3.2193 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=285 \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{3465 e^2 (d+e x)^5 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{693 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{99 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (d+e x)^8 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*e^2*(2*c*d - b*
e)*(d + e*x)^8) - (2*(6*c*e*f + 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(5/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^7) - (8*c*(6*c*e*f + 16*c*d*
g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(693*e^2*(2*c*d - b*e
)^3*(d + e*x)^6) - (16*c^2*(6*c*e*f + 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^
2*x - c*e^2*x^2)^(5/2))/(3465*e^2*(2*c*d - b*e)^4*(d + e*x)^5)

_______________________________________________________________________________________

Rubi [A]  time = 1.00168, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068 \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{3465 e^2 (d+e x)^5 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{693 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{99 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (d+e x)^8 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*e^2*(2*c*d - b*
e)*(d + e*x)^8) - (2*(6*c*e*f + 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(5/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^7) - (8*c*(6*c*e*f + 16*c*d*
g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(693*e^2*(2*c*d - b*e
)^3*(d + e*x)^6) - (16*c^2*(6*c*e*f + 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^
2*x - c*e^2*x^2)^(5/2))/(3465*e^2*(2*c*d - b*e)^4*(d + e*x)^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 107.905, size = 274, normalized size = 0.96 \[ \frac{16 c^{2} \left (11 b e g - 16 c d g - 6 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{3465 e^{2} \left (d + e x\right )^{5} \left (b e - 2 c d\right )^{4}} - \frac{8 c \left (11 b e g - 16 c d g - 6 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{693 e^{2} \left (d + e x\right )^{6} \left (b e - 2 c d\right )^{3}} + \frac{2 \left (11 b e g - 16 c d g - 6 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{99 e^{2} \left (d + e x\right )^{7} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{11 e^{2} \left (d + e x\right )^{8} \left (b e - 2 c d\right )} \]

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)**(3/2)/(e*x+d)**8,x)

[Out]

16*c**2*(11*b*e*g - 16*c*d*g - 6*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d
))**(5/2)/(3465*e**2*(d + e*x)**5*(b*e - 2*c*d)**4) - 8*c*(11*b*e*g - 16*c*d*g -
6*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(693*e**2*(d + e*x)*
*6*(b*e - 2*c*d)**3) + 2*(11*b*e*g - 16*c*d*g - 6*c*e*f)*(-b*e**2*x - c*e**2*x**
2 + d*(-b*e + c*d))**(5/2)/(99*e**2*(d + e*x)**7*(b*e - 2*c*d)**2) - 2*(d*g - e*
f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(11*e**2*(d + e*x)**8*(b*e
- 2*c*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.590672, size = 249, normalized size = 0.87 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (-35 b^3 e^3 (2 d g+9 e f+11 e g x)+10 b^2 c e^2 \left (43 d^2 g+d e (210 f+254 g x)+e^2 x (21 f+22 g x)\right )-4 b c^2 e \left (212 d^3 g+d^2 e (1185 f+1391 g x)+2 d e^2 x (135 f+128 g x)+2 e^3 x^2 (15 f+11 g x)\right )+8 c^3 \left (61 d^4 g+8 d^3 e (57 f+61 g x)+d^2 e^2 x (183 f+128 g x)+16 d e^3 x^2 (3 f+g x)+6 e^4 f x^3\right )\right )}{3465 e^2 (d+e x)^6 (b e-2 c d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-35*b^3*e^3
*(9*e*f + 2*d*g + 11*e*g*x) + 8*c^3*(61*d^4*g + 6*e^4*f*x^3 + 16*d*e^3*x^2*(3*f
+ g*x) + 8*d^3*e*(57*f + 61*g*x) + d^2*e^2*x*(183*f + 128*g*x)) + 10*b^2*c*e^2*(
43*d^2*g + e^2*x*(21*f + 22*g*x) + d*e*(210*f + 254*g*x)) - 4*b*c^2*e*(212*d^3*g
+ 2*e^3*x^2*(15*f + 11*g*x) + 2*d*e^2*x*(135*f + 128*g*x) + d^2*e*(1185*f + 139
1*g*x))))/(3465*e^2*(-2*c*d + b*e)^4*(d + e*x)^6)

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 382, normalized size = 1.3 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 88\,b{c}^{2}{e}^{4}g{x}^{3}-128\,{c}^{3}d{e}^{3}g{x}^{3}-48\,{c}^{3}{e}^{4}f{x}^{3}-220\,{b}^{2}c{e}^{4}g{x}^{2}+1024\,b{c}^{2}d{e}^{3}g{x}^{2}+120\,b{c}^{2}{e}^{4}f{x}^{2}-1024\,{c}^{3}{d}^{2}{e}^{2}g{x}^{2}-384\,{c}^{3}d{e}^{3}f{x}^{2}+385\,{b}^{3}{e}^{4}gx-2540\,{b}^{2}cd{e}^{3}gx-210\,{b}^{2}c{e}^{4}fx+5564\,b{c}^{2}{d}^{2}{e}^{2}gx+1080\,b{c}^{2}d{e}^{3}fx-3904\,{c}^{3}{d}^{3}egx-1464\,{c}^{3}{d}^{2}{e}^{2}fx+70\,{b}^{3}d{e}^{3}g+315\,{b}^{3}{e}^{4}f-430\,{b}^{2}c{d}^{2}{e}^{2}g-2100\,{b}^{2}cd{e}^{3}f+848\,b{c}^{2}{d}^{3}eg+4740\,b{c}^{2}{d}^{2}{e}^{2}f-488\,{c}^{3}{d}^{4}g-3648\,{c}^{3}{d}^{3}ef \right ) }{3465\, \left ( ex+d \right ) ^{7}{e}^{2} \left ({b}^{4}{e}^{4}-8\,{b}^{3}cd{e}^{3}+24\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}-32\,b{c}^{3}{d}^{3}e+16\,{c}^{4}{d}^{4} \right ) } \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^(3/2)/(e*x+d)^8,x)

[Out]

-2/3465*(c*e*x+b*e-c*d)*(88*b*c^2*e^4*g*x^3-128*c^3*d*e^3*g*x^3-48*c^3*e^4*f*x^3
-220*b^2*c*e^4*g*x^2+1024*b*c^2*d*e^3*g*x^2+120*b*c^2*e^4*f*x^2-1024*c^3*d^2*e^2
*g*x^2-384*c^3*d*e^3*f*x^2+385*b^3*e^4*g*x-2540*b^2*c*d*e^3*g*x-210*b^2*c*e^4*f*
x+5564*b*c^2*d^2*e^2*g*x+1080*b*c^2*d*e^3*f*x-3904*c^3*d^3*e*g*x-1464*c^3*d^2*e^
2*f*x+70*b^3*d*e^3*g+315*b^3*e^4*f-430*b^2*c*d^2*e^2*g-2100*b^2*c*d*e^3*f+848*b*
c^2*d^3*e*g+4740*b*c^2*d^2*e^2*f-488*c^3*d^4*g-3648*c^3*d^3*e*f)*(-c*e^2*x^2-b*e
^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^7/e^2/(b^4*e^4-8*b^3*c*d*e^3+24*b^2*c^2*d^2*e^2-
32*b*c^3*d^3*e+16*c^4*d^4)

_______________________________________________________________________________________

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((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 36.5666, size = 1430, normalized size = 5.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d)^8,x, algorithm="fricas")

[Out]

-2/3465*(8*(6*c^5*e^6*f + (16*c^5*d*e^5 - 11*b*c^4*e^6)*g)*x^5 + 4*(6*(12*c^5*d*
e^5 - b*c^4*e^6)*f + (192*c^5*d^2*e^4 - 148*b*c^4*d*e^5 + 11*b^2*c^3*e^6)*g)*x^4
+ (6*(124*c^5*d^2*e^4 - 28*b*c^4*d*e^5 + 3*b^2*c^3*e^6)*f + (1984*c^5*d^3*e^3 -
1812*b*c^4*d^2*e^4 + 356*b^2*c^3*d*e^5 - 33*b^3*c^2*e^6)*g)*x^3 + (3*(368*c^5*d
^3*e^3 - 180*b*c^4*d^2*e^4 + 48*b^2*c^3*d*e^5 - 5*b^3*c^2*e^6)*f - 2*(3148*c^5*d
^4*e^2 - 7508*b*c^4*d^3*e^3 + 6243*b^2*c^3*d^2*e^4 - 2158*b^3*c^2*d*e^5 + 275*b^
4*c*e^6)*g)*x^2 + 3*(1216*c^5*d^5*e - 4012*b*c^4*d^4*e^2 + 5076*b^2*c^3*d^3*e^3
- 3085*b^3*c^2*d^2*e^4 + 910*b^4*c*d*e^5 - 105*b^5*e^6)*f + 2*(244*c^5*d^6 - 912
*b*c^4*d^5*e + 1307*b^2*c^3*d^4*e^2 - 889*b^3*c^2*d^3*e^3 + 285*b^4*c*d^2*e^4 -
35*b^5*d*e^5)*g - (6*(972*c^5*d^4*e^2 - 2128*b*c^4*d^3*e^3 + 1641*b^2*c^3*d^2*e^
4 - 555*b^3*c^2*d*e^5 + 70*b^4*c*e^6)*f - (2928*c^5*d^5*e - 10700*b*c^4*d^4*e^2
+ 15016*b^2*c^3*d^3*e^3 - 10029*b^3*c^2*d^2*e^4 + 3170*b^4*c*d*e^5 - 385*b^5*e^6
)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)/(16*c^4*d^10*e^2 - 32*b*c^3*d
^9*e^3 + 24*b^2*c^2*d^8*e^4 - 8*b^3*c*d^7*e^5 + b^4*d^6*e^6 + (16*c^4*d^4*e^8 -
32*b*c^3*d^3*e^9 + 24*b^2*c^2*d^2*e^10 - 8*b^3*c*d*e^11 + b^4*e^12)*x^6 + 6*(16*
c^4*d^5*e^7 - 32*b*c^3*d^4*e^8 + 24*b^2*c^2*d^3*e^9 - 8*b^3*c*d^2*e^10 + b^4*d*e
^11)*x^5 + 15*(16*c^4*d^6*e^6 - 32*b*c^3*d^5*e^7 + 24*b^2*c^2*d^4*e^8 - 8*b^3*c*
d^3*e^9 + b^4*d^2*e^10)*x^4 + 20*(16*c^4*d^7*e^5 - 32*b*c^3*d^6*e^6 + 24*b^2*c^2
*d^5*e^7 - 8*b^3*c*d^4*e^8 + b^4*d^3*e^9)*x^3 + 15*(16*c^4*d^8*e^4 - 32*b*c^3*d^
7*e^5 + 24*b^2*c^2*d^6*e^6 - 8*b^3*c*d^5*e^7 + b^4*d^4*e^8)*x^2 + 6*(16*c^4*d^9*
e^3 - 32*b*c^3*d^8*e^4 + 24*b^2*c^2*d^7*e^5 - 8*b^3*c*d^6*e^6 + b^4*d^5*e^7)*x)

_______________________________________________________________________________________

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((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 130.813, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d)^8,x, algorithm="giac")

[Out]

sage0*x