Optimal. Leaf size=94 \[ \frac{c x \left (2 a e^2+3 c d^2\right )}{e^4}-\frac{\left (a e^2+c d^2\right )^2}{e^5 (d+e x)}-\frac{4 c d \left (a e^2+c d^2\right ) \log (d+e x)}{e^5}-\frac{c^2 d x^2}{e^3}+\frac{c^2 x^3}{3 e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0756636, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ \frac{c x \left (2 a e^2+3 c d^2\right )}{e^4}-\frac{\left (a e^2+c d^2\right )^2}{e^5 (d+e x)}-\frac{4 c d \left (a e^2+c d^2\right ) \log (d+e x)}{e^5}-\frac{c^2 d x^2}{e^3}+\frac{c^2 x^3}{3 e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^2}{(d+e x)^2} \, dx &=\int \left (\frac{c \left (3 c d^2+2 a e^2\right )}{e^4}-\frac{2 c^2 d x}{e^3}+\frac{c^2 x^2}{e^2}+\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^2}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)}\right ) \, dx\\ &=\frac{c \left (3 c d^2+2 a e^2\right ) x}{e^4}-\frac{c^2 d x^2}{e^3}+\frac{c^2 x^3}{3 e^2}-\frac{\left (c d^2+a e^2\right )^2}{e^5 (d+e x)}-\frac{4 c d \left (c d^2+a e^2\right ) \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0556821, size = 91, normalized size = 0.97 \[ \frac{3 c e x \left (2 a e^2+3 c d^2\right )-\frac{3 \left (a e^2+c d^2\right )^2}{d+e x}-12 c d \left (a e^2+c d^2\right ) \log (d+e x)-3 c^2 d e^2 x^2+c^2 e^3 x^3}{3 e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 126, normalized size = 1.3 \begin{align*}{\frac{{c}^{2}{x}^{3}}{3\,{e}^{2}}}-{\frac{{c}^{2}d{x}^{2}}{{e}^{3}}}+2\,{\frac{acx}{{e}^{2}}}+3\,{\frac{{c}^{2}{d}^{2}x}{{e}^{4}}}-4\,{\frac{cd\ln \left ( ex+d \right ) a}{{e}^{3}}}-4\,{\frac{{c}^{2}{d}^{3}\ln \left ( ex+d \right ) }{{e}^{5}}}-{\frac{{a}^{2}}{e \left ( ex+d \right ) }}-2\,{\frac{ac{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-{\frac{{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.17403, size = 151, normalized size = 1.61 \begin{align*} -\frac{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{e^{6} x + d e^{5}} + \frac{c^{2} e^{2} x^{3} - 3 \, c^{2} d e x^{2} + 3 \,{\left (3 \, c^{2} d^{2} + 2 \, a c e^{2}\right )} x}{3 \, e^{4}} - \frac{4 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.81917, size = 309, normalized size = 3.29 \begin{align*} \frac{c^{2} e^{4} x^{4} - 2 \, c^{2} d e^{3} x^{3} - 3 \, c^{2} d^{4} - 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 6 \,{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 3 \,{\left (3 \, c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x - 12 \,{\left (c^{2} d^{4} + a c d^{2} e^{2} +{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.720812, size = 105, normalized size = 1.12 \begin{align*} - \frac{c^{2} d x^{2}}{e^{3}} + \frac{c^{2} x^{3}}{3 e^{2}} - \frac{4 c d \left (a e^{2} + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}}{d e^{5} + e^{6} x} + \frac{x \left (2 a c e^{2} + 3 c^{2} d^{2}\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27953, size = 201, normalized size = 2.14 \begin{align*} \frac{1}{3} \,{\left (c^{2} - \frac{6 \, c^{2} d}{x e + d} + \frac{6 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-5\right )} + 4 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{2} d^{4} e^{3}}{x e + d} + \frac{2 \, a c d^{2} e^{5}}{x e + d} + \frac{a^{2} e^{7}}{x e + d}\right )} e^{\left (-8\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]