Optimal. Leaf size=30 \[ \frac{a c \sqrt{c x^2} \log (x)}{x}+b c \sqrt{c x^2} \]
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Rubi [A] time = 0.0052881, antiderivative size = 30, 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} \[ \frac{a c \sqrt{c x^2} \log (x)}{x}+b c \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2} (a+b x)}{x^4} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{a+b x}{x} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (b+\frac{a}{x}\right ) \, dx}{x}\\ &=b c \sqrt{c x^2}+\frac{a c \sqrt{c x^2} \log (x)}{x}\\ \end{align*}
Mathematica [A] time = 0.0043884, size = 21, normalized size = 0.7 \[ \frac{\left (c x^2\right )^{3/2} (a \log (x)+b x)}{x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 20, normalized size = 0.7 \begin{align*}{\frac{bx+a\ln \left ( x \right ) }{{x}^{3}} \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8643, size = 49, normalized size = 1.63 \begin{align*} \frac{{\left (b c x + a c \log \left (x\right )\right )} \sqrt{c x^{2}}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06045, size = 23, normalized size = 0.77 \begin{align*}{\left (b x \mathrm{sgn}\left (x\right ) + a \log \left ({\left | x \right |}\right ) \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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