Optimal. Leaf size=71 \[ \frac{2 c d \left (c d^2-a e^2\right )}{e^3 (d+e x)}-\frac{\left (c d^2-a e^2\right )^2}{2 e^3 (d+e x)^2}+\frac{c^2 d^2 \log (d+e x)}{e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0519226, antiderivative size = 71, 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, 43} \[ \frac{2 c d \left (c d^2-a e^2\right )}{e^3 (d+e x)}-\frac{\left (c d^2-a e^2\right )^2}{2 e^3 (d+e x)^2}+\frac{c^2 d^2 \log (d+e x)}{e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^5} \, dx &=\int \frac{(a e+c d x)^2}{(d+e x)^3} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^3}-\frac{2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^2}+\frac{c^2 d^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{\left (c d^2-a e^2\right )^2}{2 e^3 (d+e x)^2}+\frac{2 c d \left (c d^2-a e^2\right )}{e^3 (d+e x)}+\frac{c^2 d^2 \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0314704, size = 59, normalized size = 0.83 \[ \frac{\frac{\left (c d^2-a e^2\right ) \left (a e^2+c d (3 d+4 e x)\right )}{(d+e x)^2}+2 c^2 d^2 \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 98, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}e}{2\, \left ( ex+d \right ) ^{2}}}+{\frac{ac{d}^{2}}{e \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2}{d}^{4}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{2}{d}^{2}\ln \left ( ex+d \right ) }{{e}^{3}}}-2\,{\frac{acd}{e \left ( ex+d \right ) }}+2\,{\frac{{c}^{2}{d}^{3}}{{e}^{3} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.07287, size = 122, normalized size = 1.72 \begin{align*} \frac{c^{2} d^{2} \log \left (e x + d\right )}{e^{3}} + \frac{3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65813, size = 224, normalized size = 3.15 \begin{align*} \frac{3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x + 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.966675, size = 90, normalized size = 1.27 \begin{align*} \frac{c^{2} d^{2} \log{\left (d + e x \right )}}{e^{3}} - \frac{a^{2} e^{4} + 2 a c d^{2} e^{2} - 3 c^{2} d^{4} + x \left (4 a c d e^{3} - 4 c^{2} d^{3} e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22589, size = 159, normalized size = 2.24 \begin{align*} -c^{2} d^{2} e^{\left (-3\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{2} \,{\left (\frac{4 \, c^{2} d^{3} e^{9}}{x e + d} - \frac{c^{2} d^{4} e^{9}}{{\left (x e + d\right )}^{2}} - \frac{4 \, a c d e^{11}}{x e + d} + \frac{2 \, a c d^{2} e^{11}}{{\left (x e + d\right )}^{2}} - \frac{a^{2} e^{13}}{{\left (x e + d\right )}^{2}}\right )} e^{\left (-12\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]