frac{{e}^{3}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}}+{\frac{1}{ \left ( 3\,a{e}^{2}-3\,c{d}^{2} \right ) \left ( cdx+ae \right ) ^{3}}}+{\frac{e}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdx+ae \right ) ^{2}}}+{\frac{{e}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdx+ae \right ) }}-{\frac{{e}^{3}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.07751, size = 556, normalized size = 4. \begin{align*} -\frac{e^{3} \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac{e^{3} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac{6 \, c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{4} - 7 \, a c d^{2} e^{2} + 11 \, a^{2} e^{4} - 3 \,{\left (c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x}{6 \,{\left (a^{3} c^{3} d^{6} e^{3} - 3 \, a^{4} c^{2} d^{4} e^{5} + 3 \, a^{5} c d^{2} e^{7} - a^{6} e^{9} +{\left (c^{6} d^{9} - 3 \, a c^{5} d^{7} e^{2} + 3 \, a^{2} c^{4} d^{5} e^{4} - a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \,{\left (a c^{5} d^{8} e - 3 \, a^{2} c^{4} d^{6} e^{3} + 3 \, a^{3} c^{3} d^{4} e^{5} - a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + 3 \,{\left (a^{2} c^{4} d^{7} e^{2} - 3 \, a^{3} c^{3} d^{5} e^{4} + 3 \, a^{4} c^{2} d^{3} e^{6} - a^{5} c d e^{8}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.99822, size = 961, normalized size = 6.91 \begin{align*} -\frac{2 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \,{\left (c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x + 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (c d x + a e\right ) - 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (e x + d\right )}{6 \,{\left (a^{3} c^{4} d^{8} e^{3} - 4 \, a^{4} c^{3} d^{6} e^{5} + 6 \, a^{5} c^{2} d^{4} e^{7} - 4 \, a^{6} c d^{2} e^{9} + a^{7} e^{11} +{\left (c^{7} d^{11} - 4 \, a c^{6} d^{9} e^{2} + 6 \, a^{2} c^{5} d^{7} e^{4} - 4 \, a^{3} c^{4} d^{5} e^{6} + a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \,{\left (a c^{6} d^{10} e - 4 \, a^{2} c^{5} d^{8} e^{3} + 6 \, a^{3} c^{4} d^{6} e^{5} - 4 \, a^{4} c^{3} d^{4} e^{7} + a^{5} c^{2} d^{2} e^{9}\right )} x^{2} + 3 \,{\left (a^{2} c^{5} d^{9} e^{2} - 4 \, a^{3} c^{4} d^{7} e^{4} + 6 \, a^{4} c^{3} d^{5} e^{6} - 4 \, a^{5} c^{2} d^{3} e^{8} + a^{6} c d e^{10}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 2.6889, size = 668, normalized size = 4.81 \begin{align*} \frac{e^{3} \log{\left (x + \frac{- \frac{a^{5} e^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{5 a^{4} c d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{10 a^{3} c^{2} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{10 a^{2} c^{3} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{5 a c^{4} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + a e^{5} + \frac{c^{5} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + c d^{2} e^{3}}{2 c d e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{e^{3} \log{\left (x + \frac{\frac{a^{5} e^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{5 a^{4} c d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{10 a^{3} c^{2} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{10 a^{2} c^{3} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{5 a c^{4} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + a e^{5} - \frac{c^{5} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + c d^{2} e^{3}}{2 c d e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{11 a^{2} e^{4} - 7 a c d^{2} e^{2} + 2 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (15 a c d e^{3} - 3 c^{2} d^{3} e\right )}{6 a^{6} e^{9} - 18 a^{5} c d^{2} e^{7} + 18 a^{4} c^{2} d^{4} e^{5} - 6 a^{3} c^{3} d^{6} e^{3} + x^{3} \left (6 a^{3} c^{3} d^{3} e^{6} - 18 a^{2} c^{4} d^{5} e^{4} + 18 a c^{5} d^{7} e^{2} - 6 c^{6} d^{9}\right ) + x^{2} \left (18 a^{4} c^{2} d^{2} e^{7} - 54 a^{3} c^{3} d^{4} e^{5} + 54 a^{2} c^{4} d^{6} e^{3} - 18 a c^{5} d^{8} e\right ) + x \left (18 a^{5} c d e^{8} - 54 a^{4} c^{2} d^{3} e^{6} + 54 a^{3} c^{3} d^{5} e^{4} - 18 a^{2} c^{4} d^{7} e^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.34534, size = 973, normalized size = 7. \begin{align*} \frac{2 \,{\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} \arctan \left (-\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} - \frac{6 \, c^{5} d^{8} x^{5} e^{5} + 15 \, c^{5} d^{9} x^{4} e^{4} + 11 \, c^{5} d^{10} x^{3} e^{3} + 3 \, c^{5} d^{11} x^{2} e^{2} + 3 \, c^{5} d^{12} x e + 2 \, c^{5} d^{13} - 18 \, a c^{4} d^{6} x^{5} e^{7} - 30 \, a c^{4} d^{7} x^{4} e^{6} + 5 \, a c^{4} d^{8} x^{3} e^{5} + 15 \, a c^{4} d^{9} x^{2} e^{4} - 15 \, a c^{4} d^{10} x e^{3} - 13 \, a c^{4} d^{11} e^{2} + 18 \, a^{2} c^{3} d^{4} x^{5} e^{9} - 70 \, a^{2} c^{3} d^{6} x^{3} e^{7} - 30 \, a^{2} c^{3} d^{7} x^{2} e^{6} + 60 \, a^{2} c^{3} d^{8} x e^{5} + 38 \, a^{2} c^{3} d^{9} e^{4} - 6 \, a^{3} c^{2} d^{2} x^{5} e^{11} + 30 \, a^{3} c^{2} d^{3} x^{4} e^{10} + 70 \, a^{3} c^{2} d^{4} x^{3} e^{9} - 30 \, a^{3} c^{2} d^{5} x^{2} e^{8} - 120 \, a^{3} c^{2} d^{6} x e^{7} - 56 \, a^{3} c^{2} d^{7} e^{6} - 15 \, a^{4} c d x^{4} e^{12} - 5 \, a^{4} c d^{2} x^{3} e^{11} + 75 \, a^{4} c d^{3} x^{2} e^{10} + 105 \, a^{4} c d^{4} x e^{9} + 40 \, a^{4} c d^{5} e^{8} - 11 \, a^{5} x^{3} e^{13} - 33 \, a^{5} d x^{2} e^{12} - 33 \, a^{5} d^{2} x e^{11} - 11 \, a^{5} d^{3} e^{10}}{6 \,{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]