Optimal. Leaf size=71 \[ -\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]
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Rubi [A] time = 0.0186558, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 43} \[ -\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^4} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{a b+b^2 x}{x^4} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a b}{x^4}+\frac{b^2}{x^3}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0069367, size = 33, normalized size = 0.46 \[ -\frac{\sqrt{(a+b x)^2} (2 a+3 b x)}{6 x^3 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 30, normalized size = 0.4 \begin{align*} -{\frac{3\,bx+2\,a}{6\,{x}^{3} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89047, size = 32, normalized size = 0.45 \begin{align*} -\frac{3 \, b x + 2 \, a}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.888519, size = 14, normalized size = 0.2 \begin{align*} - \frac{2 a + 3 b x}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24129, size = 54, normalized size = 0.76 \begin{align*} \frac{b^{3} \mathrm{sgn}\left (b x + a\right )}{6 \, a^{2}} - \frac{3 \, b x \mathrm{sgn}\left (b x + a\right ) + 2 \, a \mathrm{sgn}\left (b x + a\right )}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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