Optimal. Leaf size=188 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^6}+\frac{6 c d \left (a e^2+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac{\left (a e^2+c d^2\right )^3}{8 e^7 (d+e x)^8}-\frac{c^3}{2 e^7 (d+e x)^2}+\frac{2 c^3 d}{e^7 (d+e x)^3} \]
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Rubi [A] time = 0.12541, antiderivative size = 188, 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} \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^6}+\frac{6 c d \left (a e^2+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac{\left (a e^2+c d^2\right )^3}{8 e^7 (d+e x)^8}-\frac{c^3}{2 e^7 (d+e x)^2}+\frac{2 c^3 d}{e^7 (d+e x)^3} \]
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)^9} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^9}-\frac{6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^8}+\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)^7}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^5}-\frac{6 c^3 d}{e^6 (d+e x)^4}+\frac{c^3}{e^6 (d+e x)^3}\right ) \, dx\\ &=-\frac{\left (c d^2+a e^2\right )^3}{8 e^7 (d+e x)^8}+\frac{6 c d \left (c d^2+a e^2\right )^2}{7 e^7 (d+e x)^7}-\frac{c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^6}+\frac{4 c^2 d \left (5 c d^2+3 a e^2\right )}{5 e^7 (d+e x)^5}-\frac{3 c^2 \left (5 c d^2+a e^2\right )}{4 e^7 (d+e x)^4}+\frac{2 c^3 d}{e^7 (d+e x)^3}-\frac{c^3}{2 e^7 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0616959, size = 163, normalized size = 0.87 \[ -\frac{5 a^2 c e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+35 a^3 e^6+3 a c^2 e^2 \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+5 c^3 \left (28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+8 d^5 e x+d^6+56 d e^5 x^5+28 e^6 x^6\right )}{280 e^7 (d+e x)^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 218, normalized size = 1.2 \begin{align*} -{\frac{3\,{c}^{2} \left ( a{e}^{2}+5\,c{d}^{2} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{c \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}+5\,{c}^{2}{d}^{4} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\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}}{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}+2\,{\frac{{c}^{3}d}{{e}^{7} \left ( ex+d \right ) ^{3}}}+{\frac{6\,cd \left ({a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}+{\frac{4\,{c}^{2}d \left ( 3\,a{e}^{2}+5\,c{d}^{2} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{3}}{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 = 1.2149, size = 381, normalized size = 2.03 \begin{align*} -\frac{140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} + 70 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} + 56 \,{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 28 \,{\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 5 \, a^{2} c e^{6}\right )} x^{2} + 8 \,{\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{280 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06227, size = 590, normalized size = 3.14 \begin{align*} -\frac{140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} + 70 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} + 56 \,{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 28 \,{\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 5 \, a^{2} c e^{6}\right )} x^{2} + 8 \,{\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{280 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 26.2735, size = 296, normalized size = 1.57 \begin{align*} - \frac{35 a^{3} e^{6} + 5 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + 5 c^{3} d^{6} + 280 c^{3} d e^{5} x^{5} + 140 c^{3} e^{6} x^{6} + x^{4} \left (210 a c^{2} e^{6} + 350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (168 a c^{2} d e^{5} + 280 c^{3} d^{3} e^{3}\right ) + x^{2} \left (140 a^{2} c e^{6} + 84 a c^{2} d^{2} e^{4} + 140 c^{3} d^{4} e^{2}\right ) + x \left (40 a^{2} c d e^{5} + 24 a c^{2} d^{3} e^{3} + 40 c^{3} d^{5} e\right )}{280 d^{8} e^{7} + 2240 d^{7} e^{8} x + 7840 d^{6} e^{9} x^{2} + 15680 d^{5} e^{10} x^{3} + 19600 d^{4} e^{11} x^{4} + 15680 d^{3} e^{12} x^{5} + 7840 d^{2} e^{13} x^{6} + 2240 d e^{14} x^{7} + 280 e^{15} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20293, size = 258, normalized size = 1.37 \begin{align*} -\frac{{\left (140 \, c^{3} x^{6} e^{6} + 280 \, c^{3} d x^{5} e^{5} + 350 \, c^{3} d^{2} x^{4} e^{4} + 280 \, c^{3} d^{3} x^{3} e^{3} + 140 \, c^{3} d^{4} x^{2} e^{2} + 40 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 210 \, a c^{2} x^{4} e^{6} + 168 \, a c^{2} d x^{3} e^{5} + 84 \, a c^{2} d^{2} x^{2} e^{4} + 24 \, a c^{2} d^{3} x e^{3} + 3 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c x^{2} e^{6} + 40 \, a^{2} c d x e^{5} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{280 \,{\left (x e + d\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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