Optimal. Leaf size=60 \[ \frac {\sqrt {a x^2+b x^3}}{b \sqrt {x}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{b^{3/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2049, 2054,
212} \begin {gather*} \frac {\sqrt {a x^2+b x^3}}{b \sqrt {x}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\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^{3/2}}{\sqrt {a x^2+b x^3}} \, dx &=\frac {\sqrt {a x^2+b x^3}}{b \sqrt {x}}-\frac {a \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^3}} \, dx}{2 b}\\ &=\frac {\sqrt {a x^2+b x^3}}{b \sqrt {x}}-\frac {a \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{b}\\ &=\frac {\sqrt {a x^2+b x^3}}{b \sqrt {x}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 71, normalized size = 1.18 \begin {gather*} \frac {\sqrt {b} x^{3/2} (a+b x)+a x \sqrt {a+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{b^{3/2} \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 79, normalized size = 1.32
method | result | size |
risch | \(\frac {x^{\frac {3}{2}} \left (b x +a \right )}{b \sqrt {x^{2} \left (b x +a \right )}}-\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x}\, \sqrt {x \left (b x +a \right )}}{2 b^{\frac {3}{2}} \sqrt {x^{2} \left (b x +a \right )}}\) | \(78\) |
default | \(\frac {\sqrt {x}\, \left (2 b^{\frac {5}{2}} x^{2}+2 b^{\frac {3}{2}} a x -a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) b \right )}{2 \sqrt {b \,x^{3}+a \,x^{2}}\, b^{\frac {5}{2}}}\) | \(79\) |
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 = 2.90, size = 131, 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^{3} + a x^{2}} \sqrt {b} \sqrt {x}}{x}\right ) + 2 \, \sqrt {b x^{3} + a x^{2}} b \sqrt {x}}{2 \, b^{2} x}, \frac {a \sqrt {-b} x \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-b}}{b x^{\frac {3}{2}}}\right ) + \sqrt {b x^{3} + a x^{2}} b \sqrt {x}}{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^{\frac {3}{2}}}{\sqrt {x^{2} \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 = 1.63, size = 55, normalized size = 0.92 \begin {gather*} -\frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, b^{\frac {3}{2}}} + \frac {\frac {a \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{b^{\frac {3}{2}}} + \frac {\sqrt {b x + a} \sqrt {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^{3/2}}{\sqrt {b\,x^3+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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