Optimal. Leaf size=36 \[ \frac{2 c \left (a+b x^2\right )^{3/2} \sqrt{c \sqrt{a+b x^2}}}{7 b} \]
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Rubi [A] time = 0.0195566, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1591, 15, 30} \[ \frac{2 c \left (a+b x^2\right )^{3/2} \sqrt{c \sqrt{a+b x^2}}}{7 b} \]
Antiderivative was successfully verified.
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Rule 1591
Rule 15
Rule 30
Rubi steps
\begin{align*} \int x \left (c \sqrt{a+b x^2}\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c \sqrt{x}\right )^{3/2} \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \operatorname{Subst}\left (\int x^{3/4} \, dx,x,a+b x^2\right )}{2 b \sqrt [4]{a+b x^2}}\\ &=\frac{2 c \sqrt{c \sqrt{a+b x^2}} \left (a+b x^2\right )^{3/2}}{7 b}\\ \end{align*}
Mathematica [A] time = 0.0067952, size = 31, normalized size = 0.86 \[ \frac{2 \left (a+b x^2\right ) \left (c \sqrt{a+b x^2}\right )^{3/2}}{7 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 0.7 \begin{align*}{\frac{2\,b{x}^{2}+2\,a}{7\,b} \left ( c\sqrt{b{x}^{2}+a} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993363, size = 34, normalized size = 0.94 \begin{align*} \frac{2 \,{\left (b x^{2} + a\right )} \left (\sqrt{b x^{2} + a} c\right )^{\frac{3}{2}}}{7 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6987, size = 85, normalized size = 2.36 \begin{align*} \frac{2 \,{\left (b c x^{2} + a c\right )} \sqrt{b x^{2} + a} \sqrt{\sqrt{b x^{2} + a} c}}{7 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.1388, size = 58, normalized size = 1.61 \begin{align*} \begin{cases} \frac{2 a c^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}}{7 b} + \frac{2 c^{\frac{3}{2}} x^{2} \left (a + b x^{2}\right )^{\frac{3}{4}}}{7} & \text{for}\: b \neq 0 \\\frac{x^{2} \left (\sqrt{a} c\right )^{\frac{3}{2}}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13623, size = 23, normalized size = 0.64 \begin{align*} \frac{2 \,{\left (b x^{2} + a\right )}^{\frac{7}{4}} c^{\frac{3}{2}}}{7 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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