Optimal. Leaf size=41 \[ -\frac{a}{4 c^2 x^3 \sqrt{c x^2}}-\frac{b}{3 c^2 x^2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0072754, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 43} \[ -\frac{a}{4 c^2 x^3 \sqrt{c x^2}}-\frac{b}{3 c^2 x^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{\left (c x^2\right )^{5/2}} \, dx &=\frac{x \int \frac{a+b x}{x^5} \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^5}+\frac{b}{x^4}\right ) \, dx}{c^2 \sqrt{c x^2}}\\ &=-\frac{a}{4 c^2 x^3 \sqrt{c x^2}}-\frac{b}{3 c^2 x^2 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0027211, size = 27, normalized size = 0.66 \[ -\frac{\sqrt{c x^2} (3 a+4 b x)}{12 c^3 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 19, normalized size = 0.5 \begin{align*} -{\frac{x \left ( 4\,bx+3\,a \right ) }{12} \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 = 1.06456, size = 31, normalized size = 0.76 \begin{align*} -\frac{b}{3 \, \left (c x^{2}\right )^{\frac{3}{2}} c} - \frac{a}{4 \, c^{\frac{5}{2}} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54431, size = 58, normalized size = 1.41 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (4 \, b x + 3 \, a\right )}}{12 \, c^{3} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.879129, size = 36, normalized size = 0.88 \begin{align*} - \frac{a x}{4 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b x^{2}}{3 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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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