Optimal. Leaf size=45 \[ -\frac{6 a^2 b}{7 x^{7/2}}-\frac{a^3}{4 x^4}-\frac{a b^2}{x^3}-\frac{2 b^3}{5 x^{5/2}} \]
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Rubi [A] time = 0.0203438, antiderivative size = 45, 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 = {266, 43} \[ -\frac{6 a^2 b}{7 x^{7/2}}-\frac{a^3}{4 x^4}-\frac{a b^2}{x^3}-\frac{2 b^3}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^3}{x^5} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^9} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^3}{x^9}+\frac{3 a^2 b}{x^8}+\frac{3 a b^2}{x^7}+\frac{b^3}{x^6}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^3}{4 x^4}-\frac{6 a^2 b}{7 x^{7/2}}-\frac{a b^2}{x^3}-\frac{2 b^3}{5 x^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0161908, size = 41, normalized size = 0.91 \[ -\frac{120 a^2 b \sqrt{x}+35 a^3+140 a b^2 x+56 b^3 x^{3/2}}{140 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3}}{4\,{x}^{4}}}-{\frac{6\,b{a}^{2}}{7}{x}^{-{\frac{7}{2}}}}-{\frac{{b}^{2}a}{{x}^{3}}}-{\frac{2\,{b}^{3}}{5}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973464, size = 47, normalized size = 1.04 \begin{align*} -\frac{56 \, b^{3} x^{\frac{3}{2}} + 140 \, a b^{2} x + 120 \, a^{2} b \sqrt{x} + 35 \, a^{3}}{140 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46931, size = 92, normalized size = 2.04 \begin{align*} -\frac{140 \, a b^{2} x + 35 \, a^{3} + 8 \,{\left (7 \, b^{3} x + 15 \, a^{2} b\right )} \sqrt{x}}{140 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.92839, size = 42, normalized size = 0.93 \begin{align*} - \frac{a^{3}}{4 x^{4}} - \frac{6 a^{2} b}{7 x^{\frac{7}{2}}} - \frac{a b^{2}}{x^{3}} - \frac{2 b^{3}}{5 x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1176, size = 47, normalized size = 1.04 \begin{align*} -\frac{56 \, b^{3} x^{\frac{3}{2}} + 140 \, a b^{2} x + 120 \, a^{2} b \sqrt{x} + 35 \, a^{3}}{140 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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