Optimal. Leaf size=41 \[ -\frac{a}{8 c^2 x^7 \sqrt{c x^2}}-\frac{b}{7 c^2 x^6 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0083773, 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} \[ -\frac{a}{8 c^2 x^7 \sqrt{c x^2}}-\frac{b}{7 c^2 x^6 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{x^4 \left (c x^2\right )^{5/2}} \, dx &=\frac{x \int \frac{a+b x}{x^9} \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^9}+\frac{b}{x^8}\right ) \, dx}{c^2 \sqrt{c x^2}}\\ &=-\frac{a}{8 c^2 x^7 \sqrt{c x^2}}-\frac{b}{7 c^2 x^6 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0066191, size = 24, normalized size = 0.59 \[ \frac{-7 a-8 b x}{56 x^3 \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.5 \begin{align*} -{\frac{8\,bx+7\,a}{56\,{x}^{3}} \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 = 1.0884, size = 26, normalized size = 0.63 \begin{align*} -\frac{b}{7 \, c^{\frac{5}{2}} x^{7}} - \frac{a}{8 \, c^{\frac{5}{2}} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48948, size = 58, normalized size = 1.41 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (8 \, b x + 7 \, a\right )}}{56 \, c^{3} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.7776, size = 37, normalized size = 0.9 \begin{align*} - \frac{a}{8 c^{\frac{5}{2}} x^{3} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b}{7 c^{\frac{5}{2}} x^{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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