Optimal. Leaf size=57 \[ \frac {1}{4} a^2 x^3 \sqrt {c x^2}+\frac {2}{5} a b x^4 \sqrt {c x^2}+\frac {1}{6} b^2 x^5 \sqrt {c x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 57, 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} \begin {gather*} \frac {1}{4} a^2 x^3 \sqrt {c x^2}+\frac {2}{5} a b x^4 \sqrt {c x^2}+\frac {1}{6} b^2 x^5 \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int x^2 \sqrt {c x^2} (a+b x)^2 \, dx &=\frac {\sqrt {c x^2} \int x^3 (a+b x)^2 \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (a^2 x^3+2 a b x^4+b^2 x^5\right ) \, dx}{x}\\ &=\frac {1}{4} a^2 x^3 \sqrt {c x^2}+\frac {2}{5} a b x^4 \sqrt {c x^2}+\frac {1}{6} b^2 x^5 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.61 \begin {gather*} \frac {1}{60} x^3 \sqrt {c x^2} \left (15 a^2+24 a b x+10 b^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 35, normalized size = 0.61 \begin {gather*} \frac {1}{60} x^3 \sqrt {c x^2} \left (15 a^2+24 a b x+10 b^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 33, normalized size = 0.58 \begin {gather*} \frac {1}{60} \, {\left (10 \, b^{2} x^{5} + 24 \, a b x^{4} + 15 \, a^{2} x^{3}\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 35, normalized size = 0.61 \begin {gather*} \frac {1}{60} \, {\left (10 \, b^{2} x^{6} \mathrm {sgn}\relax (x) + 24 \, a b x^{5} \mathrm {sgn}\relax (x) + 15 \, a^{2} x^{4} \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 = 0.56 \begin {gather*} \frac {\left (10 b^{2} x^{2}+24 a b x +15 a^{2}\right ) \sqrt {c \,x^{2}}\, x^{3}}{60} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 52, normalized size = 0.91 \begin {gather*} \frac {\left (c x^{2}\right )^{\frac {3}{2}} b^{2} x^{3}}{6 \, c} + \frac {2 \, \left (c x^{2}\right )^{\frac {3}{2}} a b x^{2}}{5 \, c} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} a^{2} x}{4 \, 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^2\,\sqrt {c\,x^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 = 0.46, size = 61, normalized size = 1.07 \begin {gather*} \frac {a^{2} \sqrt {c} x^{3} \sqrt {x^{2}}}{4} + \frac {2 a b \sqrt {c} x^{4} \sqrt {x^{2}}}{5} + \frac {b^{2} \sqrt {c} x^{5} \sqrt {x^{2}}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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