Optimal. Leaf size=59 \[ -\frac{a^2}{7 b^3 \left (a+b x^2\right )^{7/2}}+\frac{2 a}{5 b^3 \left (a+b x^2\right )^{5/2}}-\frac{1}{3 b^3 \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.033334, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{a^2}{7 b^3 \left (a+b x^2\right )^{7/2}}+\frac{2 a}{5 b^3 \left (a+b x^2\right )^{5/2}}-\frac{1}{3 b^3 \left (a+b x^2\right )^{3/2}} \]
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 )^{9/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{9/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{9/2}}-\frac{2 a}{b^2 (a+b x)^{7/2}}+\frac{1}{b^2 (a+b x)^{5/2}}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{7 b^3 \left (a+b x^2\right )^{7/2}}+\frac{2 a}{5 b^3 \left (a+b x^2\right )^{5/2}}-\frac{1}{3 b^3 \left (a+b x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0179678, size = 39, normalized size = 0.66 \[ \frac{-8 a^2-28 a b x^2-35 b^2 x^4}{105 b^3 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 36, normalized size = 0.6 \begin{align*} -{\frac{35\,{b}^{2}{x}^{4}+28\,ab{x}^{2}+8\,{a}^{2}}{105\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4914, size = 72, normalized size = 1.22 \begin{align*} -\frac{x^{4}}{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{4 \, a x^{2}}{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{8 \, a^{2}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31278, size = 167, normalized size = 2.83 \begin{align*} -\frac{{\left (35 \, b^{2} x^{4} + 28 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (b^{7} x^{8} + 4 \, a b^{6} x^{6} + 6 \, a^{2} b^{5} x^{4} + 4 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.39027, size = 272, normalized size = 4.61 \begin{align*} \begin{cases} - \frac{8 a^{2}}{105 a^{3} b^{3} \sqrt{a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt{a + b x^{2}} + 315 a b^{5} x^{4} \sqrt{a + b x^{2}} + 105 b^{6} x^{6} \sqrt{a + b x^{2}}} - \frac{28 a b x^{2}}{105 a^{3} b^{3} \sqrt{a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt{a + b x^{2}} + 315 a b^{5} x^{4} \sqrt{a + b x^{2}} + 105 b^{6} x^{6} \sqrt{a + b x^{2}}} - \frac{35 b^{2} x^{4}}{105 a^{3} b^{3} \sqrt{a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt{a + b x^{2}} + 315 a b^{5} x^{4} \sqrt{a + b x^{2}} + 105 b^{6} x^{6} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{9}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.4579, size = 55, normalized size = 0.93 \begin{align*} -\frac{35 \,{\left (b x^{2} + a\right )}^{2} - 42 \,{\left (b x^{2} + a\right )} a + 15 \, a^{2}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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