Optimal. Leaf size=37 \[ \frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 43} \[ \frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \left (c x^2\right )^{3/2} (a+b x) \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x^3 (a+b x) \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (a x^3+b x^4\right ) \, dx}{x}\\ &=\frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.59 \[ \frac {1}{20} x \left (c x^2\right )^{3/2} (5 a+4 b x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 24, normalized size = 0.65 \[ \frac {1}{20} \, {\left (4 \, b c x^{4} + 5 \, a c x^{3}\right )} \sqrt {c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 22, normalized size = 0.59 \[ \frac {1}{20} \, {\left (4 \, b x^{5} \mathrm {sgn}\relax (x) + 5 \, a x^{4} \mathrm {sgn}\relax (x)\right )} c^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.51 \[ \frac {\left (4 b x +5 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}} x}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 25, normalized size = 0.68 \[ \frac {1}{4} \, \left (c x^{2}\right )^{\frac {3}{2}} a x + \frac {\left (c x^{2}\right )^{\frac {5}{2}} b}{5 \, c} \]
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 \[ \int {\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 34, normalized size = 0.92 \[ \frac {a c^{\frac {3}{2}} x \left (x^{2}\right )^{\frac {3}{2}}}{4} + \frac {b c^{\frac {3}{2}} x^{2} \left (x^{2}\right )^{\frac {3}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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