Optimal. Leaf size=51 \[ \frac {2 \sqrt {a x^2+b x^3}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 51, 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 = {2046, 2033,
212} \begin {gather*} \frac {2 \sqrt {a x^2+b x^3}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2033
Rule 2046
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+b x^3}}{x^2} \, dx &=\frac {2 \sqrt {a x^2+b x^3}}{x}+a \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx\\ &=\frac {2 \sqrt {a x^2+b x^3}}{x}-(2 a) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )\\ &=\frac {2 \sqrt {a x^2+b x^3}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 53, normalized size = 1.04 \begin {gather*} \frac {2 x \left (a+b x-\sqrt {a} \sqrt {a+b x} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{\sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 51, normalized size = 1.00
method | result | size |
default | \(\frac {2 \sqrt {b \,x^{3}+a \,x^{2}}\, \left (-\sqrt {a}\, \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\sqrt {b x +a}\right )}{x \sqrt {b x +a}}\) | \(51\) |
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 = 1.80, size = 111, normalized size = 2.18 \begin {gather*} \left [\frac {\sqrt {a} x \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}}}{x}, \frac {2 \, {\left (\sqrt {-a} x \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + \sqrt {b x^{3} + a x^{2}}\right )}}{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 {\sqrt {x^{2} \left (a + b x\right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.26, size = 67, normalized size = 1.31 \begin {gather*} \frac {2 \, a \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a}} + 2 \, \sqrt {b x + a} \mathrm {sgn}\left (x\right ) - \frac {2 \, {\left (a \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + \sqrt {-a} \sqrt {a}\right )} \mathrm {sgn}\left (x\right )}{\sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.36, size = 73, normalized size = 1.43 \begin {gather*} \frac {2\,\sqrt {b\,x^3+a\,x^2}}{x}+\frac {\sqrt {a}\,\mathrm {asin}\left (\frac {\sqrt {a}\,\sqrt {\frac {1}{x}}\,1{}\mathrm {i}}{\sqrt {b}}\right )\,\sqrt {b\,x^3+a\,x^2}\,{\left (\frac {1}{x}\right )}^{3/2}\,2{}\mathrm {i}}{\sqrt {b}\,\sqrt {\frac {a}{b\,x}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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