Optimal. Leaf size=66 \[ \frac {1}{7} a^2 c^2 x^6 \sqrt {c x^2}+\frac {1}{4} a b c^2 x^7 \sqrt {c x^2}+\frac {1}{9} b^2 c^2 x^8 \sqrt {c x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 66, 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} \begin {gather*} \frac {1}{7} a^2 c^2 x^6 \sqrt {c x^2}+\frac {1}{4} a b c^2 x^7 \sqrt {c x^2}+\frac {1}{9} b^2 c^2 x^8 \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int x \left (c x^2\right )^{5/2} (a+b x)^2 \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^6 (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c^2 \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^2 x^6 \sqrt {c x^2}+\frac {1}{4} a b c^2 x^7 \sqrt {c x^2}+\frac {1}{9} b^2 c^2 x^8 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.53 \begin {gather*} \frac {1}{252} x^2 \left (c x^2\right )^{5/2} \left (36 a^2+63 a b x+28 b^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 35, normalized size = 0.53 \begin {gather*} \frac {1}{252} x^2 \left (c x^2\right )^{5/2} \left (36 a^2+63 a b x+28 b^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 42, normalized size = 0.64 \begin {gather*} \frac {1}{252} \, {\left (28 \, b^{2} c^{2} x^{8} + 63 \, a b c^{2} x^{7} + 36 \, a^{2} c^{2} x^{6}\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 44, normalized size = 0.67 \begin {gather*} \frac {1}{252} \, {\left (28 \, b^{2} c^{2} x^{9} \mathrm {sgn}\relax (x) + 63 \, a b c^{2} x^{8} \mathrm {sgn}\relax (x) + 36 \, a^{2} c^{2} x^{7} \mathrm {sgn}\relax (x)\right )} \sqrt {c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.48 \begin {gather*} \frac {\left (28 b^{2} x^{2}+63 a b x +36 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}} x^{2}}{252} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 49, normalized size = 0.74 \begin {gather*} \frac {\left (c x^{2}\right )^{\frac {7}{2}} b^{2} x^{2}}{9 \, c} + \frac {\left (c x^{2}\right )^{\frac {7}{2}} a b x}{4 \, c} + \frac {\left (c x^{2}\right )^{\frac {7}{2}} a^{2}}{7 \, 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 = -0.02 \begin {gather*} \int x\,{\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.19, size = 60, normalized size = 0.91 \begin {gather*} \frac {a^{2} c^{\frac {5}{2}} x^{2} \left (x^{2}\right )^{\frac {5}{2}}}{7} + \frac {a b c^{\frac {5}{2}} x^{3} \left (x^{2}\right )^{\frac {5}{2}}}{4} + \frac {b^{2} c^{\frac {5}{2}} x^{4} \left (x^{2}\right )^{\frac {5}{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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