3.2211 \(\int \frac{(d+e x) (f+g x)}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=149 \[ \frac{(2 c d-b e) (-3 b e g+2 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{5/2} e^2}+\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-4 c (d g+e f)-2 c e g x)}{4 c^2 e^2} \]

[Out]

((3*b*e*g - 4*c*(e*f + d*g) - 2*c*e*g*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2])/(4*c^2*e^2) + ((2*c*d - b*e)*(4*c*e*f + 2*c*d*g - 3*b*e*g)*ArcTan[(e*(b + 2*
c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(8*c^(5/2)*e^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.36539, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{(2 c d-b e) (-3 b e g+2 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{5/2} e^2}+\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-4 c (d g+e f)-2 c e g x)}{4 c^2 e^2} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]

[Out]

((3*b*e*g - 4*c*(e*f + d*g) - 2*c*e*g*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2])/(4*c^2*e^2) + ((2*c*d - b*e)*(4*c*e*f + 2*c*d*g - 3*b*e*g)*ArcTan[(e*(b + 2*
c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(8*c^(5/2)*e^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 56.5658, size = 182, normalized size = 1.22 \[ - \frac{g \left (d + e x\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{2 c e^{2}} + \frac{\left (\frac{3 b e g}{2} - c d g - 2 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{2 c^{2} e^{2}} + \frac{\left (b e - 2 c d\right ) \left (3 b e g - 2 c d g - 4 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{8 c^{\frac{5}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

-g*(d + e*x)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(2*c*e**2) + (3*b*e*
g/2 - c*d*g - 2*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(2*c**2*e*
*2) + (b*e - 2*c*d)*(3*b*e*g - 2*c*d*g - 4*c*e*f)*atan(-e*(-b - 2*c*x)/(2*sqrt(c
)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))))/(8*c**(5/2)*e**2)

_______________________________________________________________________________________

Mathematica [C]  time = 0.343243, size = 184, normalized size = 1.23 \[ \frac{-2 \sqrt{c} (d+e x) (c (d-e x)-b e) (2 c (2 d g+2 e f+e g x)-3 b e g)+i \sqrt{d+e x} (2 c d-b e) \sqrt{c (d-e x)-b e} (-3 b e g+2 c d g+4 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{8 c^{5/2} e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]

[Out]

(-2*Sqrt[c]*(d + e*x)*(-(b*e) + c*(d - e*x))*(-3*b*e*g + 2*c*(2*e*f + 2*d*g + e*
g*x)) + I*(2*c*d - b*e)*(4*c*e*f + 2*c*d*g - 3*b*e*g)*Sqrt[d + e*x]*Sqrt[-(b*e)
+ c*(d - e*x)]*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-(b*e) +
c*(d - e*x)]])/(8*c^(5/2)*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

_______________________________________________________________________________________

Maple [B]  time = 0.014, size = 460, normalized size = 3.1 \[{fd\arctan \left ({1\sqrt{c{e}^{2}} \left ( x+{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \right ){\frac{1}{\sqrt{c{e}^{2}}}}}-{\frac{dg}{c{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}-{\frac{f}{ce}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}-{\frac{bdg}{c}\arctan \left ({1\sqrt{c{e}^{2}} \left ( x+{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \right ){\frac{1}{\sqrt{c{e}^{2}}}}}-{\frac{bef}{2\,c}\arctan \left ({1\sqrt{c{e}^{2}} \left ( x+{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \right ){\frac{1}{\sqrt{c{e}^{2}}}}}-{\frac{gx}{2\,ce}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}+{\frac{3\,bg}{4\,e{c}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}+{\frac{3\,{b}^{2}eg}{8\,{c}^{2}}\arctan \left ({1\sqrt{c{e}^{2}} \left ( x+{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \right ){\frac{1}{\sqrt{c{e}^{2}}}}}+{\frac{{d}^{2}g}{2\,e}\arctan \left ({1\sqrt{c{e}^{2}} \left ( x+{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \right ){\frac{1}{\sqrt{c{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

f*d/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(1/2))-1/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*g-1/c/e*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2)*f-b/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d*g-1/2*b/c/(c*e^2)^(1/2)*arctan((c*e^2)
^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*e*f-1/2/e*g*x/c*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+3/4/e*g*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2)+3/8*e*g*b^2/c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(1/2))+1/2/e*g/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*
b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^2

_______________________________________________________________________________________

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((e*x + d)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.586916, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c e g x + 4 \, c e f +{\left (4 \, c d - 3 \, b e\right )} g\right )} \sqrt{-c} -{\left (4 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} f +{\left (4 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} g\right )} \log \left (4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{16 \, \sqrt{-c} c^{2} e^{2}}, -\frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c e g x + 4 \, c e f +{\left (4 \, c d - 3 \, b e\right )} g\right )} \sqrt{c} -{\left (4 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} f +{\left (4 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} g\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{8 \, c^{\frac{5}{2}} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="fricas")

[Out]

[-1/16*(4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*g*x + 4*c*e*f + (4*c
*d - 3*b*e)*g)*sqrt(-c) - (4*(2*c^2*d*e - b*c*e^2)*f + (4*c^2*d^2 - 8*b*c*d*e +
3*b^2*e^2)*g)*log(4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c^2*e*x + b*c*
e) + (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)*sqrt(-c)))/
(sqrt(-c)*c^2*e^2), -1/8*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*g*
x + 4*c*e*f + (4*c*d - 3*b*e)*g)*sqrt(c) - (4*(2*c^2*d*e - b*c*e^2)*f + (4*c^2*d
^2 - 8*b*c*d*e + 3*b^2*e^2)*g)*arctan(1/2*(2*c*e*x + b*e)/(sqrt(-c*e^2*x^2 - b*e
^2*x + c*d^2 - b*d*e)*sqrt(c))))/(c^(5/2)*e^2)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right ) \left (f + g x\right )}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

Integral((d + e*x)*(f + g*x)/sqrt(-(d + e*x)*(b*e - c*d + c*e*x)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.320224, size = 242, normalized size = 1.62 \[ -\frac{1}{4} \, \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left (\frac{2 \, g x e^{\left (-1\right )}}{c} + \frac{{\left (4 \, c d g e + 4 \, c f e^{2} - 3 \, b g e^{2}\right )} e^{\left (-3\right )}}{c^{2}}\right )} + \frac{{\left (4 \, c^{2} d^{2} g + 8 \, c^{2} d f e - 8 \, b c d g e - 4 \, b c f e^{2} + 3 \, b^{2} g e^{2}\right )} \sqrt{-c e^{2}} e^{\left (-3\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{-c e^{2}} x - \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt{-c e^{2}} b \right |}\right )}{8 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="giac")

[Out]

-1/4*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*g*x*e^(-1)/c + (4*c*d*g*e + 4
*c*f*e^2 - 3*b*g*e^2)*e^(-3)/c^2) + 1/8*(4*c^2*d^2*g + 8*c^2*d*f*e - 8*b*c*d*g*e
- 4*b*c*f*e^2 + 3*b^2*g*e^2)*sqrt(-c*e^2)*e^(-3)*ln(abs(-2*(sqrt(-c*e^2)*x - sq
rt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*c - sqrt(-c*e^2)*b))/c^3