Optimal. Leaf size=27 \[ \frac {x (a+b x)^3}{3 b c \sqrt {c x^2}} \]
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Rubi [A] time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \begin {gather*} \frac {x (a+b x)^3}{3 b c \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int (a+b x)^2 \, dx}{c \sqrt {c x^2}}\\ &=\frac {x (a+b x)^3}{3 b c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^3 (a+b x)^3}{3 b \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 34, normalized size = 1.26 \begin {gather*} \frac {x^4 \left (3 a^2+3 a b x+b^2 x^2\right )}{3 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 30, normalized size = 1.11 \begin {gather*} \frac {{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}}}{3 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 39, normalized size = 1.44 \begin {gather*} \frac {\sqrt {c x^{2}} {\left ({\left (\frac {b^{2} x}{c} + \frac {3 \, a b}{c}\right )} x + \frac {3 \, a^{2}}{c}\right )}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 1.15 \begin {gather*} \frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) x^{4}}{3 \left (c \,x^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 52, normalized size = 1.93 \begin {gather*} \frac {b^{2} x^{4}}{3 \, \sqrt {c x^{2}} c} + \frac {a b x^{3}}{\sqrt {c x^{2}} c} + \frac {a^{2} x^{2}}{\sqrt {c x^{2}} 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 = -0.04 \begin {gather*} \int \frac {x^3\,{\left (a+b\,x\right )}^2}{{\left (c\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.80, size = 56, normalized size = 2.07 \begin {gather*} \frac {a^{2} x^{4}}{c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {a b x^{5}}{c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} x^{6}}{3 c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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