Optimal. Leaf size=101 \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)}+\frac{2 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{4 c^2 d \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]
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Rubi [A] time = 0.0746482, antiderivative size = 101, 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} \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)}+\frac{2 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{4 c^2 d \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]
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)^4} \, dx &=\int \left (\frac{c^2}{e^4}+\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^4}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^3}+\frac{2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)^2}-\frac{4 c^2 d}{e^4 (d+e x)}\right ) \, dx\\ &=\frac{c^2 x}{e^4}-\frac{\left (c d^2+a e^2\right )^2}{3 e^5 (d+e x)^3}+\frac{2 c d \left (c d^2+a e^2\right )}{e^5 (d+e x)^2}-\frac{2 c \left (3 c d^2+a e^2\right )}{e^5 (d+e x)}-\frac{4 c^2 d \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0539365, size = 110, normalized size = 1.09 \[ -\frac{a^2 e^4+2 a c e^2 \left (d^2+3 d e x+3 e^2 x^2\right )+c^2 \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )+12 c^2 d (d+e x)^3 \log (d+e x)}{3 e^5 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 140, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}x}{{e}^{4}}}-{\frac{{a}^{2}}{3\,e \left ( ex+d \right ) ^{3}}}-{\frac{2\,ac{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-4\,{\frac{{c}^{2}d\ln \left ( ex+d \right ) }{{e}^{5}}}-2\,{\frac{ac}{{e}^{3} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}+2\,{\frac{acd}{{e}^{3} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{2}{d}^{3}}{{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 = 1.29147, size = 176, normalized size = 1.74 \begin{align*} -\frac{13 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4} + 6 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \,{\left (5 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac{c^{2} x}{e^{4}} - \frac{4 \, c^{2} d \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 = 1.94676, size = 373, normalized size = 3.69 \begin{align*} \frac{3 \, c^{2} e^{4} x^{4} + 9 \, c^{2} d e^{3} x^{3} - 13 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4} - 3 \,{\left (3 \, c^{2} d^{2} e^{2} + 2 \, a c e^{4}\right )} x^{2} - 3 \,{\left (9 \, c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x - 12 \,{\left (c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{2} e^{2} x^{2} + 3 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.43739, size = 134, normalized size = 1.33 \begin{align*} - \frac{4 c^{2} d \log{\left (d + e x \right )}}{e^{5}} + \frac{c^{2} x}{e^{4}} - \frac{a^{2} e^{4} + 2 a c d^{2} e^{2} + 13 c^{2} d^{4} + x^{2} \left (6 a c e^{4} + 18 c^{2} d^{2} e^{2}\right ) + x \left (6 a c d e^{3} + 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2176, size = 136, normalized size = 1.35 \begin{align*} -4 \, c^{2} d e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + c^{2} x e^{\left (-4\right )} - \frac{{\left (13 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 6 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + a^{2} e^{4} + 6 \,{\left (5 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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