Optimal. Leaf size=77 \[ \frac{6 \sqrt{a x+b \sqrt{x}}}{a^2}-\frac{6 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{5/2}}-\frac{4 x}{a \sqrt{a x+b \sqrt{x}}} \]
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Rubi [A] time = 0.0742663, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {2018, 668, 640, 620, 206} \[ \frac{6 \sqrt{a x+b \sqrt{x}}}{a^2}-\frac{6 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{5/2}}-\frac{4 x}{a \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 668
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\left (b \sqrt{x}+a x\right )^{3/2}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{\left (b x+a x^2\right )^{3/2}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 x}{a \sqrt{b \sqrt{x}+a x}}+\frac{6 \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{a}\\ &=-\frac{4 x}{a \sqrt{b \sqrt{x}+a x}}+\frac{6 \sqrt{b \sqrt{x}+a x}}{a^2}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=-\frac{4 x}{a \sqrt{b \sqrt{x}+a x}}+\frac{6 \sqrt{b \sqrt{x}+a x}}{a^2}-\frac{(6 b) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{a^2}\\ &=-\frac{4 x}{a \sqrt{b \sqrt{x}+a x}}+\frac{6 \sqrt{b \sqrt{x}+a x}}{a^2}-\frac{6 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0378811, size = 64, normalized size = 0.83 \[ \frac{4 x^{3/2} \sqrt{\frac{a \sqrt{x}}{b}+1} \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};-\frac{a \sqrt{x}}{b}\right )}{5 b \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 237, normalized size = 3.1 \begin{align*} -{\sqrt{b\sqrt{x}+ax} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ) x{a}^{2}b-6\,{a}^{5/2}x\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }+6\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ) \sqrt{x}a{b}^{2}-12\,{a}^{3/2}\sqrt{x}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }b+4\,{a}^{3/2} \left ( \sqrt{x} \left ( b+a\sqrt{x} \right ) \right ) ^{3/2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){b}^{3}-6\,\sqrt{a}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{2} \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}} \left ( b+a\sqrt{x} \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (a x + b \sqrt{x}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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