Optimal. Leaf size=56 \[ \frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 56, 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 = {666, 660, 207} \begin {gather*} \frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 666
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{b}\\ &=\frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{b}\\ &=\frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.66 \begin {gather*} \frac {2 \sqrt {x} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c x}{b}+1\right )}{b \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 63, normalized size = 1.12 \begin {gather*} \frac {2 \sqrt {b x+c x^2}}{b \sqrt {x} (b+c x)}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 155, normalized size = 2.77 \begin {gather*} \left [\frac {{\left (c x^{2} + b x\right )} \sqrt {b} \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} b \sqrt {x}}{b^{2} c x^{2} + b^{3} x}, \frac {2 \, {\left ({\left (c x^{2} + b x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + \sqrt {c x^{2} + b x} b \sqrt {x}\right )}}{b^{2} c x^{2} + b^{3} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 67, normalized size = 1.20 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {2 \, {\left (\sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + \sqrt {-b}\right )}}{\sqrt {-b} b^{\frac {3}{2}}} + \frac {2}{\sqrt {c x + b} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 51, normalized size = 0.91 \begin {gather*} -\frac {2 \sqrt {\left (c x +b \right ) x}\, \left (\sqrt {c x +b}\, \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-\sqrt {b}\right )}{\left (c x +b \right ) b^{\frac {3}{2}} \sqrt {x}} \end {gather*}
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*} \int \frac {\sqrt {x}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \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 {x}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \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 (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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