Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac{\left (a+b x^2\right )^{9/2}}{9 b^3}-\frac{2 a \left (a+b x^2\right )^{7/2}}{7 b^3} \]
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Rubi [A] time = 0.0351328, 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} \[ \frac{a^2 \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac{\left (a+b x^2\right )^{9/2}}{9 b^3}-\frac{2 a \left (a+b x^2\right )^{7/2}}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a+b x^2\right )^{3/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x)^{3/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{3/2}}{b^2}-\frac{2 a (a+b x)^{5/2}}{b^2}+\frac{(a+b x)^{7/2}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 \left (a+b x^2\right )^{5/2}}{5 b^3}-\frac{2 a \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac{\left (a+b x^2\right )^{9/2}}{9 b^3}\\ \end{align*}
Mathematica [A] time = 0.017978, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^2\right )^{5/2} \left (8 a^2-20 a b x^2+35 b^2 x^4\right )}{315 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 36, normalized size = 0.6 \begin{align*}{\frac{35\,{b}^{2}{x}^{4}-20\,ab{x}^{2}+8\,{a}^{2}}{315\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57227, size = 126, normalized size = 2.14 \begin{align*} \frac{{\left (35 \, b^{4} x^{8} + 50 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + 8 \, a^{4}\right )} \sqrt{b x^{2} + a}}{315 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.07804, size = 109, normalized size = 1.85 \begin{align*} \begin{cases} \frac{8 a^{4} \sqrt{a + b x^{2}}}{315 b^{3}} - \frac{4 a^{3} x^{2} \sqrt{a + b x^{2}}}{315 b^{2}} + \frac{a^{2} x^{4} \sqrt{a + b x^{2}}}{105 b} + \frac{10 a x^{6} \sqrt{a + b x^{2}}}{63} + \frac{b x^{8} \sqrt{a + b x^{2}}}{9} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.72273, size = 143, normalized size = 2.42 \begin{align*} \frac{\frac{3 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} a}{b^{2}} + \frac{35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}}{b^{2}}}{315 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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