Optimal. Leaf size=100 \[ \frac{4 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)}-\frac{\left (a e^2+c d^2\right )^2}{2 e^5 (d+e x)^2}+\frac{2 c \left (a e^2+3 c d^2\right ) \log (d+e x)}{e^5}-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^2}{2 e^3} \]
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Rubi [A] time = 0.0810327, antiderivative size = 100, 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} \[ \frac{4 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)}-\frac{\left (a e^2+c d^2\right )^2}{2 e^5 (d+e x)^2}+\frac{2 c \left (a e^2+3 c d^2\right ) \log (d+e x)}{e^5}-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^2}{2 e^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^2}{(d+e x)^3} \, dx &=\int \left (-\frac{3 c^2 d}{e^4}+\frac{c^2 x}{e^3}+\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^3}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^2}+\frac{2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^2}{2 e^3}-\frac{\left (c d^2+a e^2\right )^2}{2 e^5 (d+e x)^2}+\frac{4 c d \left (c d^2+a e^2\right )}{e^5 (d+e x)}+\frac{2 c \left (3 c d^2+a e^2\right ) \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0374773, size = 111, normalized size = 1.11 \[ \frac{-a^2 e^4+4 c (d+e x)^2 \left (a e^2+3 c d^2\right ) \log (d+e x)+2 a c d e^2 (3 d+4 e x)+c^2 \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )}{2 e^5 (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 136, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}{x}^{2}}{2\,{e}^{3}}}-3\,{\frac{x{c}^{2}d}{{e}^{4}}}+2\,{\frac{c\ln \left ( ex+d \right ) a}{{e}^{3}}}+6\,{\frac{{c}^{2}\ln \left ( ex+d \right ){d}^{2}}{{e}^{5}}}+4\,{\frac{acd}{{e}^{3} \left ( ex+d \right ) }}+4\,{\frac{{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{a}^{2}}{2\,e \left ( ex+d \right ) ^{2}}}-{\frac{ac{d}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2}{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.32838, size = 162, normalized size = 1.62 \begin{align*} \frac{7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac{c^{2} e x^{2} - 6 \, c^{2} d x}{2 \, e^{4}} + \frac{2 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93023, size = 360, normalized size = 3.6 \begin{align*} \frac{c^{2} e^{4} x^{4} - 4 \, c^{2} d e^{3} x^{3} - 11 \, c^{2} d^{2} e^{2} x^{2} + 7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 2 \,{\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x + 4 \,{\left (3 \, c^{2} d^{4} + a c d^{2} e^{2} +{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 2 \,{\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.48875, size = 122, normalized size = 1.22 \begin{align*} - \frac{3 c^{2} d x}{e^{4}} + \frac{c^{2} x^{2}}{2 e^{3}} + \frac{2 c \left (a e^{2} + 3 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{5}} + \frac{- a^{2} e^{4} + 6 a c d^{2} e^{2} + 7 c^{2} d^{4} + x \left (8 a c d e^{3} + 8 c^{2} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30426, size = 143, normalized size = 1.43 \begin{align*} 2 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (c^{2} x^{2} e^{3} - 6 \, c^{2} d x e^{2}\right )} e^{\left (-6\right )} + \frac{{\left (7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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