Optimal. Leaf size=32 \[ \frac{c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^2}}{8 b} \]
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Rubi [A] time = 0.0223109, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1591, 15, 30} \[ \frac{c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^2}}{8 b} \]
Antiderivative was successfully verified.
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Rule 1591
Rule 15
Rule 30
Rubi steps
\begin{align*} \int x \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c x^2\right )^{3/2} \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac{\left (c \sqrt{c \left (a+b x^2\right )^2}\right ) \operatorname{Subst}\left (\int x^3 \, dx,x,a+b x^2\right )}{2 b \left (a+b x^2\right )}\\ &=\frac{c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^2}}{8 b}\\ \end{align*}
Mathematica [A] time = 0.0123346, size = 29, normalized size = 0.91 \[ \frac{\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^2\right )^{3/2}}{8 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 59, normalized size = 1.8 \begin{align*}{\frac{{x}^{2} \left ({b}^{3}{x}^{6}+4\,a{b}^{2}{x}^{4}+6\,{a}^{2}b{x}^{2}+4\,{a}^{3} \right ) }{8\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( c \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{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 [B] time = 1.39286, size = 153, normalized size = 4.78 \begin{align*} \frac{{\left (b^{3} c x^{8} + 4 \, a b^{2} c x^{6} + 6 \, a^{2} b c x^{4} + 4 \, a^{3} c x^{2}\right )} \sqrt{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{8 \,{\left (b x^{2} + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23664, size = 34, normalized size = 1.06 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{4} c^{\frac{3}{2}} \mathrm{sgn}\left (b x^{2} + a\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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