Optimal. Leaf size=184 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{6 c d \left (a e^2+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac{6 c^3 d}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7} \]
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Rubi [A] time = 0.135293, antiderivative size = 184, 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} \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{6 c d \left (a e^2+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac{6 c^3 d}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7} \]
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)^7} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^7}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^6}+\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)^5}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)^4}+\frac{3 c^2 \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^3}-\frac{6 c^3 d}{e^6 (d+e x)^2}+\frac{c^3}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{\left (c d^2+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac{6 c d \left (c d^2+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{4 e^7 (d+e x)^4}+\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{3 e^7 (d+e x)^3}-\frac{3 c^2 \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^2}+\frac{6 c^3 d}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.0623787, size = 172, normalized size = 0.93 \[ \frac{-3 a^2 c e^4 \left (d^2+6 d e x+15 e^2 x^2\right )-10 a^3 e^6-6 a c^2 e^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+c^3 d \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 278, normalized size = 1.5 \begin{align*} -{\frac{3\,{a}^{2}c}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{9\,a{c}^{2}{d}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{c}^{3}{d}^{4}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{3}}{6\,e \left ( ex+d \right ) ^{6}}}-{\frac{{a}^{2}c{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{a{c}^{2}{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{d}^{6}{c}^{3}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}+4\,{\frac{a{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{20\,{c}^{3}{d}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+{\frac{{c}^{3}\ln \left ( ex+d \right ) }{{e}^{7}}}+{\frac{6\,cd{a}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}+{\frac{12\,a{c}^{2}{d}^{3}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}+{\frac{6\,{c}^{3}{d}^{5}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+6\,{\frac{{c}^{3}d}{{e}^{7} \left ( ex+d \right ) }}-{\frac{3\,a{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{c}^{3}{d}^{2}}{2\,{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 = 1.25224, size = 355, normalized size = 1.93 \begin{align*} \frac{360 \, c^{3} d e^{5} x^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \,{\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac{c^{3} \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 = 1.98179, size = 710, normalized size = 3.86 \begin{align*} \frac{360 \, c^{3} d e^{5} x^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \,{\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x + 60 \,{\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.546, size = 272, normalized size = 1.48 \begin{align*} \frac{c^{3} \log{\left (d + e x \right )}}{e^{7}} + \frac{- 10 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 6 a c^{2} d^{4} e^{2} + 147 c^{3} d^{6} + 360 c^{3} d e^{5} x^{5} + x^{4} \left (- 90 a c^{2} e^{6} + 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 120 a c^{2} d e^{5} + 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 45 a^{2} c e^{6} - 90 a c^{2} d^{2} e^{4} + 1875 c^{3} d^{4} e^{2}\right ) + x \left (- 18 a^{2} c d e^{5} - 36 a c^{2} d^{3} e^{3} + 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29102, size = 265, normalized size = 1.44 \begin{align*} c^{3} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (360 \, c^{3} d x^{5} e^{4} + 90 \,{\left (15 \, c^{3} d^{2} e^{3} - a c^{2} e^{5}\right )} x^{4} + 40 \,{\left (55 \, c^{3} d^{3} e^{2} - 3 \, a c^{2} d e^{4}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e - 6 \, a c^{2} d^{2} e^{3} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}\right )} x +{\left (147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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