Optimal. Leaf size=52 \[ \frac {2 x}{a \sqrt {a x^2+b x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{3/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 52, 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 = {2048, 2033,
212} \begin {gather*} \frac {2 x}{a \sqrt {a x^2+b x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2033
Rule 2048
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a x^2+b x^3\right )^{3/2}} \, dx &=\frac {2 x}{a \sqrt {a x^2+b x^3}}+\frac {\int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{a}\\ &=\frac {2 x}{a \sqrt {a x^2+b x^3}}-\frac {2 \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{a}\\ &=\frac {2 x}{a \sqrt {a x^2+b x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 54, normalized size = 1.04 \begin {gather*} \frac {2 x \left (\sqrt {a}-\sqrt {a+b x} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{a^{3/2} \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 54, normalized size = 1.04
method | result | size |
default | \(-\frac {2 x^{3} \left (b x +a \right ) \left (\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a \sqrt {b x +a}-a^{\frac {3}{2}}\right )}{\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {5}{2}}}\) | \(54\) |
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 = 1.47, size = 156, normalized size = 3.00 \begin {gather*} \left [\frac {{\left (b x^{2} + a x\right )} \sqrt {a} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, \sqrt {b x^{3} + a x^{2}} a}{a^{2} b x^{2} + a^{3} x}, \frac {2 \, {\left ({\left (b x^{2} + a x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + \sqrt {b x^{3} + a x^{2}} a\right )}}{a^{2} b x^{2} + a^{3} 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}}{\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 77, normalized size = 1.48 \begin {gather*} -\frac {2 \, {\left (\sqrt {a} \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + \sqrt {-a}\right )} \mathrm {sgn}\left (x\right )}{\sqrt {-a} a^{\frac {3}{2}}} + \frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (x\right )} + \frac {2}{\sqrt {b x + a} a \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}{{\left (b\,x^3+a\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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