Optimal. Leaf size=37 \[ \frac{1}{7} a c x^6 \sqrt{c x^2}+\frac{1}{8} b c x^7 \sqrt{c x^2} \]
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Rubi [A] time = 0.0133218, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac{1}{7} a c x^6 \sqrt{c x^2}+\frac{1}{8} b c x^7 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (c x^2\right )^{3/2} (a+b x) \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^6 (a+b x) \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a x^6+b x^7\right ) \, dx}{x}\\ &=\frac{1}{7} a c x^6 \sqrt{c x^2}+\frac{1}{8} b c x^7 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0066866, size = 24, normalized size = 0.65 \[ \frac{1}{56} x^4 \left (c x^2\right )^{3/2} (8 a+7 b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.6 \begin{align*}{\frac{{x}^{4} \left ( 7\,bx+8\,a \right ) }{56} \left ( c{x}^{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 [A] time = 1.49586, size = 57, normalized size = 1.54 \begin{align*} \frac{1}{56} \,{\left (7 \, b c x^{7} + 8 \, a c x^{6}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.09628, size = 36, normalized size = 0.97 \begin{align*} \frac{a c^{\frac{3}{2}} x^{4} \left (x^{2}\right )^{\frac{3}{2}}}{7} + \frac{b c^{\frac{3}{2}} x^{5} \left (x^{2}\right )^{\frac{3}{2}}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06378, size = 30, normalized size = 0.81 \begin{align*} \frac{1}{56} \,{\left (7 \, b x^{8} \mathrm{sgn}\left (x\right ) + 8 \, a x^{7} \mathrm{sgn}\left (x\right )\right )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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