Optimal. Leaf size=41 \[ -\frac{a}{7 c^2 x^6 \sqrt{c x^2}}-\frac{b}{6 c^2 x^5 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0088245, antiderivative size = 41, normalized size of antiderivative = 1., 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} \[ -\frac{a}{7 c^2 x^6 \sqrt{c x^2}}-\frac{b}{6 c^2 x^5 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{x^3 \left (c x^2\right )^{5/2}} \, dx &=\frac{x \int \frac{a+b x}{x^8} \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^8}+\frac{b}{x^7}\right ) \, dx}{c^2 \sqrt{c x^2}}\\ &=-\frac{a}{7 c^2 x^6 \sqrt{c x^2}}-\frac{b}{6 c^2 x^5 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0081205, size = 22, normalized size = 0.54 \[ \frac{c (-6 a-7 b x)}{42 \left (c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.5 \begin{align*} -{\frac{7\,bx+6\,a}{42\,{x}^{2}} \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 = 1.08664, size = 26, normalized size = 0.63 \begin{align*} -\frac{b}{6 \, c^{\frac{5}{2}} x^{6}} - \frac{a}{7 \, c^{\frac{5}{2}} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60592, size = 58, normalized size = 1.41 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (7 \, b x + 6 \, a\right )}}{42 \, c^{3} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.49165, size = 36, normalized size = 0.88 \begin{align*} - \frac{a}{7 c^{\frac{5}{2}} x^{2} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b}{6 c^{\frac{5}{2}} x \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{b x + a}{\left (c x^{2}\right )^{\frac{5}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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