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 = 0.06, 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 = {2049, 2054,
212} \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 212
Rule 2049
Rule 2054
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 \text {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 = 0.01, size = 73, normalized size = 1.30 \begin {gather*} \frac {\sqrt {b} x^2 (a+b x)+a x^{3/2} \sqrt {a+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{b^{3/2} \sqrt {x^3 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 78, normalized size = 1.39
method | result | size |
risch | \(\frac {x^{2} \left (b x +a \right )}{b \sqrt {x^{3} \left (b x +a \right )}}-\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) x \sqrt {x \left (b x +a \right )}}{2 b^{\frac {3}{2}} \sqrt {x^{3} \left (b x +a \right )}}\) | \(76\) |
default | \(\frac {x \sqrt {x \left (b x +a \right )}\, \left (2 \sqrt {b \,x^{2}+a x}\, b^{\frac {3}{2}}-a \ln \left (\frac {2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) b \right )}{2 \sqrt {b \,x^{4}+a \,x^{3}}\, b^{\frac {5}{2}}}\) | \(78\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.52, 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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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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Giac [A]
time = 0.61, size = 71, normalized size = 1.27 \begin {gather*} -\frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, b^{\frac {3}{2}}} + \frac {a \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right )}{2 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} + \frac {\sqrt {b x^{2} + a x}}{b \mathrm {sgn}\left (x\right )} \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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