Optimal. Leaf size=56 \[ -\frac {a^2}{c \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c \sqrt {c x^2}}+\frac {b^2 x^2}{c \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 56, 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} \[ -\frac {a^2}{c \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c \sqrt {c x^2}}+\frac {b^2 x^2}{c \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^2} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a^2}{c \sqrt {c x^2}}+\frac {b^2 x^2}{c \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.59 \[ \frac {x^2 \left (-a^2+2 a b x \log (x)+b^2 x^2\right )}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 34, normalized size = 0.61 \[ \frac {{\left (b^{2} x^{2} + 2 \, a b x \log \relax (x) - a^{2}\right )} \sqrt {c x^{2}}}{c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 69, normalized size = 1.23 \[ \frac {\frac {\sqrt {c x^{2}} b^{2}}{c} - \frac {2 \, {\left (a b \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right ) - \frac {a^{2} \sqrt {c}}{\sqrt {c} x - \sqrt {c x^{2}}}\right )}}{\sqrt {c}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.57 \[ \frac {\left (2 a b x \ln \relax (x )+b^{2} x^{2}-a^{2}\right ) x^{2}}{\left (c \,x^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 42, normalized size = 0.75 \[ \frac {b^{2} x^{2}}{\sqrt {c x^{2}} c} + \frac {2 \, a b \log \relax (x)}{c^{\frac {3}{2}}} - \frac {a^{2}}{\sqrt {c x^{2}} c} \]
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 \[ \int \frac {x\,{\left (a+b\,x\right )}^2}{{\left (c\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a + b x\right )^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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