3.2175 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx\)

Optimal. Leaf size=192 \[ \frac{(2 c d-b e) (-b e g-2 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e^2}+\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-2 c d g+4 c e f)}{4 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 c e^2 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.558293, antiderivative size = 192, 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 \[ \frac{(2 c d-b e) (-b e g-2 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e^2}+\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-2 c d g+4 c e f)}{4 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 c e^2 (d+e x)} \]

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),x]

[Out]

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

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Rubi in Sympy [A]  time = 58.6566, size = 177, normalized size = 0.92 \[ - \frac{g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{2 c e^{2} \left (d + e x\right )} - \frac{\left (\frac{b e g}{2} + c d g - 2 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{2 c e^{2}} + \frac{\left (b e - 2 c d\right ) \left (b e g + 2 c d g - 4 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{8 c^{\frac{3}{2}} e^{2}} \]

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),x)

[Out]

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

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Mathematica [C]  time = 0.388595, size = 157, normalized size = 0.82 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-\frac{i (2 c d-b e) (b e g+2 c d g-4 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{c^{3/2} \sqrt{d+e x} \sqrt{c (d-e x)-b e}}+\frac{2 b e g}{c}-8 d g+8 e f+4 e g x\right )}{8 e^2} \]

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),x]

[Out]

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

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Maple [B]  time = 0.027, size = 697, normalized size = 3.6 \[ \text{result too large to display} \]

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),x)

[Out]

1/2*g/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+1/4*g/e/c*(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(1/2)*b+1/8*g*e/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))*b^2-1/2*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))*b*d+1/2*g/e*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^2-1
/e^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d*g+1/e*(-c*(d/e+x)^2*e^2
+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*f+1/2/(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-1/2*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-1/e/(c*e^2)^(1/2)*arcta
n((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*c*g+1/(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*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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.598017, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c e g x + 4 \, c e f -{\left (4 \, c d - b e\right )} g\right )} \sqrt{-c} +{\left (4 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} f -{\left (4 \, c^{2} d^{2} - b^{2} e^{2}\right )} g\right )} \log \left (4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{16 \, \sqrt{-c} c e^{2}}, \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c e g x + 4 \, c e f -{\left (4 \, c d - b e\right )} g\right )} \sqrt{c} +{\left (4 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} f -{\left (4 \, c^{2} d^{2} - b^{2} e^{2}\right )} g\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{8 \, c^{\frac{3}{2}} e^{2}}\right ] \]

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),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 - b*e)*g)*sqrt(-c) + (4*(2*c^2*d*e - b*c*e^2)*f - (4*c^2*d^2 - 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*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
- b*e)*g)*sqrt(c) + (4*(2*c^2*d*e - b*c*e^2)*f - (4*c^2*d^2 - b^2*e^2)*g)*arcta
n(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^
(3/2)*e^2)]

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

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),x)

[Out]

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

[Out]

Exception raised: TypeError