Optimal. Leaf size=35 \[ \frac{a x^4}{3 \sqrt{c x^2}}+\frac{b x^5}{4 \sqrt{c x^2}} \]
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Rubi [A] time = 0.008578, antiderivative size = 35, 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} \[ \frac{a x^4}{3 \sqrt{c x^2}}+\frac{b x^5}{4 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3 (a+b x)}{\sqrt{c x^2}} \, dx &=\frac{x \int x^2 (a+b x) \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (a x^2+b x^3\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{a x^4}{3 \sqrt{c x^2}}+\frac{b x^5}{4 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0043804, size = 24, normalized size = 0.69 \[ \frac{x^4 (4 a+3 b x)}{12 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 21, normalized size = 0.6 \begin{align*}{\frac{{x}^{4} \left ( 3\,bx+4\,a \right ) }{12}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04352, size = 45, normalized size = 1.29 \begin{align*} \frac{\sqrt{c x^{2}} b x^{3}}{4 \, c} + \frac{\sqrt{c x^{2}} a x^{2}}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80944, size = 54, normalized size = 1.54 \begin{align*} \frac{{\left (3 \, b x^{3} + 4 \, a x^{2}\right )} \sqrt{c x^{2}}}{12 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.561217, size = 36, normalized size = 1.03 \begin{align*} \frac{a x^{4}}{3 \sqrt{c} \sqrt{x^{2}}} + \frac{b x^{5}}{4 \sqrt{c} \sqrt{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06321, size = 35, normalized size = 1. \begin{align*} \frac{1}{12} \, \sqrt{c x^{2}}{\left (\frac{3 \, b x}{c} + \frac{4 \, a}{c}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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