3.2212 \(\int \frac{f+g x}{(d+e x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=121 \[ \frac{g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{\sqrt{c} e^2}-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x) (2 c d-b e)} \]

[Out]

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

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Rubi [A]  time = 0.370835, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068 \[ \frac{g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{\sqrt{c} e^2}-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x) (2 c d-b e)} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 36.5469, size = 109, normalized size = 0.9 \[ - \frac{2 \left (d g - e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \left (d + e x\right ) \left (b e - 2 c d\right )} + \frac{g \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{\sqrt{c} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(d*g - e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(e**2*(d + e*x)*(b
*e - 2*c*d)) + g*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 = 0.458773, size = 156, normalized size = 1.29 \[ \frac{-2 \sqrt{c} (e f-d g) (b e-c d+c e x)-i g \sqrt{d+e x} (2 c d-b e) \sqrt{c (d-e x)-b e} \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{\sqrt{c} e^2 (b e-2 c d) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 134, normalized size = 1.1 \[{\frac{g}{e}\arctan \left ({1\sqrt{c{e}^{2}} \left ( x+{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \right ){\frac{1}{\sqrt{c{e}^{2}}}}}-2\,{\frac{-dg+ef}{{e}^{2} \left ( -b{e}^{2}+2\,dec \right ) }\sqrt{-c \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+ \left ( -b{e}^{2}+2\,dec \right ) \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.97947, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (e f - d g\right )} \sqrt{-c} -{\left ({\left (2 \, c d e - b e^{2}\right )} g x +{\left (2 \, c d^{2} - b d e\right )} g\right )} \log \left (4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{2 \,{\left (2 \, c d^{2} e^{2} - b d e^{3} +{\left (2 \, c d e^{3} - b e^{4}\right )} x\right )} \sqrt{-c}}, -\frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (e f - d g\right )} \sqrt{c} -{\left ({\left (2 \, c d e - b e^{2}\right )} g x +{\left (2 \, c d^{2} - b d e\right )} g\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{{\left (2 \, c d^{2} e^{2} - b d e^{3} +{\left (2 \, c d e^{3} - b e^{4}\right )} x\right )} \sqrt{c}}\right ] \]

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)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError