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 = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.43, size = 25, normalized size = 0.71 \[ \frac {{\left (3 \, b x^{3} + 4 \, a x^{2}\right )} \sqrt {c x^{2}}}{12 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 26, normalized size = 0.74 \[ \frac {1}{12} \, \sqrt {c x^{2}} {\left (\frac {3 \, b x}{c} + \frac {4 \, a}{c}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.60 \[ \frac {\left (3 b x +4 a \right ) x^{4}}{12 \sqrt {c \,x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 33, normalized size = 0.94 \[ \frac {\sqrt {c x^{2}} b x^{3}}{4 \, c} + \frac {\sqrt {c x^{2}} a x^{2}}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^3\,\left (a+b\,x\right )}{\sqrt {c\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 36, normalized size = 1.03 \[ \frac {a x^{4}}{3 \sqrt {c} \sqrt {x^{2}}} + \frac {b x^{5}}{4 \sqrt {c} \sqrt {x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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