Optimal. Leaf size=74 \[ \frac {2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} \sqrt {d e-c f}}+\frac {2 b \sqrt {e+f x}}{d f} \]
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Rubi [A] time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {80, 63, 208} \begin {gather*} \frac {2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} \sqrt {d e-c f}}+\frac {2 b \sqrt {e+f x}}{d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 208
Rubi steps
\begin {align*} \int \frac {a+b x}{(c+d x) \sqrt {e+f x}} \, dx &=\frac {2 b \sqrt {e+f x}}{d f}+\frac {\left (2 \left (-\frac {1}{2} b c f+\frac {a d f}{2}\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d f}\\ &=\frac {2 b \sqrt {e+f x}}{d f}-\frac {(2 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{d f}\\ &=\frac {2 b \sqrt {e+f x}}{d f}+\frac {2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} \sqrt {d e-c f}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 74, normalized size = 1.00 \begin {gather*} \frac {2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} \sqrt {d e-c f}}+\frac {2 b \sqrt {e+f x}}{d f} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 84, normalized size = 1.14 \begin {gather*} \frac {2 b \sqrt {e+f x}}{d f}-\frac {2 (a d-b c) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x} \sqrt {c f-d e}}{d e-c f}\right )}{d^{3/2} \sqrt {c f-d e}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.38, size = 209, normalized size = 2.82 \begin {gather*} \left [-\frac {\sqrt {d^{2} e - c d f} {\left (b c - a d\right )} f \log \left (\frac {d f x + 2 \, d e - c f - 2 \, \sqrt {d^{2} e - c d f} \sqrt {f x + e}}{d x + c}\right ) - 2 \, {\left (b d^{2} e - b c d f\right )} \sqrt {f x + e}}{d^{3} e f - c d^{2} f^{2}}, -\frac {2 \, {\left (\sqrt {-d^{2} e + c d f} {\left (b c - a d\right )} f \arctan \left (\frac {\sqrt {-d^{2} e + c d f} \sqrt {f x + e}}{d f x + d e}\right ) - {\left (b d^{2} e - b c d f\right )} \sqrt {f x + e}\right )}}{d^{3} e f - c d^{2} f^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 70, normalized size = 0.95 \begin {gather*} -\frac {2 \, {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{\sqrt {c d f - d^{2} e} d} + \frac {2 \, \sqrt {f x + e} b}{d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 1.30 \begin {gather*} \frac {2 a \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}}-\frac {2 b c \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}\, d}+\frac {2 \sqrt {f x +e}\, b}{d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 62, normalized size = 0.84 \begin {gather*} \frac {2\,b\,\sqrt {e+f\,x}}{d\,f}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}}{\sqrt {c\,f-d\,e}}\right )\,\left (a\,d-b\,c\right )}{d^{3/2}\,\sqrt {c\,f-d\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.34, size = 66, normalized size = 0.89 \begin {gather*} \frac {2 b \sqrt {e + f x}}{d f} - \frac {2 \left (a d - b c\right ) \operatorname {atan}{\left (\frac {1}{\sqrt {\frac {d}{c f - d e}} \sqrt {e + f x}} \right )}}{d \sqrt {\frac {d}{c f - d e}} \left (c f - d e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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