Optimal. Leaf size=109 \[ -\frac{c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)^2}+\frac{4 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac{4 c^2 d}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5} \]
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Rubi [A] time = 0.0747606, antiderivative size = 109, 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} \[ -\frac{c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)^2}+\frac{4 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac{4 c^2 d}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5} \]
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)^5} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^5}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^4}+\frac{2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)^3}-\frac{4 c^2 d}{e^4 (d+e x)^2}+\frac{c^2}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac{\left (c d^2+a e^2\right )^2}{4 e^5 (d+e x)^4}+\frac{4 c d \left (c d^2+a e^2\right )}{3 e^5 (d+e x)^3}-\frac{c \left (3 c d^2+a e^2\right )}{e^5 (d+e x)^2}+\frac{4 c^2 d}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0357054, size = 100, normalized size = 0.92 \[ \frac{-3 a^2 e^4-2 a c e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^2 d \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 146, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2}}{4\,e \left ( ex+d \right ) ^{4}}}-{\frac{ac{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{4\,acd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{c}^{2}{d}^{3}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{{c}^{2}\ln \left ( ex+d \right ) }{{e}^{5}}}+4\,{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }}-{\frac{ac}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{c}^{2}{d}^{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 = 1.17105, size = 197, normalized size = 1.81 \begin{align*} \frac{48 \, c^{2} d e^{3} x^{3} + 25 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 12 \,{\left (9 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 8 \,{\left (11 \, c^{2} d^{3} e - a c d e^{3}\right )} x}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac{c^{2} \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 = 1.84814, size = 397, normalized size = 3.64 \begin{align*} \frac{48 \, c^{2} d e^{3} x^{3} + 25 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 12 \,{\left (9 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 8 \,{\left (11 \, c^{2} d^{3} e - a c d e^{3}\right )} x + 12 \,{\left (c^{2} e^{4} x^{4} + 4 \, c^{2} d e^{3} x^{3} + 6 \, c^{2} d^{2} e^{2} x^{2} + 4 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.84963, size = 150, normalized size = 1.38 \begin{align*} \frac{c^{2} \log{\left (d + e x \right )}}{e^{5}} + \frac{- 3 a^{2} e^{4} - 2 a c d^{2} e^{2} + 25 c^{2} d^{4} + 48 c^{2} d e^{3} x^{3} + x^{2} \left (- 12 a c e^{4} + 108 c^{2} d^{2} e^{2}\right ) + x \left (- 8 a c d e^{3} + 88 c^{2} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21402, size = 220, normalized size = 2.02 \begin{align*} -c^{2} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{12} \,{\left (\frac{48 \, c^{2} d e^{15}}{x e + d} - \frac{36 \, c^{2} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac{16 \, c^{2} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac{3 \, c^{2} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac{12 \, a c e^{17}}{{\left (x e + d\right )}^{2}} + \frac{16 \, a c d e^{17}}{{\left (x e + d\right )}^{3}} - \frac{6 \, a c d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{3 \, a^{2} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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