Optimal. Leaf size=41 \[ -\frac{a}{3 c^2 x^2 \sqrt{c x^2}}-\frac{b}{2 c^2 x \sqrt{c x^2}} \]
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Rubi [A] time = 0.0082408, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {15, 43} \[ -\frac{a}{3 c^2 x^2 \sqrt{c x^2}}-\frac{b}{2 c^2 x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x (a+b x)}{\left (c x^2\right )^{5/2}} \, dx &=\frac{x \int \frac{a+b x}{x^4} \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^4}+\frac{b}{x^3}\right ) \, dx}{c^2 \sqrt{c x^2}}\\ &=-\frac{a}{3 c^2 x^2 \sqrt{c x^2}}-\frac{b}{2 c^2 x \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0037681, size = 24, normalized size = 0.59 \[ \frac{x^2 (-2 a-3 b x)}{6 \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 21, normalized size = 0.5 \begin{align*} -{\frac{{x}^{2} \left ( 3\,bx+2\,a \right ) }{6} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.069, size = 31, normalized size = 0.76 \begin{align*} -\frac{a}{3 \, \left (c x^{2}\right )^{\frac{3}{2}} c} - \frac{b}{2 \, c^{\frac{5}{2}} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59097, size = 57, normalized size = 1.39 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (3 \, b x + 2 \, a\right )}}{6 \, c^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.852636, size = 37, normalized size = 0.9 \begin{align*} - \frac{a x^{2}}{3 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b x^{3}}{2 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )} x}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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