Optimal. Leaf size=61 \[ \frac {a^2 x \log (x)}{c \sqrt {c x^2}}+\frac {2 a b x^2}{c \sqrt {c x^2}}+\frac {b^2 x^3}{2 c \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 61, 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 \log (x)}{c \sqrt {c x^2}}+\frac {2 a b x^2}{c \sqrt {c x^2}}+\frac {b^2 x^3}{2 c \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^2 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx}{c \sqrt {c x^2}}\\ &=\frac {2 a b x^2}{c \sqrt {c x^2}}+\frac {b^2 x^3}{2 c \sqrt {c x^2}}+\frac {a^2 x \log (x)}{c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 0.56 \begin {gather*} \frac {x^3 \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 39, normalized size = 0.64 \begin {gather*} \frac {a^2 x^3 \log (x)+\frac {1}{2} \left (4 a b x^4+b^2 x^5\right )}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 35, normalized size = 0.57 \begin {gather*} \frac {{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \relax (x)\right )} \sqrt {c x^{2}}}{2 \, c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 55, normalized size = 0.90 \begin {gather*} -\frac {\frac {2 \, a^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right )}{\sqrt {c}} - \sqrt {c x^{2}} {\left (\frac {b^{2} x}{c} + \frac {4 \, a b}{c}\right )}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.54 \begin {gather*} \frac {\left (b^{2} x^{2}+2 a^{2} \ln \relax (x )+4 a b x \right ) x^{3}}{2 \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 [A] time = 1.33, size = 45, normalized size = 0.74 \begin {gather*} \frac {b^{2} x^{3}}{2 \, \sqrt {c x^{2}} c} + \frac {2 \, a b x^{2}}{\sqrt {c x^{2}} c} + \frac {a^{2} \log \relax (x)}{c^{\frac {3}{2}}} \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^2\,{\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 [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b x\right )^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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