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 = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 206
Rule 2029
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.39, size = 74, normalized size = 2.31 \[ \left [\frac {\log \left (\frac {2 \, b x^{2} + a x + 2 \, \sqrt {b x^{4} + a x^{3}} \sqrt {b}}{x}\right )}{\sqrt {b}}, -\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{4} + a x^{3}} \sqrt {-b}}{b x^{2}}\right )}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 23, normalized size = 0.72 \[ -\frac {2 \, \arctan \left (\frac {\sqrt {b + \frac {a}{x}}}{\sqrt {-b}}\right )}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 56, normalized size = 1.75 \[ \frac {\sqrt {\left (b x +a \right ) x}\, x \ln \left (\frac {2 b x +a +2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}}{2 \sqrt {b}}\right )}{\sqrt {b \,x^{4}+a \,x^{3}}\, \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {b x^{4} + a x^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{\sqrt {b\,x^4+a\,x^3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {x^{3} \left (a + b x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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