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 = 0.06, 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 = {2023, 2008, 206} \[ \frac {2 x}{a \sqrt {a x^2+b x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2023
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 \operatorname {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 [C] time = 0.03, size = 35, normalized size = 0.67 \[ \frac {2 x \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x}{a}+1\right )}{a \sqrt {x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 156, normalized size = 3.00 \[ \left [\frac {{\left (b x^{2} + a x\right )} \sqrt {a} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, \sqrt {b x^{3} + a x^{2}} a}{a^{2} b x^{2} + a^{3} x}, \frac {2 \, {\left ({\left (b x^{2} + a x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + \sqrt {b x^{3} + a x^{2}} a\right )}}{a^{2} b x^{2} + a^{3} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 54, normalized size = 1.04 \[ -\frac {2 \left (b x +a \right ) \left (\sqrt {b x +a}\, a \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-a^{\frac {3}{2}}\right ) x^{3}}{\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {5}{2}}} \]
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 \[ \int \frac {x^{2}}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{{\left (b\,x^3+a\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2}}{\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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