Optimal. Leaf size=52 \[ -\frac {\sqrt {a x^2+b x^3}}{x^2}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{\sqrt {a}} \]
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Rubi [A]
time = 0.03, 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 = {2045, 2033,
212} \begin {gather*} -\frac {\sqrt {a x^2+b x^3}}{x^2}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2033
Rule 2045
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+b x^3}}{x^3} \, dx &=-\frac {\sqrt {a x^2+b x^3}}{x^2}+\frac {1}{2} b \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx\\ &=-\frac {\sqrt {a x^2+b x^3}}{x^2}-b \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )\\ &=-\frac {\sqrt {a x^2+b x^3}}{x^2}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 64, normalized size = 1.23 \begin {gather*} -\frac {\sqrt {a+b x} \left (\sqrt {a} \sqrt {a+b x}+b x \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{\sqrt {a} \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.51, size = 56, normalized size = 1.08
method | result | size |
default | \(-\frac {\sqrt {b \,x^{3}+a \,x^{2}}\, \left (\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b x +\sqrt {b x +a}\, \sqrt {a}\right )}{x^{2} \sqrt {b x +a}\, \sqrt {a}}\) | \(56\) |
risch | \(-\frac {\sqrt {x^{2} \left (b x +a \right )}}{x^{2}}-\frac {b \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {x^{2} \left (b x +a \right )}}{\sqrt {a}\, x \sqrt {b x +a}}\) | \(57\) |
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 = 1.42, size = 127, normalized size = 2.44 \begin {gather*} \left [\frac {\sqrt {a} b x^{2} \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}{2 \, a x^{2}}, \frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) - \sqrt {b x^{3} + a x^{2}} a}{a 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 {\sqrt {x^{2} \left (a + b x\right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 45, normalized size = 0.87 \begin {gather*} \frac {\frac {b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a}} - \frac {\sqrt {b x + a} b \mathrm {sgn}\left (x\right )}{x}}{b} \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 {\sqrt {b\,x^3+a\,x^2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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