Optimal. Leaf size=38 \[ \frac {a x^2}{c \sqrt {c x^2}}+\frac {b x^3}{2 c \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {15} \begin {gather*} \frac {a x^2}{c \sqrt {c x^2}}+\frac {b x^3}{2 c \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rubi steps
\begin {align*} \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int (a+b x) \, dx}{c \sqrt {c x^2}}\\ &=\frac {a x^2}{c \sqrt {c x^2}}+\frac {b x^3}{2 c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 0.61 \begin {gather*} \frac {x^4 (2 a+b x)}{2 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 23, normalized size = 0.61 \begin {gather*} \frac {x^4 (2 a+b x)}{2 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 19, normalized size = 0.50 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + 2 \, a\right )}}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 25, normalized size = 0.66 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (\frac {b x}{c} + \frac {2 \, a}{c}\right )}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.53 \begin {gather*} \frac {\left (b x +2 a \right ) x^{4}}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 32, normalized size = 0.84 \begin {gather*} \frac {b x^{3}}{2 \, \sqrt {c x^{2}} c} + \frac {a x^{2}}{\sqrt {c x^{2}} c} \end {gather*}
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 \begin {gather*} \int \frac {x^3\,\left (a+b\,x\right )}{{\left (c\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 34, normalized size = 0.89 \begin {gather*} \frac {a x^{4}}{c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {b x^{5}}{2 c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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