Optimal. Leaf size=56 \[ \frac {\sqrt {a x^3+b x^4}}{b x}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2024, 2029, 206} \begin {gather*} \frac {\sqrt {a x^3+b x^4}}{b x}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2024
Rule 2029
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a x^3+b x^4}} \, dx &=\frac {\sqrt {a x^3+b x^4}}{b x}-\frac {a \int \frac {x}{\sqrt {a x^3+b x^4}} \, dx}{2 b}\\ &=\frac {\sqrt {a x^3+b x^4}}{b x}-\frac {a \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a x^3+b x^4}}\right )}{b}\\ &=\frac {\sqrt {a x^3+b x^4}}{b x}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^3+b x^4}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 1.34 \begin {gather*} \frac {\sqrt {b} x^2 (a+b x)-a^{3/2} x^{3/2} \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2} \sqrt {x^3 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 81, normalized size = 1.45 \begin {gather*} -\frac {a \log (x)}{2 b^{3/2}}+\frac {a \log \left (-2 b^{3/2} \sqrt {a x^3+b x^4}+a b x+2 b^2 x^2\right )}{2 b^{3/2}}+\frac {\sqrt {a x^3+b x^4}}{b x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 122, normalized size = 2.18 \begin {gather*} \left [\frac {a \sqrt {b} x \log \left (\frac {2 \, b x^{2} + a x - 2 \, \sqrt {b x^{4} + a x^{3}} \sqrt {b}}{x}\right ) + 2 \, \sqrt {b x^{4} + a x^{3}} b}{2 \, b^{2} x}, \frac {a \sqrt {-b} x \arctan \left (\frac {\sqrt {b x^{4} + a x^{3}} \sqrt {-b}}{b x^{2}}\right ) + \sqrt {b x^{4} + a x^{3}} b}{b^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 48, normalized size = 0.86 \begin {gather*} \frac {\frac {a \sqrt {b + \frac {a}{x}} x}{b} + \frac {a^{2} \arctan \left (\frac {\sqrt {b + \frac {a}{x}}}{\sqrt {-b}}\right )}{\sqrt {-b} b}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 78, normalized size = 1.39 \begin {gather*} \frac {\sqrt {\left (b x +a \right ) x}\, \left (-a b \ln \left (\frac {2 b x +a +2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}}{2 \sqrt {b}}\right )+2 \sqrt {b \,x^{2}+a x}\, b^{\frac {3}{2}}\right ) x}{2 \sqrt {b \,x^{4}+a \,x^{3}}\, b^{\frac {5}{2}}} \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\sqrt {b x^{4} + a x^{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2}{\sqrt {b\,x^4+a\,x^3}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\sqrt {x^{3} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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