3.47 \(\int \frac{A+B x+C x^2}{\sqrt{1-d x} \sqrt{1+d x} (e+f x)^2} \, dx\)
Optimal. Leaf size=163 \[ \frac{\sqrt{1-d^2 x^2} \left (A f^2-B e f+C e^2\right )}{f \left (d^2 e^2-f^2\right ) (e+f x)}-\frac{\tan ^{-1}\left (\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right ) \left (-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right )}{f^2 \left (d^2 e^2-f^2\right )^{3/2}}+\frac{C \sin ^{-1}(d x)}{d f^2} \]
[Out]
((C*e^2 - B*e*f + A*f^2)*Sqrt[1 - d^2*x^2])/(f*(d^2*e^2 - f^2)*(e + f*x)) + (C*A
rcSin[d*x])/(d*f^2) - ((C*d^2*e^3 - 2*C*e*f^2 - A*d^2*e*f^2 + B*f^3)*ArcTan[(f +
d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(f^2*(d^2*e^2 - f^2)^(3/2))
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Rubi [A] time = 0.584769, antiderivative size = 163, 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
\[ \frac{\sqrt{1-d^2 x^2} \left (A f^2-B e f+C e^2\right )}{f \left (d^2 e^2-f^2\right ) (e+f x)}-\frac{\tan ^{-1}\left (\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right ) \left (-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right )}{f^2 \left (d^2 e^2-f^2\right )^{3/2}}+\frac{C \sin ^{-1}(d x)}{d f^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2),x]
[Out]
((C*e^2 - B*e*f + A*f^2)*Sqrt[1 - d^2*x^2])/(f*(d^2*e^2 - f^2)*(e + f*x)) + (C*A
rcSin[d*x])/(d*f^2) - ((C*d^2*e^3 - 2*C*e*f^2 - A*d^2*e*f^2 + B*f^3)*ArcTan[(f +
d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(f^2*(d^2*e^2 - f^2)^(3/2))
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Rubi in Sympy [A] time = 90.9699, size = 146, normalized size = 0.9 \[ \frac{C \operatorname{asin}{\left (d x \right )}}{d f^{2}} - \frac{\sqrt{- d^{2} x^{2} + 1} \left (A f^{2} - B e f + C e^{2}\right )}{f \left (e + f x\right ) \left (- d^{2} e^{2} + f^{2}\right )} - \frac{\left (- A d^{2} e f^{2} + B f^{3} + C d^{2} e^{3} - 2 C e f^{2}\right ) \operatorname{atanh}{\left (\frac{d^{2} e x + f}{\sqrt{- d e + f} \sqrt{d e + f} \sqrt{- d^{2} x^{2} + 1}} \right )}}{f^{2} \left (- d e + f\right )^{\frac{3}{2}} \left (d e + f\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((C*x**2+B*x+A)/(f*x+e)**2/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
C*asin(d*x)/(d*f**2) - sqrt(-d**2*x**2 + 1)*(A*f**2 - B*e*f + C*e**2)/(f*(e + f*
x)*(-d**2*e**2 + f**2)) - (-A*d**2*e*f**2 + B*f**3 + C*d**2*e**3 - 2*C*e*f**2)*a
tanh((d**2*e*x + f)/(sqrt(-d*e + f)*sqrt(d*e + f)*sqrt(-d**2*x**2 + 1)))/(f**2*(
-d*e + f)**(3/2)*(d*e + f)**(3/2))
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Mathematica [A] time = 0.526313, size = 211, normalized size = 1.29 \[ \frac{-\frac{f \sqrt{1-d^2 x^2} \left (f (A f-B e)+C e^2\right )}{\left (f^2-d^2 e^2\right ) (e+f x)}-\frac{\log \left (\sqrt{1-d^2 x^2} \sqrt{f^2-d^2 e^2}+d^2 e x+f\right ) \left (-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right )}{\left (f^2-d^2 e^2\right )^{3/2}}+\frac{\log (e+f x) \left (-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right )}{\left (f^2-d^2 e^2\right )^{3/2}}+\frac{C \sin ^{-1}(d x)}{d}}{f^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2),x]
[Out]
(-((f*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[1 - d^2*x^2])/((-(d^2*e^2) + f^2)*(e + f*x
))) + (C*ArcSin[d*x])/d + ((C*d^2*e^3 - 2*C*e*f^2 - A*d^2*e*f^2 + B*f^3)*Log[e +
f*x])/(-(d^2*e^2) + f^2)^(3/2) - ((C*d^2*e^3 - 2*C*e*f^2 - A*d^2*e*f^2 + B*f^3)
*Log[f + d^2*e*x + Sqrt[-(d^2*e^2) + f^2]*Sqrt[1 - d^2*x^2]])/(-(d^2*e^2) + f^2)
^(3/2))/f^2
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Maple [C] time = 0., size = 899, normalized size = 5.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((C*x^2+B*x+A)/(f*x+e)^2/(-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))*x*d^3*e*f^3+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))*x*d^3*e^3*f-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^3*e^2*f^2+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^3*e^4+C*arctan(csgn(d)*d
*x/(-d^2*x^2+1)^(1/2))*x*d^2*e^2*f^2*(-(d^2*e^2-f^2)/f^2)^(1/2)+A*csgn(d)*d*f^4*
(-(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)+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))*x*d*f^4-B*csgn(d)*d*e*f^3*(-(d
^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)-2*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))*x*d*e*f^3+C*csgn(d)*d*e^2*f^2*(-
(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)+C*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1
/2))*d^2*e^3*f*(-(d^2*e^2-f^2)/f^2)^(1/2)+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*e*f^3-2*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*e^2*f^2-C*arctan(c
sgn(d)*d*x/(-d^2*x^2+1)^(1/2))*x*f^4*(-(d^2*e^2-f^2)/f^2)^(1/2)-C*arctan(csgn(d)
*d*x/(-d^2*x^2+1)^(1/2))*e*f^3*(-(d^2*e^2-f^2)/f^2)^(1/2))*csgn(d)*(d*x+1)^(1/2)
*(-d*x+1)^(1/2)/(-d^2*x^2+1)^(1/2)/(d*e-f)/(d*e+f)/d/(-(d^2*e^2-f^2)/f^2)^(1/2)/
f^3/(f*x+e)
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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)^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 30.3043, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
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)^2),x, algorithm="fricas")
[Out]
[((C*d*e^2*f^2 - B*d*e*f^3 + A*d*f^4)*sqrt(-d^2*e^2 + f^2)*sqrt(d*x + 1)*sqrt(-d
*x + 1)*x - 2*((C*d^2*e^4 - C*e^2*f^2 + (C*d^2*e^3*f - C*e*f^3)*x)*sqrt(-d^2*e^2
+ f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1) - (C*d^2*e^4 - C*e^2*f^2 + (C*d^2*e^3*f - C
*e*f^3)*x)*sqrt(-d^2*e^2 + f^2))*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)
) - (C*d^3*e^5 + B*d*e^2*f^3 - (A*d^3 + 2*C*d)*e^3*f^2 - (C*d^3*e^5 + B*d*e^2*f^
3 - (A*d^3 + 2*C*d)*e^3*f^2 + (C*d^3*e^4*f + B*d*e*f^4 - (A*d^3 + 2*C*d)*e^2*f^3
)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + (C*d^3*e^4*f + B*d*e*f^4 - (A*d^3 + 2*C*d)*e
^2*f^3)*x)*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 - s
qrt(-d^2*e^2 + f^2)*(e*f*x + e^2)*sqrt(-d*x + 1))*sqrt(d*x + 1))/(sqrt(d*x + 1)*
sqrt(-d*x + 1)*(f*x + e) - f*x - e)) - sqrt(-d^2*e^2 + f^2)*((C*d^3*e^3*f - B*d^
3*e^2*f^2 + A*d^3*e*f^3)*x^2 + (C*d*e^2*f^2 - B*d*e*f^3 + A*d*f^4)*x))/((d^3*e^4
*f^2 - d*e^2*f^4 + (d^3*e^3*f^3 - d*e*f^5)*x)*sqrt(-d^2*e^2 + f^2)*sqrt(d*x + 1)
*sqrt(-d*x + 1) - (d^3*e^4*f^2 - d*e^2*f^4 + (d^3*e^3*f^3 - d*e*f^5)*x)*sqrt(-d^
2*e^2 + f^2)), ((C*d*e^2*f^2 - B*d*e*f^3 + A*d*f^4)*sqrt(d^2*e^2 - f^2)*sqrt(d*x
+ 1)*sqrt(-d*x + 1)*x + 2*(C*d^3*e^5 + B*d*e^2*f^3 - (A*d^3 + 2*C*d)*e^3*f^2 -
(C*d^3*e^5 + B*d*e^2*f^3 - (A*d^3 + 2*C*d)*e^3*f^2 + (C*d^3*e^4*f + B*d*e*f^4 -
(A*d^3 + 2*C*d)*e^2*f^3)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + (C*d^3*e^4*f + B*d*e*
f^4 - (A*d^3 + 2*C*d)*e^2*f^3)*x)*arctan(-(sqrt(d^2*e^2 - f^2)*sqrt(d*x + 1)*sqr
t(-d*x + 1)*e - sqrt(d^2*e^2 - f^2)*(f*x + e))/((d^2*e^2 - f^2)*x)) - 2*((C*d^2*
e^4 - C*e^2*f^2 + (C*d^2*e^3*f - C*e*f^3)*x)*sqrt(d^2*e^2 - f^2)*sqrt(d*x + 1)*s
qrt(-d*x + 1) - (C*d^2*e^4 - C*e^2*f^2 + (C*d^2*e^3*f - C*e*f^3)*x)*sqrt(d^2*e^2
- f^2))*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)) - sqrt(d^2*e^2 - f^2)*
((C*d^3*e^3*f - B*d^3*e^2*f^2 + A*d^3*e*f^3)*x^2 + (C*d*e^2*f^2 - B*d*e*f^3 + A*
d*f^4)*x))/((d^3*e^4*f^2 - d*e^2*f^4 + (d^3*e^3*f^3 - d*e*f^5)*x)*sqrt(d^2*e^2 -
f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1) - (d^3*e^4*f^2 - d*e^2*f^4 + (d^3*e^3*f^3 - d
*e*f^5)*x)*sqrt(d^2*e^2 - f^2))]
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Sympy [F(-2)] 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)/(f*x+e)**2/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
Exception raised: ValueError
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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)^2),x, algorithm="giac")
[Out]
Exception raised: TypeError