Optimal. Leaf size=35 \[ -\frac{a}{4 x^3 \sqrt{c x^2}}-\frac{b}{3 x^2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0067296, antiderivative size = 35, 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} \[ -\frac{a}{4 x^3 \sqrt{c x^2}}-\frac{b}{3 x^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{x^4 \sqrt{c x^2}} \, dx &=\frac{x \int \frac{a+b x}{x^5} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^5}+\frac{b}{x^4}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{a}{4 x^3 \sqrt{c x^2}}-\frac{b}{3 x^2 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0049902, size = 24, normalized size = 0.69 \[ \frac{-3 a-4 b x}{12 x^3 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.6 \begin{align*} -{\frac{4\,bx+3\,a}{12\,{x}^{3}}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03886, size = 26, normalized size = 0.74 \begin{align*} -\frac{b}{3 \, \sqrt{c} x^{3}} - \frac{a}{4 \, \sqrt{c} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58097, size = 55, normalized size = 1.57 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (4 \, b x + 3 \, a\right )}}{12 \, c x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.729381, size = 37, normalized size = 1.06 \begin{align*} - \frac{a}{4 \sqrt{c} x^{3} \sqrt{x^{2}}} - \frac{b}{3 \sqrt{c} x^{2} \sqrt{x^{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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