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 = 0.05, 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 = {2020, 2008, 206} \[ -\frac {\sqrt {a x^2+b x^3}}{x^2}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
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 \operatorname {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 = 48, normalized size = 0.92 \[ -\frac {b x \sqrt {\frac {b x}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x}{a}+1}\right )+a+b x}{\sqrt {x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 127, normalized size = 2.44 \[ \left [\frac {\sqrt {a} b x^{2} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) - 2 \, \sqrt {b x^{3} + a x^{2}} a}{2 \, a x^{2}}, \frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) - \sqrt {b x^{3} + a x^{2}} a}{a x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 45, normalized size = 0.87 \[ \frac {\frac {b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-a}} - \frac {\sqrt {b x + a} b \mathrm {sgn}\relax (x)}{x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 56, normalized size = 1.08 \[ -\frac {\sqrt {b \,x^{3}+a \,x^{2}}\, \left (b x \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\sqrt {b x +a}\, \sqrt {a}\right )}{\sqrt {b x +a}\, \sqrt {a}\, x^{2}} \]
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 \[ \int \frac {\sqrt {b x^{3} + a x^{2}}}{x^{3}}\,{d x} \]
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 \[ \int \frac {\sqrt {b\,x^3+a\,x^2}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {x^{2} \left (a + b x\right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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