Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac{\left (a+b x^2\right )^{11/2}}{11 b^3}-\frac{2 a \left (a+b x^2\right )^{9/2}}{9 b^3} \]
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Rubi [A] time = 0.0338526, 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} \[ \frac{a^2 \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac{\left (a+b x^2\right )^{11/2}}{11 b^3}-\frac{2 a \left (a+b x^2\right )^{9/2}}{9 b^3} \]
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 )^{5/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x)^{5/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{5/2}}{b^2}-\frac{2 a (a+b x)^{7/2}}{b^2}+\frac{(a+b x)^{9/2}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 \left (a+b x^2\right )^{7/2}}{7 b^3}-\frac{2 a \left (a+b x^2\right )^{9/2}}{9 b^3}+\frac{\left (a+b x^2\right )^{11/2}}{11 b^3}\\ \end{align*}
Mathematica [A] time = 0.0198446, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^2\right )^{7/2} \left (8 a^2-28 a b x^2+63 b^2 x^4\right )}{693 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.6 \begin{align*}{\frac{63\,{b}^{2}{x}^{4}-28\,ab{x}^{2}+8\,{a}^{2}}{693\,{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 [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 = 1.5803, size = 153, normalized size = 2.59 \begin{align*} \frac{{\left (63 \, b^{5} x^{10} + 161 \, a b^{4} x^{8} + 113 \, a^{2} b^{3} x^{6} + 3 \, a^{3} b^{2} x^{4} - 4 \, a^{4} b x^{2} + 8 \, a^{5}\right )} \sqrt{b x^{2} + a}}{693 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.51374, size = 133, normalized size = 2.25 \begin{align*} \begin{cases} \frac{8 a^{5} \sqrt{a + b x^{2}}}{693 b^{3}} - \frac{4 a^{4} x^{2} \sqrt{a + b x^{2}}}{693 b^{2}} + \frac{a^{3} x^{4} \sqrt{a + b x^{2}}}{231 b} + \frac{113 a^{2} x^{6} \sqrt{a + b x^{2}}}{693} + \frac{23 a b x^{8} \sqrt{a + b x^{2}}}{99} + \frac{b^{2} x^{10} \sqrt{a + b x^{2}}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{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 = 2.9664, size = 243, normalized size = 4.12 \begin{align*} \frac{\frac{33 \,{\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^{2}}{b^{2}} + \frac{22 \,{\left (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}\right )} a}{b^{2}} + \frac{315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4}}{b^{2}}}{3465 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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