Optimal. Leaf size=75 \[ \frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{5/2}}-\frac {3 \sqrt {a x^2+b x^3}}{a^2 x^2}+\frac {2}{a \sqrt {a x^2+b x^3}} \]
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Rubi [A] time = 0.09, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2023, 2025, 2008, 206} \begin {gather*} -\frac {3 \sqrt {a x^2+b x^3}}{a^2 x^2}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{5/2}}+\frac {2}{a \sqrt {a x^2+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2023
Rule 2025
Rubi steps
\begin {align*} \int \frac {x}{\left (a x^2+b x^3\right )^{3/2}} \, dx &=\frac {2}{a \sqrt {a x^2+b x^3}}+\frac {3 \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx}{a}\\ &=\frac {2}{a \sqrt {a x^2+b x^3}}-\frac {3 \sqrt {a x^2+b x^3}}{a^2 x^2}-\frac {(3 b) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{2 a^2}\\ &=\frac {2}{a \sqrt {a x^2+b x^3}}-\frac {3 \sqrt {a x^2+b x^3}}{a^2 x^2}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{a^2}\\ &=\frac {2}{a \sqrt {a x^2+b x^3}}-\frac {3 \sqrt {a x^2+b x^3}}{a^2 x^2}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 36, normalized size = 0.48 \begin {gather*} -\frac {2 b x \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {b x}{a}+1\right )}{a^2 \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.79, size = 76, normalized size = 1.01 \begin {gather*} \frac {x \sqrt {a+b x} \left (\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2 a-3 (a+b x)}{a^2 x \sqrt {a+b x}}\right )}{\sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 189, normalized size = 2.52 \begin {gather*} \left [\frac {3 \, {\left (b^{2} x^{3} + a b x^{2}\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}} {\left (3 \, a b x + a^{2}\right )}}{2 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, -\frac {3 \, {\left (b^{2} x^{3} + a b x^{2}\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}} {\left (3 \, a b x + a^{2}\right )}}{a^{3} b x^{3} + a^{4} x^{2}}\right ] \end {gather*}
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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 62, normalized size = 0.83 \begin {gather*} \frac {\left (b x +a \right ) \left (3 \sqrt {b x +a}\, b x \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-3 \sqrt {a}\, b x -a^{\frac {3}{2}}\right ) x^{2}}{\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\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}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{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 = -0.01 \begin {gather*} \int \frac {x}{{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\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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