Optimal. Leaf size=41 \[ \frac{1}{4} a c^2 x^3 \sqrt{c x^2}+\frac{1}{5} b c^2 x^4 \sqrt{c x^2} \]
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Rubi [A] time = 0.011029, antiderivative size = 41, 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} \[ \frac{1}{4} a c^2 x^3 \sqrt{c x^2}+\frac{1}{5} b c^2 x^4 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int x^3 (a+b x) \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \left (a x^3+b x^4\right ) \, dx}{x}\\ &=\frac{1}{4} a c^2 x^3 \sqrt{c x^2}+\frac{1}{5} b c^2 x^4 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.002578, size = 23, normalized size = 0.56 \[ \frac{1}{20} c x \left (c x^2\right )^{3/2} (5 a+4 b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 21, normalized size = 0.5 \begin{align*}{\frac{4\,bx+5\,a}{20\,x} \left ( c{x}^{2} \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 = 1.8442, size = 62, normalized size = 1.51 \begin{align*} \frac{1}{20} \,{\left (4 \, b c^{2} x^{4} + 5 \, a c^{2} x^{3}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.35926, size = 31, normalized size = 0.76 \begin{align*} \frac{a c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{4 x} + \frac{b c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04246, size = 38, normalized size = 0.93 \begin{align*} \frac{1}{20} \,{\left (4 \, b c^{2} x^{5} \mathrm{sgn}\left (x\right ) + 5 \, a c^{2} x^{4} \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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