Optimal. Leaf size=47 \[ -\frac{6 a^2 b}{5 x^{5/2}}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{2 x^2}-\frac{2 b^3}{3 x^{3/2}} \]
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Rubi [A] time = 0.0201946, antiderivative size = 47, 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} \[ -\frac{6 a^2 b}{5 x^{5/2}}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{2 x^2}-\frac{2 b^3}{3 x^{3/2}} \]
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^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^7} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^3}{x^7}+\frac{3 a^2 b}{x^6}+\frac{3 a b^2}{x^5}+\frac{b^3}{x^4}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^3}{3 x^3}-\frac{6 a^2 b}{5 x^{5/2}}-\frac{3 a b^2}{2 x^2}-\frac{2 b^3}{3 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0159845, size = 41, normalized size = 0.87 \[ -\frac{36 a^2 b \sqrt{x}+10 a^3+45 a b^2 x+20 b^3 x^{3/2}}{30 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 36, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3}}{3\,{x}^{3}}}-{\frac{6\,b{a}^{2}}{5}{x}^{-{\frac{5}{2}}}}-{\frac{3\,{b}^{2}a}{2\,{x}^{2}}}-{\frac{2\,{b}^{3}}{3}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977509, size = 47, normalized size = 1. \begin{align*} -\frac{20 \, b^{3} x^{\frac{3}{2}} + 45 \, a b^{2} x + 36 \, a^{2} b \sqrt{x} + 10 \, a^{3}}{30 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4627, size = 88, normalized size = 1.87 \begin{align*} -\frac{45 \, a b^{2} x + 10 \, a^{3} + 4 \,{\left (5 \, b^{3} x + 9 \, a^{2} b\right )} \sqrt{x}}{30 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.35406, size = 46, normalized size = 0.98 \begin{align*} - \frac{a^{3}}{3 x^{3}} - \frac{6 a^{2} b}{5 x^{\frac{5}{2}}} - \frac{3 a b^{2}}{2 x^{2}} - \frac{2 b^{3}}{3 x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08464, size = 47, normalized size = 1. \begin{align*} -\frac{20 \, b^{3} x^{\frac{3}{2}} + 45 \, a b^{2} x + 36 \, a^{2} b \sqrt{x} + 10 \, a^{3}}{30 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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