Optimal. Leaf size=111 \[ -\frac{8 a^3 x^{3/2}}{3 b^5}+\frac{3 a^2 x^2}{2 b^4}+\frac{2 a^7}{b^8 \left (a+b \sqrt{x}\right )}-\frac{12 a^5 \sqrt{x}}{b^7}+\frac{5 a^4 x}{b^6}+\frac{14 a^6 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{4 a x^{5/2}}{5 b^3}+\frac{x^3}{3 b^2} \]
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Rubi [A] time = 0.0827748, antiderivative size = 111, 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} \[ -\frac{8 a^3 x^{3/2}}{3 b^5}+\frac{3 a^2 x^2}{2 b^4}+\frac{2 a^7}{b^8 \left (a+b \sqrt{x}\right )}-\frac{12 a^5 \sqrt{x}}{b^7}+\frac{5 a^4 x}{b^6}+\frac{14 a^6 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{4 a x^{5/2}}{5 b^3}+\frac{x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b \sqrt{x}\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^7}{(a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{6 a^5}{b^7}+\frac{5 a^4 x}{b^6}-\frac{4 a^3 x^2}{b^5}+\frac{3 a^2 x^3}{b^4}-\frac{2 a x^4}{b^3}+\frac{x^5}{b^2}-\frac{a^7}{b^7 (a+b x)^2}+\frac{7 a^6}{b^7 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^7}{b^8 \left (a+b \sqrt{x}\right )}-\frac{12 a^5 \sqrt{x}}{b^7}+\frac{5 a^4 x}{b^6}-\frac{8 a^3 x^{3/2}}{3 b^5}+\frac{3 a^2 x^2}{2 b^4}-\frac{4 a x^{5/2}}{5 b^3}+\frac{x^3}{3 b^2}+\frac{14 a^6 \log \left (a+b \sqrt{x}\right )}{b^8}\\ \end{align*}
Mathematica [A] time = 0.079838, size = 102, normalized size = 0.92 \[ \frac{-80 a^3 b^3 x^{3/2}+45 a^2 b^4 x^2+150 a^4 b^2 x+\frac{60 a^7}{a+b \sqrt{x}}-360 a^5 b \sqrt{x}+420 a^6 \log \left (a+b \sqrt{x}\right )-24 a b^5 x^{5/2}+10 b^6 x^3}{30 b^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 94, normalized size = 0.9 \begin{align*} 5\,{\frac{{a}^{4}x}{{b}^{6}}}-{\frac{8\,{a}^{3}}{3\,{b}^{5}}{x}^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}{x}^{2}}{2\,{b}^{4}}}-{\frac{4\,a}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}+{\frac{{x}^{3}}{3\,{b}^{2}}}+14\,{\frac{{a}^{6}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{8}}}-12\,{\frac{{a}^{5}\sqrt{x}}{{b}^{7}}}+2\,{\frac{{a}^{7}}{{b}^{8} \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 = 0.964366, size = 174, normalized size = 1.57 \begin{align*} \frac{14 \, a^{6} \log \left (b \sqrt{x} + a\right )}{b^{8}} + \frac{{\left (b \sqrt{x} + a\right )}^{6}}{3 \, b^{8}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{5} a}{5 \, b^{8}} + \frac{21 \,{\left (b \sqrt{x} + a\right )}^{4} a^{2}}{2 \, b^{8}} - \frac{70 \,{\left (b \sqrt{x} + a\right )}^{3} a^{3}}{3 \, b^{8}} + \frac{35 \,{\left (b \sqrt{x} + a\right )}^{2} a^{4}}{b^{8}} - \frac{42 \,{\left (b \sqrt{x} + a\right )} a^{5}}{b^{8}} + \frac{2 \, a^{7}}{{\left (b \sqrt{x} + a\right )} b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31609, size = 285, normalized size = 2.57 \begin{align*} \frac{10 \, b^{8} x^{4} + 35 \, a^{2} b^{6} x^{3} + 105 \, a^{4} b^{4} x^{2} - 150 \, a^{6} b^{2} x - 60 \, a^{8} + 420 \,{\left (a^{6} b^{2} x - a^{8}\right )} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (6 \, a b^{7} x^{3} + 14 \, a^{3} b^{5} x^{2} + 70 \, a^{5} b^{3} x - 105 \, a^{7} b\right )} \sqrt{x}}{30 \,{\left (b^{10} x - a^{2} b^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.40098, size = 272, normalized size = 2.45 \begin{align*} \begin{cases} \frac{420 a^{7} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{420 a^{7}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{420 a^{6} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt{x}} - \frac{210 a^{5} b^{2} x}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{70 a^{4} b^{3} x^{\frac{3}{2}}}{30 a b^{8} + 30 b^{9} \sqrt{x}} - \frac{35 a^{3} b^{4} x^{2}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{21 a^{2} b^{5} x^{\frac{5}{2}}}{30 a b^{8} + 30 b^{9} \sqrt{x}} - \frac{14 a b^{6} x^{3}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{10 b^{7} x^{\frac{7}{2}}}{30 a b^{8} + 30 b^{9} \sqrt{x}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09456, size = 135, normalized size = 1.22 \begin{align*} \frac{14 \, a^{6} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{8}} + \frac{2 \, a^{7}}{{\left (b \sqrt{x} + a\right )} b^{8}} + \frac{10 \, b^{10} x^{3} - 24 \, a b^{9} x^{\frac{5}{2}} + 45 \, a^{2} b^{8} x^{2} - 80 \, a^{3} b^{7} x^{\frac{3}{2}} + 150 \, a^{4} b^{6} x - 360 \, a^{5} b^{5} \sqrt{x}}{30 \, b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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