Optimal. Leaf size=34 \[ \frac{a}{18 b^2 \left (a+b x^2\right )^9}-\frac{1}{16 b^2 \left (a+b x^2\right )^8} \]
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Rubi [A] time = 0.0251604, antiderivative size = 34, 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} \[ \frac{a}{18 b^2 \left (a+b x^2\right )^9}-\frac{1}{16 b^2 \left (a+b x^2\right )^8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{10}}+\frac{1}{b (a+b x)^9}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{18 b^2 \left (a+b x^2\right )^9}-\frac{1}{16 b^2 \left (a+b x^2\right )^8}\\ \end{align*}
Mathematica [A] time = 0.0079257, size = 24, normalized size = 0.71 \[ -\frac{a+9 b x^2}{144 b^2 \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 31, normalized size = 0.9 \begin{align*}{\frac{a}{18\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{1}{16\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.60832, size = 153, normalized size = 4.5 \begin{align*} -\frac{9 \, b x^{2} + a}{144 \,{\left (b^{11} x^{18} + 9 \, a b^{10} x^{16} + 36 \, a^{2} b^{9} x^{14} + 84 \, a^{3} b^{8} x^{12} + 126 \, a^{4} b^{7} x^{10} + 126 \, a^{5} b^{6} x^{8} + 84 \, a^{6} b^{5} x^{6} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{2} + a^{9} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.19908, size = 247, normalized size = 7.26 \begin{align*} -\frac{9 \, b x^{2} + a}{144 \,{\left (b^{11} x^{18} + 9 \, a b^{10} x^{16} + 36 \, a^{2} b^{9} x^{14} + 84 \, a^{3} b^{8} x^{12} + 126 \, a^{4} b^{7} x^{10} + 126 \, a^{5} b^{6} x^{8} + 84 \, a^{6} b^{5} x^{6} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{2} + a^{9} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.96263, size = 119, normalized size = 3.5 \begin{align*} - \frac{a + 9 b x^{2}}{144 a^{9} b^{2} + 1296 a^{8} b^{3} x^{2} + 5184 a^{7} b^{4} x^{4} + 12096 a^{6} b^{5} x^{6} + 18144 a^{5} b^{6} x^{8} + 18144 a^{4} b^{7} x^{10} + 12096 a^{3} b^{8} x^{12} + 5184 a^{2} b^{9} x^{14} + 1296 a b^{10} x^{16} + 144 b^{11} x^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.69398, size = 30, normalized size = 0.88 \begin{align*} -\frac{9 \, b x^{2} + a}{144 \,{\left (b x^{2} + a\right )}^{9} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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