Optimal. Leaf size=57 \[ \frac {a^2 x^4}{3 \sqrt {c x^2}}+\frac {a b x^5}{2 \sqrt {c x^2}}+\frac {b^2 x^6}{5 \sqrt {c x^2}} \]
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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 {a^2 x^4}{3 \sqrt {c x^2}}+\frac {a b x^5}{2 \sqrt {c x^2}}+\frac {b^2 x^6}{5 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^3 (a+b x)^2}{\sqrt {c x^2}} \, dx &=\frac {x \int x^2 (a+b x)^2 \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (a^2 x^2+2 a b x^3+b^2 x^4\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {a^2 x^4}{3 \sqrt {c x^2}}+\frac {a b x^5}{2 \sqrt {c x^2}}+\frac {b^2 x^6}{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 {x^4 \left (10 a^2+15 a b x+6 b^2 x^2\right )}{30 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 38, normalized size = 0.67 \begin {gather*} \frac {x^2 \sqrt {c x^2} \left (10 a^2+15 a b x+6 b^2 x^2\right )}{30 c} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 36, normalized size = 0.63 \begin {gather*} \frac {{\left (6 \, b^{2} x^{4} + 15 \, a b x^{3} + 10 \, a^{2} x^{2}\right )} \sqrt {c x^{2}}}{30 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 41, normalized size = 0.72 \begin {gather*} \frac {1}{30} \, \sqrt {c x^{2}} {\left (3 \, {\left (\frac {2 \, b^{2} x}{c} + \frac {5 \, a b}{c}\right )} x + \frac {10 \, a^{2}}{c}\right )} x^{2} \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 (6 b^{2} x^{2}+15 a b x +10 a^{2}\right ) x^{4}}{30 \sqrt {c \,x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 54, normalized size = 0.95 \begin {gather*} \frac {\sqrt {c x^{2}} b^{2} x^{4}}{5 \, c} + \frac {\sqrt {c x^{2}} a b x^{3}}{2 \, c} + \frac {\sqrt {c x^{2}} a^{2} x^{2}}{3 \, 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 \frac {x^3\,{\left (a+b\,x\right )}^2}{\sqrt {c\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.79, size = 60, normalized size = 1.05 \begin {gather*} \frac {a^{2} x^{4}}{3 \sqrt {c} \sqrt {x^{2}}} + \frac {a b x^{5}}{2 \sqrt {c} \sqrt {x^{2}}} + \frac {b^{2} x^{6}}{5 \sqrt {c} \sqrt {x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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