Optimal. Leaf size=53 \[ \frac {2 \sqrt {b x+c x^2}}{\sqrt {x}}-2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 53, 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 = {678, 674, 213}
\begin {gather*} \frac {2 \sqrt {b x+c x^2}}{\sqrt {x}}-2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 674
Rule 678
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+c x^2}}{x^{3/2}} \, dx &=\frac {2 \sqrt {b x+c x^2}}{\sqrt {x}}+b \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx\\ &=\frac {2 \sqrt {b x+c x^2}}{\sqrt {x}}+(2 b) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )\\ &=\frac {2 \sqrt {b x+c x^2}}{\sqrt {x}}-2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 60, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {x} \sqrt {b+c x} \left (\sqrt {b+c x}-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{\sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 48, normalized size = 0.91
method | result | size |
default | \(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (\sqrt {b}\, \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-\sqrt {c x +b}\right )}{\sqrt {x}\, \sqrt {c x +b}}\) | \(48\) |
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 = 1.71, size = 111, normalized size = 2.09 \begin {gather*} \left [\frac {\sqrt {b} x \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, \sqrt {c x^{2} + b x} \sqrt {x}}{x}, \frac {2 \, {\left (\sqrt {-b} x \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + \sqrt {c x^{2} + b x} \sqrt {x}\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 \left (b + c x\right )}}{x^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 61, normalized size = 1.15 \begin {gather*} \frac {2 \, b \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + 2 \, \sqrt {c x + b} - \frac {2 \, {\left (b \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + \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 = -0.02 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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