Optimal. Leaf size=41 \[ \frac {1}{5} a c^2 x^4 \sqrt {c x^2}+\frac {1}{6} b c^2 x^5 \sqrt {c x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 41, 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} \begin {gather*} \frac {1}{5} a c^2 x^4 \sqrt {c x^2}+\frac {1}{6} b c^2 x^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 {\left (c x^2\right )^{5/2} (a+b x)}{x} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^4 (a+b x) \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a x^4+b x^5\right ) \, dx}{x}\\ &=\frac {1}{5} a c^2 x^4 \sqrt {c x^2}+\frac {1}{6} b c^2 x^5 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 0.61 \begin {gather*} \frac {1}{30} c x^2 \left (c x^2\right )^{3/2} (6 a+5 b x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 21, normalized size = 0.51 \begin {gather*} \frac {1}{30} \left (c x^2\right )^{5/2} (6 a+5 b x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 28, normalized size = 0.68 \begin {gather*} \frac {1}{30} \, {\left (5 \, b c^{2} x^{5} + 6 \, a c^{2} x^{4}\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 28, normalized size = 0.68 \begin {gather*} \frac {1}{30} \, {\left (5 \, b c^{2} x^{6} \mathrm {sgn}\relax (x) + 6 \, a c^{2} x^{5} \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 = 0.00, size = 18, normalized size = 0.44 \begin {gather*} \frac {\left (5 b x +6 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 22, normalized size = 0.54 \begin {gather*} \frac {1}{6} \, \left (c x^{2}\right )^{\frac {5}{2}} b x + \frac {1}{5} \, \left (c x^{2}\right )^{\frac {5}{2}} a \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 )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.43, size = 31, normalized size = 0.76 \begin {gather*} \frac {a c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{5} + \frac {b c^{\frac {5}{2}} x \left (x^{2}\right )^{\frac {5}{2}}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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