Optimal. Leaf size=41 \[ \frac {1}{3} a c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b c^2 x^3 \sqrt {c x^2} \]
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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} \[ \frac {1}{3} a c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b c^2 x^3 \sqrt {c x^2} \]
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^3} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^2 (a+b x) \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a x^2+b x^3\right ) \, dx}{x}\\ &=\frac {1}{3} a c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b c^2 x^3 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 27, normalized size = 0.66 \[ \frac {1}{12} c^2 x^2 \sqrt {c x^2} (4 a+3 b x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 28, normalized size = 0.68 \[ \frac {1}{12} \, {\left (3 \, b c^{2} x^{3} + 4 \, a c^{2} x^{2}\right )} \sqrt {c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 28, normalized size = 0.68 \[ \frac {1}{12} \, {\left (3 \, b c^{2} x^{4} \mathrm {sgn}\relax (x) + 4 \, a c^{2} x^{3} \mathrm {sgn}\relax (x)\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.51 \[ \frac {\left (3 b x +4 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{12 x^{2}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 25, normalized size = 0.61 \[ \frac {c^{5/2}\,\left (4\,a\,\sqrt {x^6}+3\,b\,x^3\,\sqrt {x^2}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.58, size = 34, normalized size = 0.83 \[ \frac {a c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{3 x^{2}} + \frac {b c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{4 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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