Optimal. Leaf size=131 \[ -\frac {2 (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2 \sqrt {2 c d-b e}}-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 \sqrt {d+e x}} \]
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Rubi [A] time = 0.20, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {794, 660, 208} \begin {gather*} -\frac {2 (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2 \sqrt {2 c d-b e}}-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 660
Rule 794
Rubi steps
\begin {align*} \int \frac {f+g x}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 \sqrt {d+e x}}-\frac {\left (2 \left (\frac {1}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {1}{2} \left (c e^3 f-\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c e^3}\\ &=-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 \sqrt {d+e x}}+(2 (e f-d g)) \operatorname {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 \sqrt {d+e x}}-\frac {2 (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2 \sqrt {2 c d-b e}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 148, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {d+e x} \left (g (2 c d-b e) (c (d-e x)-b e)-c \sqrt {2 c d-b e} (d g-e f) \sqrt {c (d-e x)-b e} \tanh ^{-1}\left (\frac {\sqrt {-b e+c d-c e x}}{\sqrt {2 c d-b e}}\right )\right )}{c e^2 (b e-2 c d) \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.45, size = 144, normalized size = 1.10 \begin {gather*} -\frac {2 (d g-e f) \tan ^{-1}\left (\frac {\sqrt {b e-2 c d} \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2}}{\sqrt {d+e x} (b e+c (d+e x)-2 c d)}\right )}{e^2 \sqrt {b e-2 c d}}-\frac {2 g \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2}}{c e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 442, normalized size = 3.37 \begin {gather*} \left [-\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c d - b e\right )} \sqrt {e x + d} g + {\left (c d e f - c d^{2} g + {\left (c e^{2} f - c d e g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, c^{2} d^{2} e^{2} - b c d e^{3} + {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x}, -\frac {2 \, {\left (\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c d - b e\right )} \sqrt {e x + d} g + {\left (c d e f - c d^{2} g + {\left (c e^{2} f - c d e g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right )\right )}}{2 \, c^{2} d^{2} e^{2} - b c d e^{3} + {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {g x + f}{\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 161, normalized size = 1.23 \begin {gather*} -\frac {2 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (c d g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-c e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+\sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, g \right )}{\sqrt {e x +d}\, \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,e^{2}} \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 {g x + f}{\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}\,{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 = -0.01 \begin {gather*} \int \frac {f+g\,x}{\sqrt {d+e\,x}\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \,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 {f + g x}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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