Optimal. Leaf size=53 \[ -\frac{a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac{a}{8 b^3 \left (a+b x^2\right )^8}-\frac{1}{14 b^3 \left (a+b x^2\right )^7} \]
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Rubi [A] time = 0.0388855, antiderivative size = 53, 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} \[ -\frac{a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac{a}{8 b^3 \left (a+b x^2\right )^8}-\frac{1}{14 b^3 \left (a+b x^2\right )^7} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{10}}-\frac{2 a}{b^2 (a+b x)^9}+\frac{1}{b^2 (a+b x)^8}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac{a}{8 b^3 \left (a+b x^2\right )^8}-\frac{1}{14 b^3 \left (a+b x^2\right )^7}\\ \end{align*}
Mathematica [A] time = 0.0126977, size = 35, normalized size = 0.66 \[ -\frac{a^2+9 a b x^2+36 b^2 x^4}{504 b^3 \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 48, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{18\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{a}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{1}{14\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.34149, size = 167, normalized size = 3.15 \begin{align*} -\frac{36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \,{\left (b^{12} x^{18} + 9 \, a b^{11} x^{16} + 36 \, a^{2} b^{10} x^{14} + 84 \, a^{3} b^{9} x^{12} + 126 \, a^{4} b^{8} x^{10} + 126 \, a^{5} b^{7} x^{8} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{4} + 9 \, a^{8} b^{4} x^{2} + a^{9} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.18631, size = 271, normalized size = 5.11 \begin{align*} -\frac{36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \,{\left (b^{12} x^{18} + 9 \, a b^{11} x^{16} + 36 \, a^{2} b^{10} x^{14} + 84 \, a^{3} b^{9} x^{12} + 126 \, a^{4} b^{8} x^{10} + 126 \, a^{5} b^{7} x^{8} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{4} + 9 \, a^{8} b^{4} x^{2} + a^{9} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.05524, size = 131, normalized size = 2.47 \begin{align*} - \frac{a^{2} + 9 a b x^{2} + 36 b^{2} x^{4}}{504 a^{9} b^{3} + 4536 a^{8} b^{4} x^{2} + 18144 a^{7} b^{5} x^{4} + 42336 a^{6} b^{6} x^{6} + 63504 a^{5} b^{7} x^{8} + 63504 a^{4} b^{8} x^{10} + 42336 a^{3} b^{9} x^{12} + 18144 a^{2} b^{10} x^{14} + 4536 a b^{11} x^{16} + 504 b^{12} x^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.32654, size = 45, normalized size = 0.85 \begin{align*} -\frac{36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \,{\left (b x^{2} + a\right )}^{9} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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