Optimal. Leaf size=58 \[ \frac {a^2 c^2 \sqrt {c x^2} \log (x)}{x}+2 a b c^2 \sqrt {c x^2}+\frac {1}{2} b^2 c^2 x \sqrt {c x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 58, 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 c^2 \sqrt {c x^2} \log (x)}{x}+2 a b c^2 \sqrt {c x^2}+\frac {1}{2} b^2 c^2 x \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^6} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {(a+b x)^2}{x} \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx}{x}\\ &=2 a b c^2 \sqrt {c x^2}+\frac {1}{2} b^2 c^2 x \sqrt {c x^2}+\frac {a^2 c^2 \sqrt {c x^2} \log (x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.60 \begin {gather*} \frac {c^3 x \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 37, normalized size = 0.64 \begin {gather*} \left (c x^2\right )^{5/2} \left (\frac {a^2 \log (x)}{x^5}+\frac {4 a b+b^2 x}{2 x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 41, normalized size = 0.71 \begin {gather*} \frac {{\left (b^{2} c^{2} x^{2} + 4 \, a b c^{2} x + 2 \, a^{2} c^{2} \log \relax (x)\right )} \sqrt {c x^{2}}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 41, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, {\left (b^{2} c^{2} x^{2} \mathrm {sgn}\relax (x) + 4 \, a b c^{2} x \mathrm {sgn}\relax (x) + 2 \, a^{2} c^{2} \log \left ({\left | x \right |}\right ) \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 = 0.01, size = 33, normalized size = 0.57 \begin {gather*} \frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (b^{2} x^{2}+2 a^{2} \ln \relax (x )+4 a b x \right )}{2 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 {{\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2}{x^6} \,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 {\left (c x^{2}\right )^{\frac {5}{2}} \left (a + b x\right )^{2}}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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