Optimal. Leaf size=172 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{2 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^2}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{3 c d \left (a e^2+c d^2\right )^2}{2 e^7 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{6 c^3 d \log (d+e x)}{e^7}+\frac{c^3 x}{e^6} \]
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Rubi [A] time = 0.149449, antiderivative size = 172, 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} \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{2 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^2}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{3 c d \left (a e^2+c d^2\right )^2}{2 e^7 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{6 c^3 d \log (d+e x)}{e^7}+\frac{c^3 x}{e^6} \]
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)^6} \, dx &=\int \left (\frac{c^3}{e^6}+\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^6}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^5}+\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)^4}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)^3}+\frac{3 c^2 \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^2}-\frac{6 c^3 d}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{c^3 x}{e^6}-\frac{\left (c d^2+a e^2\right )^3}{5 e^7 (d+e x)^5}+\frac{3 c d \left (c d^2+a e^2\right )^2}{2 e^7 (d+e x)^4}-\frac{c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^7 (d+e x)^3}+\frac{2 c^2 d \left (5 c d^2+3 a e^2\right )}{e^7 (d+e x)^2}-\frac{3 c^2 \left (5 c d^2+a e^2\right )}{e^7 (d+e x)}-\frac{6 c^3 d \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.0915828, size = 182, normalized size = 1.06 \[ -\frac{a^2 c e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+2 a^3 e^6+6 a c^2 e^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )+c^3 \left (600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4+375 d^5 e x+87 d^6-50 d e^5 x^5-10 e^6 x^6\right )+60 c^3 d (d+e x)^5 \log (d+e x)}{10 e^7 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 272, normalized size = 1.6 \begin{align*}{\frac{{c}^{3}x}{{e}^{6}}}+{\frac{3\,cd{a}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}+3\,{\frac{{c}^{2}{d}^{3}a}{{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{3\,{c}^{3}{d}^{5}}{2\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{2}c}{{e}^{3} \left ( ex+d \right ) ^{3}}}-6\,{\frac{a{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}-5\,{\frac{{c}^{3}{d}^{4}}{{e}^{7} \left ( ex+d \right ) ^{3}}}-6\,{\frac{{c}^{3}d\ln \left ( ex+d \right ) }{{e}^{7}}}-{\frac{{a}^{3}}{5\,e \left ( ex+d \right ) ^{5}}}-{\frac{3\,{a}^{2}c{d}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{3\,{d}^{4}a{c}^{2}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{d}^{6}{c}^{3}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-3\,{\frac{a{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-15\,{\frac{{c}^{3}{d}^{2}}{{e}^{7} \left ( ex+d \right ) }}+6\,{\frac{a{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{c}^{3}{d}^{3}}{{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 = 1.17573, size = 332, normalized size = 1.93 \begin{align*} -\frac{87 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} + 30 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 20 \,{\left (25 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 10 \,{\left (65 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 5 \,{\left (77 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac{c^{3} x}{e^{6}} - \frac{6 \, c^{3} d \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 = 2.14604, size = 672, normalized size = 3.91 \begin{align*} \frac{10 \, c^{3} e^{6} x^{6} + 50 \, c^{3} d e^{5} x^{5} - 87 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} - 10 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} - 20 \,{\left (20 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} - 10 \,{\left (60 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} - 5 \,{\left (75 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x - 60 \,{\left (c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{2} e^{4} x^{4} + 10 \, c^{3} d^{3} e^{3} x^{3} + 10 \, c^{3} d^{4} e^{2} x^{2} + 5 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.58859, size = 257, normalized size = 1.49 \begin{align*} - \frac{6 c^{3} d \log{\left (d + e x \right )}}{e^{7}} + \frac{c^{3} x}{e^{6}} - \frac{2 a^{3} e^{6} + a^{2} c d^{2} e^{4} + 6 a c^{2} d^{4} e^{2} + 87 c^{3} d^{6} + x^{4} \left (30 a c^{2} e^{6} + 150 c^{3} d^{2} e^{4}\right ) + x^{3} \left (60 a c^{2} d e^{5} + 500 c^{3} d^{3} e^{3}\right ) + x^{2} \left (10 a^{2} c e^{6} + 60 a c^{2} d^{2} e^{4} + 650 c^{3} d^{4} e^{2}\right ) + x \left (5 a^{2} c d e^{5} + 30 a c^{2} d^{3} e^{3} + 385 c^{3} d^{5} e\right )}{10 d^{5} e^{7} + 50 d^{4} e^{8} x + 100 d^{3} e^{9} x^{2} + 100 d^{2} e^{10} x^{3} + 50 d e^{11} x^{4} + 10 e^{12} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30574, size = 254, normalized size = 1.48 \begin{align*} -6 \, c^{3} d e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + c^{3} x e^{\left (-6\right )} - \frac{{\left (87 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 30 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 20 \,{\left (25 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 2 \, a^{3} e^{6} + 10 \,{\left (65 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 5 \,{\left (77 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{10 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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