Optimal. Leaf size=178 \[ -\frac{c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^4}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac{c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^3}{7 e^7 (d+e x)^7}-\frac{c^3}{e^7 (d+e x)}+\frac{3 c^3 d}{e^7 (d+e x)^2} \]
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Rubi [A] time = 0.121561, antiderivative size = 178, 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} \[ -\frac{c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^4}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac{c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^3}{7 e^7 (d+e x)^7}-\frac{c^3}{e^7 (d+e x)}+\frac{3 c^3 d}{e^7 (d+e x)^2} \]
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)^8} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^8}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^7}+\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)^6}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)^5}+\frac{3 c^2 \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^4}-\frac{6 c^3 d}{e^6 (d+e x)^3}+\frac{c^3}{e^6 (d+e x)^2}\right ) \, dx\\ &=-\frac{\left (c d^2+a e^2\right )^3}{7 e^7 (d+e x)^7}+\frac{c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^6}-\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{5 e^7 (d+e x)^5}+\frac{c^2 d \left (5 c d^2+3 a e^2\right )}{e^7 (d+e x)^4}-\frac{c^2 \left (5 c d^2+a e^2\right )}{e^7 (d+e x)^3}+\frac{3 c^3 d}{e^7 (d+e x)^2}-\frac{c^3}{e^7 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.052798, size = 161, normalized size = 0.9 \[ -\frac{a^2 c e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+5 a^3 e^6+a c^2 e^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )}{35 e^7 (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 216, normalized size = 1.2 \begin{align*}{\frac{{c}^{2}d \left ( 3\,a{e}^{2}+5\,c{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{4}}}+{\frac{cd \left ({a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{2} \left ( a{e}^{2}+5\,c{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{3}{e}^{6}+3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{d}^{4}{e}^{2}a{c}^{2}+{d}^{6}{c}^{3}}{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{3\,c \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}+5\,{c}^{2}{d}^{4} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}+3\,{\frac{{c}^{3}d}{{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 = 1.21563, size = 355, normalized size = 1.99 \begin{align*} -\frac{35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 35 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 35 \,{\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 21 \,{\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 7 \,{\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{35 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0345, size = 540, normalized size = 3.03 \begin{align*} -\frac{35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 35 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 35 \,{\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 21 \,{\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 7 \,{\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{35 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.2541, size = 280, normalized size = 1.57 \begin{align*} - \frac{5 a^{3} e^{6} + a^{2} c d^{2} e^{4} + a c^{2} d^{4} e^{2} + 5 c^{3} d^{6} + 105 c^{3} d e^{5} x^{5} + 35 c^{3} e^{6} x^{6} + x^{4} \left (35 a c^{2} e^{6} + 175 c^{3} d^{2} e^{4}\right ) + x^{3} \left (35 a c^{2} d e^{5} + 175 c^{3} d^{3} e^{3}\right ) + x^{2} \left (21 a^{2} c e^{6} + 21 a c^{2} d^{2} e^{4} + 105 c^{3} d^{4} e^{2}\right ) + x \left (7 a^{2} c d e^{5} + 7 a c^{2} d^{3} e^{3} + 35 c^{3} d^{5} e\right )}{35 d^{7} e^{7} + 245 d^{6} e^{8} x + 735 d^{5} e^{9} x^{2} + 1225 d^{4} e^{10} x^{3} + 1225 d^{3} e^{11} x^{4} + 735 d^{2} e^{12} x^{5} + 245 d e^{13} x^{6} + 35 e^{14} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22163, size = 255, normalized size = 1.43 \begin{align*} -\frac{{\left (35 \, c^{3} x^{6} e^{6} + 105 \, c^{3} d x^{5} e^{5} + 175 \, c^{3} d^{2} x^{4} e^{4} + 175 \, c^{3} d^{3} x^{3} e^{3} + 105 \, c^{3} d^{4} x^{2} e^{2} + 35 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 35 \, a c^{2} x^{4} e^{6} + 35 \, a c^{2} d x^{3} e^{5} + 21 \, a c^{2} d^{2} x^{2} e^{4} + 7 \, a c^{2} d^{3} x e^{3} + a c^{2} d^{4} e^{2} + 21 \, a^{2} c x^{2} e^{6} + 7 \, a^{2} c d x e^{5} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{35 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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