Optimal. Leaf size=32 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {674, 213}
\begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 674
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx &=2 \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 48, normalized size = 1.50 \begin {gather*} -\frac {2 \sqrt {x} \sqrt {b+c x} \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 37, normalized size = 1.16
method | result | size |
default | \(-\frac {2 \sqrt {x \left (c x +b \right )}\, \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )}{\sqrt {x}\, \sqrt {c x +b}\, \sqrt {b}}\) | \(37\) |
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 = 1.38, size = 71, normalized size = 2.22 \begin {gather*} \left [\frac {\log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right )}{\sqrt {b}}, \frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right )}{b}\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 {1}{\sqrt {x} \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.90, size = 39, normalized size = 1.22 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {2 \, \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right )}{\sqrt {-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 = -0.03 \begin {gather*} \int \frac {1}{\sqrt {x}\,\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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