Optimal. Leaf size=121 \[ \frac {g \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{\sqrt {c} e^2}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x) (2 c d-b e)} \]
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Rubi [A] time = 0.16, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {792, 621, 204} \begin {gather*} \frac {g \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{\sqrt {c} e^2}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x) (2 c d-b e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 621
Rule 792
Rubi steps
\begin {align*} \int \frac {f+g x}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)}+\frac {g \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{e}\\ &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)}+\frac {(2 g) \operatorname {Subst}\left (\int \frac {1}{-4 c e^2-x^2} \, dx,x,\frac {-b e^2-2 c e^2 x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{e}\\ &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)}+\frac {g \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{\sqrt {c} e^2}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 189, normalized size = 1.56 \begin {gather*} -\frac {2 \left (\sqrt {c} \sqrt {e (2 c d-b e)} (e f-d g) (b e-c d+c e x)+\sqrt {e} g \sqrt {d+e x} (b e-2 c d)^2 \sqrt {\frac {b e-c d+c e x}{b e-2 c d}} \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )\right )}{\sqrt {c} e^2 \sqrt {e (2 c d-b e)} (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 [B] time = 1.63, size = 250, normalized size = 2.07 \begin {gather*} \frac {g \sqrt {-c e^2} \log \left (b^2 e^2-8 c x \sqrt {-c e^2} \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}-4 b c d e-4 b c e^2 x+4 c^2 d^2-8 c^2 e^2 x^2\right )}{2 c e^3}+\frac {2 (e f-d g) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{e^2 (d+e x) (b e-2 c d)}+\frac {g \tan ^{-1}\left (\frac {2 \sqrt {c} x \sqrt {-c e^2}}{b e}-\frac {2 \sqrt {c} \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{b e}\right )}{\sqrt {c} e^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 405, normalized size = 3.35 \begin {gather*} \left [-\frac {{\left ({\left (2 \, c d e - b e^{2}\right )} g x + {\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c e f - c d g\right )}}{2 \, {\left (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 )}}, -\frac {{\left ({\left (2 \, c d e - b e^{2}\right )} g x + {\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c e f - c d g\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(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 134, normalized size = 1.11 \begin {gather*} \frac {g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{\sqrt {c \,e^{2}}\, e}-\frac {2 \left (-d g +e f \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} c \,e^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right ) e^{2}} \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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {f+g\,x}{\left (d+e\,x\right )\,\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 )} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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