Optimal. Leaf size=122 \[ \frac{\left (A f^2-B e f+C e^2\right ) \tan ^{-1}\left (\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right )}{f^2 \sqrt{d^2 e^2-f^2}}-\frac{\sin ^{-1}(d x) (C e-B f)}{d f^2}-\frac{C \sqrt{1-d^2 x^2}}{d^2 f} \]
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Rubi [A] time = 0.310779, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {1609, 1654, 844, 216, 725, 204} \[ \frac{\left (A f^2-B e f+C e^2\right ) \tan ^{-1}\left (\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right )}{f^2 \sqrt{d^2 e^2-f^2}}-\frac{\sin ^{-1}(d x) (C e-B f)}{d f^2}-\frac{C \sqrt{1-d^2 x^2}}{d^2 f} \]
Antiderivative was successfully verified.
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Rule 1609
Rule 1654
Rule 844
Rule 216
Rule 725
Rule 204
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{\sqrt{1-d x} \sqrt{1+d x} (e+f x)} \, dx &=\int \frac{A+B x+C x^2}{(e+f x) \sqrt{1-d^2 x^2}} \, dx\\ &=-\frac{C \sqrt{1-d^2 x^2}}{d^2 f}-\frac{\int \frac{-A d^2 f^2+d^2 f (C e-B f) x}{(e+f x) \sqrt{1-d^2 x^2}} \, dx}{d^2 f^2}\\ &=-\frac{C \sqrt{1-d^2 x^2}}{d^2 f}-\frac{(C e-B f) \int \frac{1}{\sqrt{1-d^2 x^2}} \, dx}{f^2}+\frac{\left (C e^2-B e f+A f^2\right ) \int \frac{1}{(e+f x) \sqrt{1-d^2 x^2}} \, dx}{f^2}\\ &=-\frac{C \sqrt{1-d^2 x^2}}{d^2 f}-\frac{(C e-B f) \sin ^{-1}(d x)}{d f^2}-\frac{\left (C e^2-B e f+A f^2\right ) \operatorname{Subst}\left (\int \frac{1}{-d^2 e^2+f^2-x^2} \, dx,x,\frac{f+d^2 e x}{\sqrt{1-d^2 x^2}}\right )}{f^2}\\ &=-\frac{C \sqrt{1-d^2 x^2}}{d^2 f}-\frac{(C e-B f) \sin ^{-1}(d x)}{d f^2}+\frac{\left (C e^2-B e f+A f^2\right ) \tan ^{-1}\left (\frac{f+d^2 e x}{\sqrt{d^2 e^2-f^2} \sqrt{1-d^2 x^2}}\right )}{f^2 \sqrt{d^2 e^2-f^2}}\\ \end{align*}
Mathematica [A] time = 0.145378, size = 117, normalized size = 0.96 \[ \frac{\frac{\left (f (A f-B e)+C e^2\right ) \tan ^{-1}\left (\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right )}{\sqrt{d^2 e^2-f^2}}+\frac{\sin ^{-1}(d x) (B f-C e)}{d}-\frac{C f \sqrt{1-d^2 x^2}}{d^2}}{f^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.04, size = 373, normalized size = 3.1 \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{{f}^{3}{d}^{2}} \left ( -A{\it csgn} \left ( d \right ) \ln \left ( 2\,{\frac{1}{fx+e} \left ({d}^{2}ex+\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}\sqrt{-{d}^{2}{x}^{2}+1}f+f \right ) } \right ){d}^{2}{f}^{2}+B{\it csgn} \left ( d \right ) \ln \left ( 2\,{\frac{1}{fx+e} \left ({d}^{2}ex+\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}\sqrt{-{d}^{2}{x}^{2}+1}f+f \right ) } \right ){d}^{2}ef-C{\it csgn} \left ( d \right ) \ln \left ( 2\,{\frac{1}{fx+e} \left ({d}^{2}ex+\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}\sqrt{-{d}^{2}{x}^{2}+1}f+f \right ) } \right ){d}^{2}{e}^{2}+B\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \right ) d{f}^{2}\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}-C{\it csgn} \left ( d \right ){f}^{2}\sqrt{-{d}^{2}{x}^{2}+1}\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}-C\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \right ) def\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}} \right ) \sqrt{-dx+1}\sqrt{dx+1}{\frac{1}{\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}}}{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 29.8818, size = 1019, normalized size = 8.35 \begin{align*} \left [-\frac{{\left (C d^{2} e^{2} - B d^{2} e f + A d^{2} f^{2}\right )} \sqrt{-d^{2} e^{2} + f^{2}} \log \left (\frac{d^{2} e f x + f^{2} - \sqrt{-d^{2} e^{2} + f^{2}}{\left (d^{2} e x + f\right )} -{\left (\sqrt{-d^{2} e^{2} + f^{2}} \sqrt{-d x + 1} f +{\left (d^{2} e^{2} - f^{2}\right )} \sqrt{-d x + 1}\right )} \sqrt{d x + 1}}{f x + e}\right ) +{\left (C d^{2} e^{2} f - C f^{3}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 2 \,{\left (C d^{3} e^{3} - B d^{3} e^{2} f - C d e f^{2} + B d f^{3}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{d^{4} e^{2} f^{2} - d^{2} f^{4}}, \frac{2 \,{\left (C d^{2} e^{2} - B d^{2} e f + A d^{2} f^{2}\right )} \sqrt{d^{2} e^{2} - f^{2}} \arctan \left (-\frac{\sqrt{d^{2} e^{2} - f^{2}} \sqrt{d x + 1} \sqrt{-d x + 1} e - \sqrt{d^{2} e^{2} - f^{2}}{\left (f x + e\right )}}{{\left (d^{2} e^{2} - f^{2}\right )} x}\right ) -{\left (C d^{2} e^{2} f - C f^{3}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 2 \,{\left (C d^{3} e^{3} - B d^{3} e^{2} f - C d e f^{2} + B d f^{3}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{d^{4} e^{2} f^{2} - d^{2} f^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x + C x^{2}}{\left (e + f x\right ) \sqrt{- d x + 1} \sqrt{d x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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