Optimal. Leaf size=95 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a x^2+b x^3}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{a x^2+b x^3}}{4 b^2 \sqrt{x}}+\frac{\sqrt{x} \sqrt{a x^2+b x^3}}{2 b} \]
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Rubi [A] time = 0.125441, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2024, 2029, 206} \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a x^2+b x^3}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{a x^2+b x^3}}{4 b^2 \sqrt{x}}+\frac{\sqrt{x} \sqrt{a x^2+b x^3}}{2 b} \]
Antiderivative was successfully verified.
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Rule 2024
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{\sqrt{a x^2+b x^3}} \, dx &=\frac{\sqrt{x} \sqrt{a x^2+b x^3}}{2 b}-\frac{(3 a) \int \frac{x^{3/2}}{\sqrt{a x^2+b x^3}} \, dx}{4 b}\\ &=-\frac{3 a \sqrt{a x^2+b x^3}}{4 b^2 \sqrt{x}}+\frac{\sqrt{x} \sqrt{a x^2+b x^3}}{2 b}+\frac{\left (3 a^2\right ) \int \frac{\sqrt{x}}{\sqrt{a x^2+b x^3}} \, dx}{8 b^2}\\ &=-\frac{3 a \sqrt{a x^2+b x^3}}{4 b^2 \sqrt{x}}+\frac{\sqrt{x} \sqrt{a x^2+b x^3}}{2 b}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^{3/2}}{\sqrt{a x^2+b x^3}}\right )}{4 b^2}\\ &=-\frac{3 a \sqrt{a x^2+b x^3}}{4 b^2 \sqrt{x}}+\frac{\sqrt{x} \sqrt{a x^2+b x^3}}{2 b}+\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a x^2+b x^3}}\right )}{4 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0519435, size = 90, normalized size = 0.95 \[ \frac{\sqrt{b} x^{3/2} \left (-3 a^2-a b x+2 b^2 x^2\right )+3 a^{5/2} x \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{5/2} \sqrt{x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 92, normalized size = 1. \begin{align*}{\frac{1}{8}\sqrt{x} \left ( 4\,{b}^{7/2}{x}^{3}-2\,{b}^{5/2}{x}^{2}a-6\,{b}^{3/2}x{a}^{2}+3\,\sqrt{x \left ( bx+a \right ) }\ln \left ( 1/2\,{\frac{2\,\sqrt{b{x}^{2}+ax}\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{2}b \right ){\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}{b}^{-{\frac{7}{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{5}{2}}}{\sqrt{b x^{3} + a x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.830067, size = 375, normalized size = 3.95 \begin{align*} \left [\frac{3 \, a^{2} \sqrt{b} x \log \left (\frac{2 \, b x^{2} + a x + 2 \, \sqrt{b x^{3} + a x^{2}} \sqrt{b} \sqrt{x}}{x}\right ) + 2 \, \sqrt{b x^{3} + a x^{2}}{\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt{x}}{8 \, b^{3} x}, -\frac{3 \, a^{2} \sqrt{-b} x \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-b}}{b x^{\frac{3}{2}}}\right ) - \sqrt{b x^{3} + a x^{2}}{\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt{x}}{4 \, b^{3} x}\right ] \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{5}{2}}}{\sqrt{x^{2} \left (a + b x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29259, size = 70, normalized size = 0.74 \begin{align*} \frac{1}{4} \, \sqrt{b x + a} \sqrt{x}{\left (\frac{2 \, x}{b} - \frac{3 \, a}{b^{2}}\right )} - \frac{3 \, a^{2} \log \left ({\left | -\sqrt{b} \sqrt{x} + \sqrt{b x + a} \right |}\right )}{4 \, b^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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