Optimal. Leaf size=38 \[ \frac{\left (a+b x^2\right )^{7/2}}{7 b^2}-\frac{a \left (a+b x^2\right )^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.0243898, antiderivative size = 38, 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} \[ \frac{\left (a+b x^2\right )^{7/2}}{7 b^2}-\frac{a \left (a+b x^2\right )^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (a+b x^2\right )^{3/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^{3/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{3/2}}{b}+\frac{(a+b x)^{5/2}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac{a \left (a+b x^2\right )^{5/2}}{5 b^2}+\frac{\left (a+b x^2\right )^{7/2}}{7 b^2}\\ \end{align*}
Mathematica [A] time = 0.0137865, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^2\right )^{5/2} \left (5 b x^2-2 a\right )}{35 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-5\,b{x}^{2}+2\,a}{35\,{b}^{2}} \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 = 1.55671, size = 97, normalized size = 2.55 \begin{align*} \frac{{\left (5 \, b^{3} x^{6} + 8 \, a b^{2} x^{4} + a^{2} b x^{2} - 2 \, a^{3}\right )} \sqrt{b x^{2} + a}}{35 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.16344, size = 85, normalized size = 2.24 \begin{align*} \begin{cases} - \frac{2 a^{3} \sqrt{a + b x^{2}}}{35 b^{2}} + \frac{a^{2} x^{2} \sqrt{a + b x^{2}}}{35 b} + \frac{8 a x^{4} \sqrt{a + b x^{2}}}{35} + \frac{b x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.44151, size = 105, normalized size = 2.76 \begin{align*} \frac{\frac{7 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} a}{b} + \frac{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}}{b}}{105 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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