Optimal. Leaf size=32 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0343985, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2029, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a x^3+b x^4}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a x^3+b x^4}}\right )\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0166366, size = 59, normalized size = 1.84 \[ \frac{2 \sqrt{a} x^{3/2} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{x^3 (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 56, normalized size = 1.8 \begin{align*}{x\sqrt{x \left ( bx+a \right ) }\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{b{x}^{2}+ax}\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ){\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}{\frac{1}{\sqrt{b}}}} \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}{\sqrt{b x^{4} + a x^{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.820835, size = 173, normalized size = 5.41 \begin{align*} \left [\frac{\log \left (\frac{2 \, b x^{2} + a x + 2 \, \sqrt{b x^{4} + a x^{3}} \sqrt{b}}{x}\right )}{\sqrt{b}}, -\frac{2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{4} + a x^{3}} \sqrt{-b}}{b x^{2}}\right )}{b}\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}{\sqrt{x^{3} \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 = 1.39866, size = 31, normalized size = 0.97 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{b + \frac{a}{x}}}{\sqrt{-b}}\right )}{\sqrt{-b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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