3.2214 \(\int \frac{f+g x}{(d+e x)^3 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=210 \[ -\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (d+e x)^3 (2 c d-b e)}-\frac{4 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x) (2 c d-b e)^3}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x)^2 (2 c d-b e)^2} \]

[Out]

(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(5*e^2*(2*c*d - b*e)*
(d + e*x)^3) - (2*(4*c*e*f + 6*c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2])/(15*e^2*(2*c*d - b*e)^2*(d + e*x)^2) - (4*c*(4*c*e*f + 6*c*d*g - 5*b*
e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(15*e^2*(2*c*d - b*e)^3*(d + e*x
))

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Rubi [A]  time = 0.774399, antiderivative size = 210, 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 \[ -\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (d+e x)^3 (2 c d-b e)}-\frac{4 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x) (2 c d-b e)^3}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x)^2 (2 c d-b e)^2} \]

Antiderivative was successfully verified.

[In]  Int[(f + g*x)/((d + e*x)^3*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])/(5*e^2*(2*c*d - b*e)*
(d + e*x)^3) - (2*(4*c*e*f + 6*c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2])/(15*e^2*(2*c*d - b*e)^2*(d + e*x)^2) - (4*c*(4*c*e*f + 6*c*d*g - 5*b*
e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(15*e^2*(2*c*d - b*e)^3*(d + e*x
))

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Rubi in Sympy [A]  time = 82.2715, size = 197, normalized size = 0.94 \[ - \frac{4 c \left (5 b e g - 6 c d g - 4 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{15 e^{2} \left (d + e x\right ) \left (b e - 2 c d\right )^{3}} + \frac{2 \left (5 b e g - 6 c d g - 4 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{15 e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{5 e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)/(e*x+d)**3/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

-4*c*(5*b*e*g - 6*c*d*g - 4*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d)
)/(15*e**2*(d + e*x)*(b*e - 2*c*d)**3) + 2*(5*b*e*g - 6*c*d*g - 4*c*e*f)*sqrt(-b
*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(15*e**2*(d + e*x)**2*(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))/(5*e**2*(d + e*x)
**3*(b*e - 2*c*d))

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Mathematica [A]  time = 0.357301, size = 156, normalized size = 0.74 \[ \frac{(d+e x) (-b e+c d-c e x) \left (\frac{2 (e f-d g)}{5 e^2 (d+e x)^3 (b e-2 c d)}+\frac{4 c (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x) (b e-2 c d)^3}-\frac{2 (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x)^2 (b e-2 c d)^2}\right )}{\sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)*(c*d - b*e - c*e*x)*((2*(e*f - d*g))/(5*e^2*(-2*c*d + b*e)*(d + e*x)^
3) - (2*(4*c*e*f + 6*c*d*g - 5*b*e*g))/(15*e^2*(-2*c*d + b*e)^2*(d + e*x)^2) + (
4*c*(4*c*e*f + 6*c*d*g - 5*b*e*g))/(15*e^2*(-2*c*d + b*e)^3*(d + e*x))))/Sqrt[(d
+ e*x)*(-(b*e) + c*(d - e*x))]

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Maple [A]  time = 0.014, size = 236, normalized size = 1.1 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -10\,bc{e}^{3}g{x}^{2}+12\,{c}^{2}d{e}^{2}g{x}^{2}+8\,{c}^{2}{e}^{3}f{x}^{2}+5\,{b}^{2}{e}^{3}gx-36\,bcd{e}^{2}gx-4\,bc{e}^{3}fx+36\,{c}^{2}{d}^{2}egx+24\,{c}^{2}d{e}^{2}fx+2\,{b}^{2}d{e}^{2}g+3\,{b}^{2}{e}^{3}f-14\,bc{d}^{2}eg-16\,bcd{e}^{2}f+12\,{c}^{2}{d}^{3}g+28\,{c}^{2}{d}^{2}ef \right ) }{15\,{e}^{2} \left ({b}^{3}{e}^{3}-6\,{b}^{2}cd{e}^{2}+12\,b{c}^{2}{d}^{2}e-8\,{c}^{3}{d}^{3} \right ) \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(c*e*x+b*e-c*d)*(-10*b*c*e^3*g*x^2+12*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x^2+5*b^
2*e^3*g*x-36*b*c*d*e^2*g*x-4*b*c*e^3*f*x+36*c^2*d^2*e*g*x+24*c^2*d*e^2*f*x+2*b^2
*d*e^2*g+3*b^2*e^3*f-14*b*c*d^2*e*g-16*b*c*d*e^2*f+12*c^2*d^3*g+28*c^2*d^2*e*f)/
(e*x+d)^2/(b^3*e^3-6*b^2*c*d*e^2+12*b*c^2*d^2*e-8*c^3*d^3)/e^2/(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(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)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60532, size = 497, normalized size = 2.37 \[ -\frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \,{\left (4 \, c^{2} e^{3} f +{\left (6 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} g\right )} x^{2} +{\left (28 \, c^{2} d^{2} e - 16 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + 2 \,{\left (6 \, c^{2} d^{3} - 7 \, b c d^{2} e + b^{2} d e^{2}\right )} g +{\left (4 \,{\left (6 \, c^{2} d e^{2} - b c e^{3}\right )} f +{\left (36 \, c^{2} d^{2} e - 36 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )} g\right )} x\right )}}{15 \,{\left (8 \, c^{3} d^{6} e^{2} - 12 \, b c^{2} d^{5} e^{3} + 6 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5} +{\left (8 \, c^{3} d^{3} e^{5} - 12 \, b c^{2} d^{2} e^{6} + 6 \, b^{2} c d e^{7} - b^{3} e^{8}\right )} x^{3} + 3 \,{\left (8 \, c^{3} d^{4} e^{4} - 12 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} - b^{3} d e^{7}\right )} x^{2} + 3 \,{\left (8 \, c^{3} d^{5} e^{3} - 12 \, b c^{2} d^{4} e^{4} + 6 \, b^{2} c d^{3} e^{5} - b^{3} d^{2} e^{6}\right )} x\right )}} \]

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)^3),x, algorithm="fricas")

[Out]

-2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^2*e^3*f + (6*c^2*d*e^2
- 5*b*c*e^3)*g)*x^2 + (28*c^2*d^2*e - 16*b*c*d*e^2 + 3*b^2*e^3)*f + 2*(6*c^2*d^3
- 7*b*c*d^2*e + b^2*d*e^2)*g + (4*(6*c^2*d*e^2 - b*c*e^3)*f + (36*c^2*d^2*e - 3
6*b*c*d*e^2 + 5*b^2*e^3)*g)*x)/(8*c^3*d^6*e^2 - 12*b*c^2*d^5*e^3 + 6*b^2*c*d^4*e
^4 - b^3*d^3*e^5 + (8*c^3*d^3*e^5 - 12*b*c^2*d^2*e^6 + 6*b^2*c*d*e^7 - b^3*e^8)*
x^3 + 3*(8*c^3*d^4*e^4 - 12*b*c^2*d^3*e^5 + 6*b^2*c*d^2*e^6 - b^3*d*e^7)*x^2 + 3
*(8*c^3*d^5*e^3 - 12*b*c^2*d^4*e^4 + 6*b^2*c*d^3*e^5 - b^3*d^2*e^6)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f + g x}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)/(e*x+d)**3/(-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)**3), x)

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GIAC/XCAS [A]  time = 0.698936, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

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)^3),x, algorithm="giac")

[Out]

sage0*x