Optimal. Leaf size=75 \[ \frac {2}{a \sqrt {a x^2+b x^3}}-\frac {3 \sqrt {a x^2+b x^3}}{a^2 x^2}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{5/2}} \]
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Rubi [A]
time = 0.06, 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 = {2048, 2050,
2033, 212} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2033
Rule 2048
Rule 2050
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) \text {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 [A]
time = 0.06, size = 62, normalized size = 0.83 \begin {gather*} \frac {-\sqrt {a} (a+3 b x)+3 b x \sqrt {a+b x} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 62, normalized size = 0.83
method | result | size |
default | \(\frac {x^{2} \left (b x +a \right ) \left (3 \sqrt {b x +a}\, \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b x -3 b x \sqrt {a}-a^{\frac {3}{2}}\right )}{\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {5}{2}}}\) | \(62\) |
risch | \(-\frac {b x +a}{a^{2} \sqrt {x^{2} \left (b x +a \right )}}-\frac {b \left (-\frac {6 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {4}{\sqrt {b x +a}}\right ) \sqrt {b x +a}\, x}{2 a^{2} \sqrt {x^{2} \left (b x +a \right )}}\) | \(75\) |
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 = 1.60, 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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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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Giac [A]
time = 1.18, size = 72, normalized size = 0.96 \begin {gather*} -\frac {3 \, b \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (x\right )} - \frac {3 \, {\left (b x + a\right )} b - 2 \, a b}{{\left ({\left (b x + a\right )}^{\frac {3}{2}} - \sqrt {b x + a} a\right )} a^{2} \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 = -0.01 \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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