Optimal. Leaf size=60 \[ \frac {1}{7} a^2 c x^6 \sqrt {c x^2}+\frac {1}{4} a b c x^7 \sqrt {c x^2}+\frac {1}{9} b^2 c x^8 \sqrt {c x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ \frac {1}{7} a^2 c x^6 \sqrt {c x^2}+\frac {1}{4} a b c x^7 \sqrt {c x^2}+\frac {1}{9} b^2 c x^8 \sqrt {c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int x^3 \left (c x^2\right )^{3/2} (a+b x)^2 \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x^6 (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (a^2 x^6+2 a b x^7+b^2 x^8\right ) \, dx}{x}\\ &=\frac {1}{7} a^2 c x^6 \sqrt {c x^2}+\frac {1}{4} a b c x^7 \sqrt {c x^2}+\frac {1}{9} b^2 c x^8 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.58 \[ \frac {1}{252} x^4 \left (c x^2\right )^{3/2} \left (36 a^2+63 a b x+28 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 36, normalized size = 0.60 \[ \frac {1}{252} \, {\left (28 \, b^{2} c x^{8} + 63 \, a b c x^{7} + 36 \, a^{2} c x^{6}\right )} \sqrt {c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 35, normalized size = 0.58 \[ \frac {1}{252} \, {\left (28 \, b^{2} x^{9} \mathrm {sgn}\relax (x) + 63 \, a b x^{8} \mathrm {sgn}\relax (x) + 36 \, a^{2} x^{7} \mathrm {sgn}\relax (x)\right )} c^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.53 \[ \frac {\left (28 b^{2} x^{2}+63 a b x +36 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {3}{2}} x^{4}}{252} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 54, normalized size = 0.90 \[ \frac {\left (c x^{2}\right )^{\frac {5}{2}} b^{2} x^{4}}{9 \, c} + \frac {\left (c x^{2}\right )^{\frac {5}{2}} a b x^{3}}{4 \, c} + \frac {\left (c x^{2}\right )^{\frac {5}{2}} a^{2} x^{2}}{7 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^3\,{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.50, size = 60, normalized size = 1.00 \[ \frac {a^{2} c^{\frac {3}{2}} x^{4} \left (x^{2}\right )^{\frac {3}{2}}}{7} + \frac {a b c^{\frac {3}{2}} x^{5} \left (x^{2}\right )^{\frac {3}{2}}}{4} + \frac {b^{2} c^{\frac {3}{2}} x^{6} \left (x^{2}\right )^{\frac {3}{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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