3.2176 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=200 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^2 (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-4 c d g+2 c e f)}{e^2 (2 c d-b e)}-\frac{(b e g-4 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 \sqrt{c} e^2} \]

[Out]

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

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Rubi [A]  time = 0.728305, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^2 (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-4 c d g+2 c e f)}{e^2 (2 c d-b e)}-\frac{(b e g-4 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 \sqrt{c} e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 68.3405, size = 189, normalized size = 0.94 \[ \frac{2 \left (\frac{b e g}{2} - 2 c d g + c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \left (b e - 2 c d\right )} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )} - \frac{2 \left (\frac{b e g}{4} - c d g + \frac{c e f}{2}\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{\sqrt{c} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.39099, size = 147, normalized size = 0.74 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-\frac{i (b e g-4 c d g+2 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{\sqrt{c} \sqrt{d+e x} \sqrt{c (d-e x)-b e}}+\frac{4 (d g-e f)}{d+e x}+2 g\right )}{2 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 880, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.797548, size = 1, normalized size = 0. \[ \left [\frac{4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (e g x - 2 \, e f + 3 \, d g\right )} \sqrt{-c} +{\left (2 \, c d e f -{\left (4 \, c d^{2} - b d e\right )} g +{\left (2 \, c e^{2} f -{\left (4 \, c d e - b e^{2}\right )} g\right )} x\right )} \log \left (-4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{4 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{-c}}, \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (e g x - 2 \, e f + 3 \, d g\right )} \sqrt{c} -{\left (2 \, c d e f -{\left (4 \, c d^{2} - b d e\right )} g +{\left (2 \, c e^{2} f -{\left (4 \, c d e - b e^{2}\right )} g\right )} x\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{2 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*g*x - 2*e*f + 3*d*g)*sqrt(
-c) + (2*c*d*e*f - (4*c*d^2 - b*d*e)*g + (2*c*e^2*f - (4*c*d*e - b*e^2)*g)*x)*lo
g(-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)))/((e^3*x + d*e^2
)*sqrt(-c)), 1/2*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*g*x - 2*e*f +
3*d*g)*sqrt(c) - (2*c*d*e*f - (4*c*d^2 - b*d*e)*g + (2*c*e^2*f - (4*c*d*e - b*e^
2)*g)*x)*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))))/((e^3*x + d*e^2)*sqrt(c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out