3.2213 \(\int \frac{f+g x}{(d+e x)^2 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e^2 (d+e x)^2 (2 c d-b e)}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+4 c d g+2 c e f)}{3 e^2 (d+e x) (2 c d-b e)^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.513538, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e^2 (d+e x)^2 (2 c d-b e)}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+4 c d g+2 c e f)}{3 e^2 (d+e x) (2 c d-b e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 53.0349, size = 124, normalized size = 0.91 \[ \frac{2 \left (3 b e g - 4 c d g - 2 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{3 e^{2} \left (d + e x\right ) \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{3 e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.245818, size = 89, normalized size = 0.65 \[ -\frac{2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (2 c \left (d^2 g+2 d e (f+g x)+e^2 f x\right )-b e (2 d g+e (f+3 g x))\right )}{3 e^2 (d+e x)^2 (b e-2 c d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 127, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 3\,b{e}^{2}gx-4\,cdegx-2\,c{e}^{2}fx+2\,bdeg+b{e}^{2}f-2\,c{d}^{2}g-4\,cdef \right ) }{ \left ( 3\,ex+3\,d \right ){e}^{2} \left ({b}^{2}{e}^{2}-4\,bcde+4\,{c}^{2}{d}^{2} \right ) }{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.985901, size = 246, normalized size = 1.8 \[ -\frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left ({\left (4 \, c d e - b e^{2}\right )} f + 2 \,{\left (c d^{2} - b d e\right )} g +{\left (2 \, c e^{2} f +{\left (4 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )}}{3 \,{\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4} +{\left (4 \, c^{2} d^{2} e^{4} - 4 \, b c d e^{5} + b^{2} e^{6}\right )} x^{2} + 2 \,{\left (4 \, c^{2} d^{3} e^{3} - 4 \, b c d^{2} e^{4} + b^{2} d e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((4*c*d*e - b*e^2)*f + 2*(c*d^2
- b*d*e)*g + (2*c*e^2*f + (4*c*d*e - 3*b*e^2)*g)*x)/(4*c^2*d^4*e^2 - 4*b*c*d^3*e
^3 + b^2*d^2*e^4 + (4*c^2*d^2*e^4 - 4*b*c*d*e^5 + b^2*e^6)*x^2 + 2*(4*c^2*d^3*e^
3 - 4*b*c*d^2*e^4 + b^2*d*e^5)*x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((f + g*x)/(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)**2), x)

_______________________________________________________________________________________

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

[Out]

Exception raised: TypeError