3.2277 \(\int \frac{(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=371 \[ -\frac{32 \sqrt{d+e x} (2 c d-b e)^2 (-8 b e g+11 c d g+5 c e f)}{15 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (d+e x)^{3/2} (2 c d-b e) (-8 b e g+11 c d g+5 c e f)}{15 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{4 (d+e x)^{5/2} (-8 b e g+11 c d g+5 c e f)}{15 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{7/2} (-8 b e g+11 c d g+5 c e f)}{15 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(11/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (32*(2*c*d - b*e)^2*(5*c*e*f + 11*c*d*g - 8*b
*e*g)*Sqrt[d + e*x])/(15*c^5*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (1
6*(2*c*d - b*e)*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(3/2))/(15*c^4*e^2*Sqrt
[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (4*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d +
e*x)^(5/2))/(15*c^3*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (2*(5*c*e*f
+ 11*c*d*g - 8*b*e*g)*(d + e*x)^(7/2))/(15*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2])

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Rubi [A]  time = 1.23442, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{32 \sqrt{d+e x} (2 c d-b e)^2 (-8 b e g+11 c d g+5 c e f)}{15 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (d+e x)^{3/2} (2 c d-b e) (-8 b e g+11 c d g+5 c e f)}{15 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{4 (d+e x)^{5/2} (-8 b e g+11 c d g+5 c e f)}{15 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{7/2} (-8 b e g+11 c d g+5 c e f)}{15 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^(11/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(11/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (32*(2*c*d - b*e)^2*(5*c*e*f + 11*c*d*g - 8*b
*e*g)*Sqrt[d + e*x])/(15*c^5*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (1
6*(2*c*d - b*e)*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(3/2))/(15*c^4*e^2*Sqrt
[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (4*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d +
e*x)^(5/2))/(15*c^3*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (2*(5*c*e*f
+ 11*c*d*g - 8*b*e*g)*(d + e*x)^(7/2))/(15*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2])

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Rubi in Sympy [A]  time = 137.881, size = 359, normalized size = 0.97 \[ \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (b e g - c d g - c e f\right )}{3 c e^{2} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (8 b e g - 11 c d g - 5 c e f\right )}{15 c^{2} e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{4 \left (d + e x\right )^{\frac{5}{2}} \left (8 b e g - 11 c d g - 5 c e f\right )}{15 c^{3} e^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{16 \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (8 b e g - 11 c d g - 5 c e f\right )}{15 c^{4} e^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{32 \sqrt{d + e x} \left (b e - 2 c d\right )^{2} \left (8 b e g - 11 c d g - 5 c e f\right )}{15 c^{5} e^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(11/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

2*(d + e*x)**(11/2)*(b*e*g - c*d*g - c*e*f)/(3*c*e**2*(b*e - 2*c*d)*(-b*e**2*x -
c*e**2*x**2 + d*(-b*e + c*d))**(3/2)) + 2*(d + e*x)**(7/2)*(8*b*e*g - 11*c*d*g
- 5*c*e*f)/(15*c**2*e**2*(b*e - 2*c*d)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e +
c*d))) - 4*(d + e*x)**(5/2)*(8*b*e*g - 11*c*d*g - 5*c*e*f)/(15*c**3*e**2*sqrt(-b
*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) + 16*(d + e*x)**(3/2)*(b*e - 2*c*d)*(8*
b*e*g - 11*c*d*g - 5*c*e*f)/(15*c**4*e**2*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e
+ c*d))) + 32*sqrt(d + e*x)*(b*e - 2*c*d)**2*(8*b*e*g - 11*c*d*g - 5*c*e*f)/(15
*c**5*e**2*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d)))

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Mathematica [A]  time = 0.542241, size = 263, normalized size = 0.71 \[ \frac{2 \sqrt{d+e x} \left (128 b^4 e^4 g-16 b^3 c e^3 (47 d g+5 e f-12 e g x)+24 b^2 c^2 e^2 \left (67 d^2 g+3 d e (5 f-13 g x)+e^2 x (2 g x-5 f)\right )-2 b c^3 e \left (741 d^3 g+3 d^2 e (85 f-246 g x)+3 d e^2 x (31 g x-70 f)+e^3 x^2 (15 f+4 g x)\right )+c^4 \left (498 d^4 g+9 d^3 e (25 f-83 g x)+3 d^2 e^2 x (61 g x-115 f)+d e^3 x^2 (75 f+23 g x)+e^4 x^3 (5 f+3 g x)\right )\right )}{15 c^5 e^2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^(11/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[d + e*x]*(128*b^4*e^4*g - 16*b^3*c*e^3*(5*e*f + 47*d*g - 12*e*g*x) + 24*
b^2*c^2*e^2*(67*d^2*g + 3*d*e*(5*f - 13*g*x) + e^2*x*(-5*f + 2*g*x)) - 2*b*c^3*e
*(741*d^3*g + 3*d^2*e*(85*f - 246*g*x) + e^3*x^2*(15*f + 4*g*x) + 3*d*e^2*x*(-70
*f + 31*g*x)) + c^4*(498*d^4*g + 9*d^3*e*(25*f - 83*g*x) + e^4*x^3*(5*f + 3*g*x)
+ d*e^3*x^2*(75*f + 23*g*x) + 3*d^2*e^2*x*(-115*f + 61*g*x))))/(15*c^5*e^2*(-(c
*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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Maple [A]  time = 0.012, size = 367, normalized size = 1. \[{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 3\,g{e}^{4}{x}^{4}{c}^{4}-8\,b{c}^{3}{e}^{4}g{x}^{3}+23\,{c}^{4}d{e}^{3}g{x}^{3}+5\,{c}^{4}{e}^{4}f{x}^{3}+48\,{b}^{2}{c}^{2}{e}^{4}g{x}^{2}-186\,b{c}^{3}d{e}^{3}g{x}^{2}-30\,b{c}^{3}{e}^{4}f{x}^{2}+183\,{c}^{4}{d}^{2}{e}^{2}g{x}^{2}+75\,{c}^{4}d{e}^{3}f{x}^{2}+192\,{b}^{3}c{e}^{4}gx-936\,{b}^{2}{c}^{2}d{e}^{3}gx-120\,{b}^{2}{c}^{2}{e}^{4}fx+1476\,b{c}^{3}{d}^{2}{e}^{2}gx+420\,b{c}^{3}d{e}^{3}fx-747\,{c}^{4}{d}^{3}egx-345\,{c}^{4}{d}^{2}{e}^{2}fx+128\,{b}^{4}{e}^{4}g-752\,{b}^{3}cd{e}^{3}g-80\,{b}^{3}c{e}^{4}f+1608\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}g+360\,{b}^{2}{c}^{2}d{e}^{3}f-1482\,b{c}^{3}{d}^{3}eg-510\,b{c}^{3}{d}^{2}{e}^{2}f+498\,{c}^{4}{d}^{4}g+225\,f{d}^{3}{c}^{4}e \right ) }{15\,{c}^{5}{e}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

2/15*(c*e*x+b*e-c*d)*(3*c^4*e^4*g*x^4-8*b*c^3*e^4*g*x^3+23*c^4*d*e^3*g*x^3+5*c^4
*e^4*f*x^3+48*b^2*c^2*e^4*g*x^2-186*b*c^3*d*e^3*g*x^2-30*b*c^3*e^4*f*x^2+183*c^4
*d^2*e^2*g*x^2+75*c^4*d*e^3*f*x^2+192*b^3*c*e^4*g*x-936*b^2*c^2*d*e^3*g*x-120*b^
2*c^2*e^4*f*x+1476*b*c^3*d^2*e^2*g*x+420*b*c^3*d*e^3*f*x-747*c^4*d^3*e*g*x-345*c
^4*d^2*e^2*f*x+128*b^4*e^4*g-752*b^3*c*d*e^3*g-80*b^3*c*e^4*f+1608*b^2*c^2*d^2*e
^2*g+360*b^2*c^2*d*e^3*f-1482*b*c^3*d^3*e*g-510*b*c^3*d^2*e^2*f+498*c^4*d^4*g+22
5*c^4*d^3*e*f)*(e*x+d)^(5/2)/c^5/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)

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Maxima [A]  time = 0.752901, size = 490, normalized size = 1.32 \[ \frac{2 \,{\left (c^{3} e^{3} x^{3} + 45 \, c^{3} d^{3} - 102 \, b c^{2} d^{2} e + 72 \, b^{2} c d e^{2} - 16 \, b^{3} e^{3} + 3 \,{\left (5 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} x^{2} - 3 \,{\left (23 \, c^{3} d^{2} e - 28 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} f}{3 \,{\left (c^{5} e^{2} x - c^{5} d e + b c^{4} e^{2}\right )} \sqrt{-c e x + c d - b e}} + \frac{2 \,{\left (3 \, c^{4} e^{4} x^{4} + 498 \, c^{4} d^{4} - 1482 \, b c^{3} d^{3} e + 1608 \, b^{2} c^{2} d^{2} e^{2} - 752 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} +{\left (23 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} x^{3} + 3 \,{\left (61 \, c^{4} d^{2} e^{2} - 62 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} x^{2} - 3 \,{\left (249 \, c^{4} d^{3} e - 492 \, b c^{3} d^{2} e^{2} + 312 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} x\right )} g}{15 \,{\left (c^{6} e^{3} x - c^{6} d e^{2} + b c^{5} e^{3}\right )} \sqrt{-c e x + c d - b e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(11/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="maxima")

[Out]

2/3*(c^3*e^3*x^3 + 45*c^3*d^3 - 102*b*c^2*d^2*e + 72*b^2*c*d*e^2 - 16*b^3*e^3 +
3*(5*c^3*d*e^2 - 2*b*c^2*e^3)*x^2 - 3*(23*c^3*d^2*e - 28*b*c^2*d*e^2 + 8*b^2*c*e
^3)*x)*f/((c^5*e^2*x - c^5*d*e + b*c^4*e^2)*sqrt(-c*e*x + c*d - b*e)) + 2/15*(3*
c^4*e^4*x^4 + 498*c^4*d^4 - 1482*b*c^3*d^3*e + 1608*b^2*c^2*d^2*e^2 - 752*b^3*c*
d*e^3 + 128*b^4*e^4 + (23*c^4*d*e^3 - 8*b*c^3*e^4)*x^3 + 3*(61*c^4*d^2*e^2 - 62*
b*c^3*d*e^3 + 16*b^2*c^2*e^4)*x^2 - 3*(249*c^4*d^3*e - 492*b*c^3*d^2*e^2 + 312*b
^2*c^2*d*e^3 - 64*b^3*c*e^4)*x)*g/((c^6*e^3*x - c^6*d*e^2 + b*c^5*e^3)*sqrt(-c*e
*x + c*d - b*e))

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Fricas [A]  time = 0.282279, size = 649, normalized size = 1.75 \[ \frac{2 \,{\left (3 \, c^{4} e^{5} g x^{5} +{\left (5 \, c^{4} e^{5} f + 2 \,{\left (13 \, c^{4} d e^{4} - 4 \, b c^{3} e^{5}\right )} g\right )} x^{4} + 2 \,{\left (5 \,{\left (8 \, c^{4} d e^{4} - 3 \, b c^{3} e^{5}\right )} f +{\left (103 \, c^{4} d^{2} e^{3} - 97 \, b c^{3} d e^{4} + 24 \, b^{2} c^{2} e^{5}\right )} g\right )} x^{3} - 6 \,{\left (5 \,{\left (9 \, c^{4} d^{2} e^{3} - 13 \, b c^{3} d e^{4} + 4 \, b^{2} c^{2} e^{5}\right )} f +{\left (94 \, c^{4} d^{3} e^{2} - 215 \, b c^{3} d^{2} e^{3} + 148 \, b^{2} c^{2} d e^{4} - 32 \, b^{3} c e^{5}\right )} g\right )} x^{2} + 5 \,{\left (45 \, c^{4} d^{4} e - 102 \, b c^{3} d^{3} e^{2} + 72 \, b^{2} c^{2} d^{2} e^{3} - 16 \, b^{3} c d e^{4}\right )} f + 2 \,{\left (249 \, c^{4} d^{5} - 741 \, b c^{3} d^{4} e + 804 \, b^{2} c^{2} d^{3} e^{2} - 376 \, b^{3} c d^{2} e^{3} + 64 \, b^{4} d e^{4}\right )} g -{\left (10 \,{\left (12 \, c^{4} d^{3} e^{2} + 9 \, b c^{3} d^{2} e^{3} - 24 \, b^{2} c^{2} d e^{4} + 8 \, b^{3} c e^{5}\right )} f +{\left (249 \, c^{4} d^{4} e + 6 \, b c^{3} d^{3} e^{2} - 672 \, b^{2} c^{2} d^{2} e^{3} + 560 \, b^{3} c d e^{4} - 128 \, b^{4} e^{5}\right )} g\right )} x\right )}}{15 \,{\left (c^{6} e^{3} x - c^{6} d e^{2} + b c^{5} e^{3}\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(11/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*c^4*e^5*g*x^5 + (5*c^4*e^5*f + 2*(13*c^4*d*e^4 - 4*b*c^3*e^5)*g)*x^4 + 2
*(5*(8*c^4*d*e^4 - 3*b*c^3*e^5)*f + (103*c^4*d^2*e^3 - 97*b*c^3*d*e^4 + 24*b^2*c
^2*e^5)*g)*x^3 - 6*(5*(9*c^4*d^2*e^3 - 13*b*c^3*d*e^4 + 4*b^2*c^2*e^5)*f + (94*c
^4*d^3*e^2 - 215*b*c^3*d^2*e^3 + 148*b^2*c^2*d*e^4 - 32*b^3*c*e^5)*g)*x^2 + 5*(4
5*c^4*d^4*e - 102*b*c^3*d^3*e^2 + 72*b^2*c^2*d^2*e^3 - 16*b^3*c*d*e^4)*f + 2*(24
9*c^4*d^5 - 741*b*c^3*d^4*e + 804*b^2*c^2*d^3*e^2 - 376*b^3*c*d^2*e^3 + 64*b^4*d
*e^4)*g - (10*(12*c^4*d^3*e^2 + 9*b*c^3*d^2*e^3 - 24*b^2*c^2*d*e^4 + 8*b^3*c*e^5
)*f + (249*c^4*d^4*e + 6*b*c^3*d^3*e^2 - 672*b^2*c^2*d^2*e^3 + 560*b^3*c*d*e^4 -
128*b^4*e^5)*g)*x)/((c^6*e^3*x - c^6*d*e^2 + b*c^5*e^3)*sqrt(-c*e^2*x^2 - b*e^2
*x + c*d^2 - b*d*e)*sqrt(e*x + d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(11/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.875322, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(11/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="giac")

[Out]

sage0*x