Optimal. Leaf size=72 \[ \frac{a^3}{18 b^4 \left (a+b x^2\right )^9}-\frac{3 a^2}{16 b^4 \left (a+b x^2\right )^8}+\frac{3 a}{14 b^4 \left (a+b x^2\right )^7}-\frac{1}{12 b^4 \left (a+b x^2\right )^6} \]
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Rubi [A] time = 0.0530002, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^3}{18 b^4 \left (a+b x^2\right )^9}-\frac{3 a^2}{16 b^4 \left (a+b x^2\right )^8}+\frac{3 a}{14 b^4 \left (a+b x^2\right )^7}-\frac{1}{12 b^4 \left (a+b x^2\right )^6} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{10}}+\frac{3 a^2}{b^3 (a+b x)^9}-\frac{3 a}{b^3 (a+b x)^8}+\frac{1}{b^3 (a+b x)^7}\right ) \, dx,x,x^2\right )\\ &=\frac{a^3}{18 b^4 \left (a+b x^2\right )^9}-\frac{3 a^2}{16 b^4 \left (a+b x^2\right )^8}+\frac{3 a}{14 b^4 \left (a+b x^2\right )^7}-\frac{1}{12 b^4 \left (a+b x^2\right )^6}\\ \end{align*}
Mathematica [A] time = 0.0105129, size = 46, normalized size = 0.64 \[ -\frac{9 a^2 b x^2+a^3+36 a b^2 x^4+84 b^3 x^6}{1008 b^4 \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 65, normalized size = 0.9 \begin{align*}{\frac{{a}^{3}}{18\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{3\,{a}^{2}}{16\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{8}}}+{\frac{3\,a}{14\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{7}}}-{\frac{1}{12\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.81793, size = 182, normalized size = 2.53 \begin{align*} -\frac{84 \, b^{3} x^{6} + 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + a^{3}}{1008 \,{\left (b^{13} x^{18} + 9 \, a b^{12} x^{16} + 36 \, a^{2} b^{11} x^{14} + 84 \, a^{3} b^{10} x^{12} + 126 \, a^{4} b^{9} x^{10} + 126 \, a^{5} b^{8} x^{8} + 84 \, a^{6} b^{7} x^{6} + 36 \, a^{7} b^{6} x^{4} + 9 \, a^{8} b^{5} x^{2} + a^{9} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.19305, size = 297, normalized size = 4.12 \begin{align*} -\frac{84 \, b^{3} x^{6} + 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + a^{3}}{1008 \,{\left (b^{13} x^{18} + 9 \, a b^{12} x^{16} + 36 \, a^{2} b^{11} x^{14} + 84 \, a^{3} b^{10} x^{12} + 126 \, a^{4} b^{9} x^{10} + 126 \, a^{5} b^{8} x^{8} + 84 \, a^{6} b^{7} x^{6} + 36 \, a^{7} b^{6} x^{4} + 9 \, a^{8} b^{5} x^{2} + a^{9} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.12123, size = 143, normalized size = 1.99 \begin{align*} - \frac{a^{3} + 9 a^{2} b x^{2} + 36 a b^{2} x^{4} + 84 b^{3} x^{6}}{1008 a^{9} b^{4} + 9072 a^{8} b^{5} x^{2} + 36288 a^{7} b^{6} x^{4} + 84672 a^{6} b^{7} x^{6} + 127008 a^{5} b^{8} x^{8} + 127008 a^{4} b^{9} x^{10} + 84672 a^{3} b^{10} x^{12} + 36288 a^{2} b^{11} x^{14} + 9072 a b^{12} x^{16} + 1008 b^{13} x^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.53198, size = 59, normalized size = 0.82 \begin{align*} -\frac{84 \, b^{3} x^{6} + 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + a^{3}}{1008 \,{\left (b x^{2} + a\right )}^{9} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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