Optimal. Leaf size=29 \[ -\frac {(a+b x)^2}{2 a c^2 x \sqrt {c x^2}} \]
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Rubi [A] time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 37} \[ -\frac {(a+b x)^2}{2 a c^2 x \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {a+b x}{x^3} \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {(a+b x)^2}{2 a c^2 x \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.83 \[ \frac {x^3 (-a-2 b x)}{2 \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 21, normalized size = 0.72 \[ -\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )}}{2 \, c^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.66 \[ -\frac {\left (2 b x +a \right ) x^{3}}{2 \left (c \,x^{2}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 26, normalized size = 0.90 \[ -\frac {b x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} c} - \frac {a}{2 \, c^{\frac {5}{2}} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 25, normalized size = 0.86 \[ -\frac {2\,b\,x^3+a\,x^2}{2\,c^{5/2}\,x\,{\left (x^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.93, size = 36, normalized size = 1.24 \[ - \frac {a x^{3}}{2 c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}} - \frac {b x^{4}}{c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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