Optimal. Leaf size=64 \[ \frac{a^3}{b^4 \left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{b^4 \left (a+b \sqrt{x}\right )}-\frac{6 a \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 \sqrt{x}}{b^3} \]
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Rubi [A] time = 0.0369614, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^3}{b^4 \left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{b^4 \left (a+b \sqrt{x}\right )}-\frac{6 a \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 \sqrt{x}}{b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b \sqrt{x}\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{b^3}-\frac{a^3}{b^3 (a+b x)^3}+\frac{3 a^2}{b^3 (a+b x)^2}-\frac{3 a}{b^3 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^3}{b^4 \left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{b^4 \left (a+b \sqrt{x}\right )}+\frac{2 \sqrt{x}}{b^3}-\frac{6 a \log \left (a+b \sqrt{x}\right )}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0405811, size = 57, normalized size = 0.89 \[ \frac{\frac{a^3}{\left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{a+b \sqrt{x}}-6 a \log \left (a+b \sqrt{x}\right )+2 b \sqrt{x}}{b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 57, normalized size = 0.9 \begin{align*} -6\,{\frac{a\ln \left ( a+b\sqrt{x} \right ) }{{b}^{4}}}+2\,{\frac{\sqrt{x}}{{b}^{3}}}+{\frac{{a}^{3}}{{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-2}}-6\,{\frac{{a}^{2}}{{b}^{4} \left ( a+b\sqrt{x} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0219, size = 81, normalized size = 1.27 \begin{align*} -\frac{6 \, a \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}}{b^{4}} - \frac{6 \, a^{2}}{{\left (b \sqrt{x} + a\right )} b^{4}} + \frac{a^{3}}{{\left (b \sqrt{x} + a\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28328, size = 213, normalized size = 3.33 \begin{align*} \frac{7 \, a^{3} b^{2} x - 5 \, a^{5} - 6 \,{\left (a b^{4} x^{2} - 2 \, a^{3} b^{2} x + a^{5}\right )} \log \left (b \sqrt{x} + a\right ) + 2 \,{\left (b^{5} x^{2} - 5 \, a^{2} b^{3} x + 3 \, a^{4} b\right )} \sqrt{x}}{b^{8} x^{2} - 2 \, a^{2} b^{6} x + a^{4} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.13798, size = 264, normalized size = 4.12 \begin{align*} \begin{cases} - \frac{6 a^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{8 a^{3}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{12 a^{2} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{10 a^{2} b \sqrt{x}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{6 a b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} + \frac{a b^{2} x}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} + \frac{2 b^{3} x^{\frac{3}{2}}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09492, size = 72, normalized size = 1.12 \begin{align*} -\frac{6 \, a \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{4}} + \frac{2 \, \sqrt{x}}{b^{3}} - \frac{6 \, a^{2} b \sqrt{x} + 5 \, a^{3}}{{\left (b \sqrt{x} + a\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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