Optimal. Leaf size=113 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{2 a^{7/2}}-\frac{15 b \sqrt{a x+b \sqrt{x}}}{2 a^3}+\frac{5 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{a^2}-\frac{4 x^{3/2}}{a \sqrt{a x+b \sqrt{x}}} \]
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Rubi [A] time = 0.102718, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2018, 668, 670, 640, 620, 206} \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{2 a^{7/2}}-\frac{15 b \sqrt{a x+b \sqrt{x}}}{2 a^3}+\frac{5 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{a^2}-\frac{4 x^{3/2}}{a \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 668
Rule 670
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\left (b \sqrt{x}+a x\right )^{3/2}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^4}{\left (b x+a x^2\right )^{3/2}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 x^{3/2}}{a \sqrt{b \sqrt{x}+a x}}+\frac{10 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{a}\\ &=-\frac{4 x^{3/2}}{a \sqrt{b \sqrt{x}+a x}}+\frac{5 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{a^2}-\frac{(15 b) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{2 a^2}\\ &=-\frac{4 x^{3/2}}{a \sqrt{b \sqrt{x}+a x}}-\frac{15 b \sqrt{b \sqrt{x}+a x}}{2 a^3}+\frac{5 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{a^2}+\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{4 a^3}\\ &=-\frac{4 x^{3/2}}{a \sqrt{b \sqrt{x}+a x}}-\frac{15 b \sqrt{b \sqrt{x}+a x}}{2 a^3}+\frac{5 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{a^2}+\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{2 a^3}\\ &=-\frac{4 x^{3/2}}{a \sqrt{b \sqrt{x}+a x}}-\frac{15 b \sqrt{b \sqrt{x}+a x}}{2 a^3}+\frac{5 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{a^2}+\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{2 a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.049401, size = 62, normalized size = 0.55 \[ \frac{4 x^2 \sqrt{\frac{a \sqrt{x}}{b}+1} \, _2F_1\left (\frac{3}{2},\frac{7}{2};\frac{9}{2};-\frac{a \sqrt{x}}{b}\right )}{7 b \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 440, normalized size = 3.9 \begin{align*}{\frac{1}{4}\sqrt{b\sqrt{x}+ax} \left ( 4\,\sqrt{b\sqrt{x}+ax}{a}^{9/2}{x}^{3/2}+10\,\sqrt{b\sqrt{x}+ax}{a}^{7/2}xb-32\,{a}^{7/2}x\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }b+16\,{a}^{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{b}^{2}+8\,\sqrt{b\sqrt{x}+ax}{a}^{5/2}\sqrt{x}{b}^{2}-64\,{a}^{5/2}\sqrt{x}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{2}+16\,{a}^{5/2} \left ( \sqrt{x} \left ( b+a\sqrt{x} \right ) \right ) ^{3/2}b+32\,{a}^{2}\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}{b}^{3}-\ln \left ({\frac{1}{2} \left ( 2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b \right ){\frac{1}{\sqrt{a}}}} \right ) x{a}^{3}{b}^{2}+2\,\sqrt{b\sqrt{x}+ax}{a}^{3/2}{b}^{3}-32\,{a}^{3/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{3}+16\,a\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}^{4}-2\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ) \sqrt{x}{a}^{2}{b}^{3}-\ln \left ({\frac{1}{2} \left ( 2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b \right ){\frac{1}{\sqrt{a}}}} \right ) a{b}^{4} \right ){a}^{-{\frac{9}{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^{\frac{3}{2}}}{{\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^{\frac{3}{2}}}{\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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