Optimal. Leaf size=165 \[ \frac{c^2 x \left (3 a e^2+10 c d^2\right )}{e^6}-\frac{4 c^2 d \left (3 a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{3 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^3}-\frac{2 c^3 d x^2}{e^5}+\frac{c^3 x^3}{3 e^4} \]
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Rubi [A] time = 0.164905, antiderivative size = 165, 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} \[ \frac{c^2 x \left (3 a e^2+10 c d^2\right )}{e^6}-\frac{4 c^2 d \left (3 a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{3 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^3}-\frac{2 c^3 d x^2}{e^5}+\frac{c^3 x^3}{3 e^4} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac{c^2 \left (10 c d^2+3 a e^2\right )}{e^6}-\frac{4 c^3 d x}{e^5}+\frac{c^3 x^2}{e^4}+\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^4}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^3}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^2}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{c^2 \left (10 c d^2+3 a e^2\right ) x}{e^6}-\frac{2 c^3 d x^2}{e^5}+\frac{c^3 x^3}{3 e^4}-\frac{\left (c d^2+a e^2\right )^3}{3 e^7 (d+e x)^3}+\frac{3 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^2}-\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^7 (d+e x)}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.0657386, size = 197, normalized size = 1.19 \[ \frac{-3 a^2 c e^4 \left (d^2+3 d e x+3 e^2 x^2\right )-a^3 e^6+3 a c^2 e^2 \left (-9 d^2 e^2 x^2-27 d^3 e x-13 d^4+9 d e^3 x^3+3 e^4 x^4\right )-12 c^2 d (d+e x)^3 \left (3 a e^2+5 c d^2\right ) \log (d+e x)+c^3 \left (39 d^4 e^2 x^2+73 d^3 e^3 x^3+15 d^2 e^4 x^4-51 d^5 e x-37 d^6-3 d e^5 x^5+e^6 x^6\right )}{3 e^7 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 258, normalized size = 1.6 \begin{align*}{\frac{{x}^{3}{c}^{3}}{3\,{e}^{4}}}-2\,{\frac{{c}^{3}d{x}^{2}}{{e}^{5}}}+3\,{\frac{a{c}^{2}x}{{e}^{4}}}+10\,{\frac{x{c}^{3}{d}^{2}}{{e}^{6}}}-{\frac{{a}^{3}}{3\,e \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}c{d}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{4}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{6}{c}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}-12\,{\frac{{c}^{2}d\ln \left ( ex+d \right ) a}{{e}^{5}}}-20\,{\frac{{c}^{3}{d}^{3}\ln \left ( ex+d \right ) }{{e}^{7}}}-3\,{\frac{{a}^{2}c}{{e}^{3} \left ( ex+d \right ) }}-18\,{\frac{a{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}-15\,{\frac{{c}^{3}{d}^{4}}{{e}^{7} \left ( ex+d \right ) }}+3\,{\frac{cd{a}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}+6\,{\frac{a{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.102, size = 305, normalized size = 1.85 \begin{align*} -\frac{37 \, c^{3} d^{6} + 39 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 9 \,{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (9 \, c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac{c^{3} e^{2} x^{3} - 6 \, c^{3} d e x^{2} + 3 \,{\left (10 \, c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x}{3 \, e^{6}} - \frac{4 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10702, size = 679, normalized size = 4.12 \begin{align*} \frac{c^{3} e^{6} x^{6} - 3 \, c^{3} d e^{5} x^{5} - 37 \, c^{3} d^{6} - 39 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 3 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} +{\left (73 \, c^{3} d^{3} e^{3} + 27 \, a c^{2} d e^{5}\right )} x^{3} + 3 \,{\left (13 \, c^{3} d^{4} e^{2} - 9 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} - 3 \,{\left (17 \, c^{3} d^{5} e + 27 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 12 \,{\left (5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} +{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 3 \,{\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \,{\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.88631, size = 235, normalized size = 1.42 \begin{align*} - \frac{2 c^{3} d x^{2}}{e^{5}} + \frac{c^{3} x^{3}}{3 e^{4}} - \frac{4 c^{2} d \left (3 a e^{2} + 5 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 39 a c^{2} d^{4} e^{2} + 37 c^{3} d^{6} + x^{2} \left (9 a^{2} c e^{6} + 54 a c^{2} d^{2} e^{4} + 45 c^{3} d^{4} e^{2}\right ) + x \left (9 a^{2} c d e^{5} + 90 a c^{2} d^{3} e^{3} + 81 c^{3} d^{5} e\right )}{3 d^{3} e^{7} + 9 d^{2} e^{8} x + 9 d e^{9} x^{2} + 3 e^{10} x^{3}} + \frac{x \left (3 a c^{2} e^{2} + 10 c^{3} d^{2}\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30372, size = 259, normalized size = 1.57 \begin{align*} -4 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{3} \,{\left (c^{3} x^{3} e^{8} - 6 \, c^{3} d x^{2} e^{7} + 30 \, c^{3} d^{2} x e^{6} + 9 \, a c^{2} x e^{8}\right )} e^{\left (-12\right )} - \frac{{\left (37 \, c^{3} d^{6} + 39 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 9 \,{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (9 \, c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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