Optimal. Leaf size=173 \[ \frac{c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{2 e^5}-\frac{c d x \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{e^6}+\frac{c^2 x^4 \left (3 a e^2+c d^2\right )}{4 e^3}-\frac{c^2 d x^3 \left (3 a e^2+c d^2\right )}{3 e^4}+\frac{\left (a e^2+c d^2\right )^3 \log (d+e x)}{e^7}-\frac{c^3 d x^5}{5 e^2}+\frac{c^3 x^6}{6 e} \]
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Rubi [A] time = 0.144567, antiderivative size = 173, 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} \[ \frac{c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{2 e^5}-\frac{c d x \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{e^6}+\frac{c^2 x^4 \left (3 a e^2+c d^2\right )}{4 e^3}-\frac{c^2 d x^3 \left (3 a e^2+c d^2\right )}{3 e^4}+\frac{\left (a e^2+c d^2\right )^3 \log (d+e x)}{e^7}-\frac{c^3 d x^5}{5 e^2}+\frac{c^3 x^6}{6 e} \]
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} \, dx &=\int \left (-\frac{c d \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right )}{e^6}+\frac{c \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x}{e^5}-\frac{c^2 d \left (c d^2+3 a e^2\right ) x^2}{e^4}+\frac{c^2 \left (c d^2+3 a e^2\right ) x^3}{e^3}-\frac{c^3 d x^4}{e^2}+\frac{c^3 x^5}{e}+\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{c d \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x}{e^6}+\frac{c \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x^2}{2 e^5}-\frac{c^2 d \left (c d^2+3 a e^2\right ) x^3}{3 e^4}+\frac{c^2 \left (c d^2+3 a e^2\right ) x^4}{4 e^3}-\frac{c^3 d x^5}{5 e^2}+\frac{c^3 x^6}{6 e}+\frac{\left (c d^2+a e^2\right )^3 \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.0539716, size = 142, normalized size = 0.82 \[ \frac{c e x \left (90 a^2 e^4 (e x-2 d)+15 a c e^2 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+c^2 \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 \left (a e^2+c d^2\right )^3 \log (d+e x)}{60 e^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 220, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}{x}^{6}}{6\,e}}-{\frac{{c}^{3}d{x}^{5}}{5\,{e}^{2}}}+{\frac{3\,{c}^{2}{x}^{4}a}{4\,e}}+{\frac{{c}^{3}{x}^{4}{d}^{2}}{4\,{e}^{3}}}-{\frac{{c}^{2}{x}^{3}ad}{{e}^{2}}}-{\frac{{x}^{3}{c}^{3}{d}^{3}}{3\,{e}^{4}}}+{\frac{3\,c{x}^{2}{a}^{2}}{2\,e}}+{\frac{3\,{c}^{2}{x}^{2}a{d}^{2}}{2\,{e}^{3}}}+{\frac{{c}^{3}{x}^{2}{d}^{4}}{2\,{e}^{5}}}-3\,{\frac{cd{a}^{2}x}{{e}^{2}}}-3\,{\frac{a{c}^{2}{d}^{3}x}{{e}^{4}}}-{\frac{{c}^{3}{d}^{5}x}{{e}^{6}}}+{\frac{\ln \left ( ex+d \right ){a}^{3}}{e}}+3\,{\frac{\ln \left ( ex+d \right ){a}^{2}c{d}^{2}}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ){d}^{4}a{c}^{2}}{{e}^{5}}}+{\frac{\ln \left ( ex+d \right ){d}^{6}{c}^{3}}{{e}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14583, size = 267, normalized size = 1.54 \begin{align*} \frac{10 \, c^{3} e^{5} x^{6} - 12 \, c^{3} d e^{4} x^{5} + 15 \,{\left (c^{3} d^{2} e^{3} + 3 \, a c^{2} e^{5}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{2} + 3 \, a c^{2} d e^{4}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3} + 3 \, a^{2} c e^{5}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} + 3 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x}{60 \, e^{6}} + \frac{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\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 [A] time = 2.06353, size = 410, normalized size = 2.37 \begin{align*} \frac{10 \, c^{3} e^{6} x^{6} - 12 \, c^{3} d e^{5} x^{5} + 15 \,{\left (c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 3 \, a^{2} c e^{6}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x + 60 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.719874, size = 173, normalized size = 1. \begin{align*} - \frac{c^{3} d x^{5}}{5 e^{2}} + \frac{c^{3} x^{6}}{6 e} + \frac{x^{4} \left (3 a c^{2} e^{2} + c^{3} d^{2}\right )}{4 e^{3}} - \frac{x^{3} \left (3 a c^{2} d e^{2} + c^{3} d^{3}\right )}{3 e^{4}} + \frac{x^{2} \left (3 a^{2} c e^{4} + 3 a c^{2} d^{2} e^{2} + c^{3} d^{4}\right )}{2 e^{5}} - \frac{x \left (3 a^{2} c d e^{4} + 3 a c^{2} d^{3} e^{2} + c^{3} d^{5}\right )}{e^{6}} + \frac{\left (a e^{2} + c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32853, size = 259, normalized size = 1.5 \begin{align*}{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, c^{3} x^{6} e^{5} - 12 \, c^{3} d x^{5} e^{4} + 15 \, c^{3} d^{2} x^{4} e^{3} - 20 \, c^{3} d^{3} x^{3} e^{2} + 30 \, c^{3} d^{4} x^{2} e - 60 \, c^{3} d^{5} x + 45 \, a c^{2} x^{4} e^{5} - 60 \, a c^{2} d x^{3} e^{4} + 90 \, a c^{2} d^{2} x^{2} e^{3} - 180 \, a c^{2} d^{3} x e^{2} + 90 \, a^{2} c x^{2} e^{5} - 180 \, a^{2} c d x e^{4}\right )} e^{\left (-6\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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