3.39 \(\int \frac{A+B x+C x^2}{\sqrt{1-d x} \sqrt{1+d x} (e+f x)} \, dx\)

Optimal. Leaf size=122 \[ \frac{\left (A f^2-B e f+C e^2\right ) \tan ^{-1}\left (\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right )}{f^2 \sqrt{d^2 e^2-f^2}}-\frac{\sin ^{-1}(d x) (C e-B f)}{d f^2}-\frac{C \sqrt{1-d^2 x^2}}{d^2 f} \]

[Out]

-((C*Sqrt[1 - d^2*x^2])/(d^2*f)) - ((C*e - B*f)*ArcSin[d*x])/(d*f^2) + ((C*e^2 -
B*e*f + A*f^2)*ArcTan[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(
f^2*Sqrt[d^2*e^2 - f^2])

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Rubi [A]  time = 0.577429, 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 \[ \frac{\left (A f^2-B e f+C e^2\right ) \tan ^{-1}\left (\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right )}{f^2 \sqrt{d^2 e^2-f^2}}-\frac{\sin ^{-1}(d x) (C e-B f)}{d f^2}-\frac{C \sqrt{1-d^2 x^2}}{d^2 f} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)),x]

[Out]

-((C*Sqrt[1 - d^2*x^2])/(d^2*f)) - ((C*e - B*f)*ArcSin[d*x])/(d*f^2) + ((C*e^2 -
B*e*f + A*f^2)*ArcTan[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(
f^2*Sqrt[d^2*e^2 - f^2])

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Rubi in Sympy [A]  time = 69.1554, size = 109, normalized size = 0.89 \[ - \frac{C \sqrt{- d^{2} x^{2} + 1}}{d^{2} f} - \frac{\left (A f^{2} - B e f + C e^{2}\right ) \operatorname{atanh}{\left (\frac{d^{2} e x + f}{\sqrt{- d e + f} \sqrt{d e + f} \sqrt{- d^{2} x^{2} + 1}} \right )}}{f^{2} \sqrt{- d e + f} \sqrt{d e + f}} + \frac{\left (B f - C e\right ) \operatorname{asin}{\left (d x \right )}}{d f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((C*x**2+B*x+A)/(f*x+e)/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

-C*sqrt(-d**2*x**2 + 1)/(d**2*f) - (A*f**2 - B*e*f + C*e**2)*atanh((d**2*e*x + f
)/(sqrt(-d*e + f)*sqrt(d*e + f)*sqrt(-d**2*x**2 + 1)))/(f**2*sqrt(-d*e + f)*sqrt
(d*e + f)) + (B*f - C*e)*asin(d*x)/(d*f**2)

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Mathematica [A]  time = 0.326682, size = 155, normalized size = 1.27 \[ \frac{-\frac{\left (f (A f-B e)+C e^2\right ) \log \left (\sqrt{1-d^2 x^2} \sqrt{f^2-d^2 e^2}+d^2 e x+f\right )}{\sqrt{f^2-d^2 e^2}}+\frac{\log (e+f x) \left (f (A f-B e)+C e^2\right )}{\sqrt{f^2-d^2 e^2}}+\frac{\sin ^{-1}(d x) (B f-C e)}{d}-\frac{C f \sqrt{1-d^2 x^2}}{d^2}}{f^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)),x]

[Out]

(-((C*f*Sqrt[1 - d^2*x^2])/d^2) + ((-(C*e) + B*f)*ArcSin[d*x])/d + ((C*e^2 + f*(
-(B*e) + A*f))*Log[e + f*x])/Sqrt[-(d^2*e^2) + f^2] - ((C*e^2 + f*(-(B*e) + A*f)
)*Log[f + d^2*e*x + Sqrt[-(d^2*e^2) + f^2]*Sqrt[1 - d^2*x^2]])/Sqrt[-(d^2*e^2) +
f^2])/f^2

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Maple [C]  time = 0.059, size = 373, normalized size = 3.1 \[{\frac{{\it csgn} \left ( d \right ) }{{f}^{3}{d}^{2}} \left ( -A{\it csgn} \left ( d \right ) \ln \left ( 2\,{\frac{1}{fx+e} \left ({d}^{2}ex+\sqrt{-{d}^{2}{x}^{2}+1}\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}f+f \right ) } \right ){d}^{2}{f}^{2}+B{\it csgn} \left ( d \right ) \ln \left ( 2\,{\frac{1}{fx+e} \left ({d}^{2}ex+\sqrt{-{d}^{2}{x}^{2}+1}\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}f+f \right ) } \right ){d}^{2}ef-C{\it csgn} \left ( d \right ) \ln \left ( 2\,{\frac{1}{fx+e} \left ({d}^{2}ex+\sqrt{-{d}^{2}{x}^{2}+1}\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}f+f \right ) } \right ){d}^{2}{e}^{2}+B\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \right ) d{f}^{2}\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}-C{\it csgn} \left ( d \right ){f}^{2}\sqrt{-{d}^{2}{x}^{2}+1}\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}-C\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \right ) def\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}} \right ) \sqrt{-dx+1}\sqrt{dx+1}{\frac{1}{\sqrt{-{\frac{{d}^{2}{e}^{2}-{f}^{2}}{{f}^{2}}}}}}{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+B*x+A)/(f*x+e)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x)

[Out]

(-A*csgn(d)*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*
x+e))*d^2*f^2+B*csgn(d)*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1
/2)*f+f)/(f*x+e))*d^2*e*f-C*csgn(d)*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-
f^2)/f^2)^(1/2)*f+f)/(f*x+e))*d^2*e^2+B*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d
*f^2*(-(d^2*e^2-f^2)/f^2)^(1/2)-C*csgn(d)*f^2*(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)
/f^2)^(1/2)-C*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d*e*f*(-(d^2*e^2-f^2)/f^2)^
(1/2))*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*csgn(d)/(-(d^2*e^2-f^2)/f^2)^(1/2)/f^3/(-d^2
*x^2+1)^(1/2)/d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)/(sqrt(d*x + 1)*sqrt(-d*x + 1)*(f*x + e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.89625, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{-d^{2} e^{2} + f^{2}} C d f x^{2} + 2 \,{\left (\sqrt{-d^{2} e^{2} + f^{2}}{\left (C e - B f\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - \sqrt{-d^{2} e^{2} + f^{2}}{\left (C e - B f\right )}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right ) -{\left (C d e^{2} - B d e f + A d f^{2} -{\left (C d e^{2} - B d e f + A d f^{2}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}\right )} \log \left (-\frac{{\left (d^{2} e^{2} f - f^{3}\right )} x^{2} +{\left (d^{2} e^{3} - e f^{2}\right )} x + \sqrt{-d^{2} e^{2} + f^{2}}{\left (e f x -{\left (d^{2} e^{2} - f^{2}\right )} x^{2} + e^{2}\right )} -{\left ({\left (d^{2} e^{3} - e f^{2}\right )} \sqrt{-d x + 1} x + \sqrt{-d^{2} e^{2} + f^{2}}{\left (e f x + e^{2}\right )} \sqrt{-d x + 1}\right )} \sqrt{d x + 1}}{\sqrt{d x + 1} \sqrt{-d x + 1}{\left (f x + e\right )} - f x - e}\right )}{\sqrt{-d^{2} e^{2} + f^{2}} \sqrt{d x + 1} \sqrt{-d x + 1} d f^{2} - \sqrt{-d^{2} e^{2} + f^{2}} d f^{2}}, \frac{\sqrt{d^{2} e^{2} - f^{2}} C d f x^{2} - 2 \,{\left (C d e^{2} - B d e f + A d f^{2} -{\left (C d e^{2} - B d e f + A d f^{2}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}\right )} \arctan \left (-\frac{\sqrt{d^{2} e^{2} - f^{2}} \sqrt{d x + 1} \sqrt{-d x + 1} e - \sqrt{d^{2} e^{2} - f^{2}}{\left (f x + e\right )}}{{\left (d^{2} e^{2} - f^{2}\right )} x}\right ) + 2 \,{\left (\sqrt{d^{2} e^{2} - f^{2}}{\left (C e - B f\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - \sqrt{d^{2} e^{2} - f^{2}}{\left (C e - B f\right )}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{\sqrt{d^{2} e^{2} - f^{2}} \sqrt{d x + 1} \sqrt{-d x + 1} d f^{2} - \sqrt{d^{2} e^{2} - f^{2}} d f^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)/(sqrt(d*x + 1)*sqrt(-d*x + 1)*(f*x + e)),x, algorithm="fricas")

[Out]

[(sqrt(-d^2*e^2 + f^2)*C*d*f*x^2 + 2*(sqrt(-d^2*e^2 + f^2)*(C*e - B*f)*sqrt(d*x
+ 1)*sqrt(-d*x + 1) - sqrt(-d^2*e^2 + f^2)*(C*e - B*f))*arctan((sqrt(d*x + 1)*sq
rt(-d*x + 1) - 1)/(d*x)) - (C*d*e^2 - B*d*e*f + A*d*f^2 - (C*d*e^2 - B*d*e*f + A
*d*f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1))*log(-((d^2*e^2*f - f^3)*x^2 + (d^2*e^3 - e
*f^2)*x + sqrt(-d^2*e^2 + f^2)*(e*f*x - (d^2*e^2 - f^2)*x^2 + e^2) - ((d^2*e^3 -
e*f^2)*sqrt(-d*x + 1)*x + sqrt(-d^2*e^2 + f^2)*(e*f*x + e^2)*sqrt(-d*x + 1))*sq
rt(d*x + 1))/(sqrt(d*x + 1)*sqrt(-d*x + 1)*(f*x + e) - f*x - e)))/(sqrt(-d^2*e^2
+ f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1)*d*f^2 - sqrt(-d^2*e^2 + f^2)*d*f^2), (sqrt(
d^2*e^2 - f^2)*C*d*f*x^2 - 2*(C*d*e^2 - B*d*e*f + A*d*f^2 - (C*d*e^2 - B*d*e*f +
A*d*f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1))*arctan(-(sqrt(d^2*e^2 - f^2)*sqrt(d*x +
1)*sqrt(-d*x + 1)*e - sqrt(d^2*e^2 - f^2)*(f*x + e))/((d^2*e^2 - f^2)*x)) + 2*(s
qrt(d^2*e^2 - f^2)*(C*e - B*f)*sqrt(d*x + 1)*sqrt(-d*x + 1) - sqrt(d^2*e^2 - f^2
)*(C*e - B*f))*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(sqrt(d^2*e^2 -
f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1)*d*f^2 - sqrt(d^2*e^2 - f^2)*d*f^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x + C x^{2}}{\left (e + f x\right ) \sqrt{- d x + 1} \sqrt{d x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x**2+B*x+A)/(f*x+e)/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

Integral((A + B*x + C*x**2)/((e + f*x)*sqrt(-d*x + 1)*sqrt(d*x + 1)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)/(sqrt(d*x + 1)*sqrt(-d*x + 1)*(f*x + e)),x, algorithm="giac")

[Out]

Exception raised: TypeError