3.2182 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=439 \[ -\frac{256 c^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{45045 e^2 (d+e x)^3 (2 c d-b e)^6}+\frac{128 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{15015 e^2 (d+e x)^4 (2 c d-b e)^5}-\frac{32 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{3003 e^2 (d+e x)^5 (2 c d-b e)^4}+\frac{16 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{1287 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{143 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (d+e x)^8 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(13*e^2*(2*c*d - b*
e)*(d + e*x)^8) - (2*(10*c*e*f + 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(3/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^7) + (16*c*(13*b*e*g - 2*c
*(5*e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(1287*e^2*(2*c*d
- b*e)^3*(d + e*x)^6) - (32*c^2*(10*c*e*f + 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(3/2))/(3003*e^2*(2*c*d - b*e)^4*(d + e*x)^5) + (128*c^3*
(13*b*e*g - 2*c*(5*e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(1
5015*e^2*(2*c*d - b*e)^5*(d + e*x)^4) - (256*c^4*(10*c*e*f + 16*c*d*g - 13*b*e*g
)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(45045*e^2*(2*c*d - b*e)^6*(d + e
*x)^3)

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Rubi [A]  time = 1.57107, antiderivative size = 439, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068 \[ -\frac{256 c^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{45045 e^2 (d+e x)^3 (2 c d-b e)^6}+\frac{128 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{15015 e^2 (d+e x)^4 (2 c d-b e)^5}-\frac{32 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{3003 e^2 (d+e x)^5 (2 c d-b e)^4}+\frac{16 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (13 b e g-2 c (8 d g+5 e f))}{1287 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+16 c d g+10 c e f)}{143 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (d+e x)^8 (2 c d-b e)} \]

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)^8,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(13*e^2*(2*c*d - b*
e)*(d + e*x)^8) - (2*(10*c*e*f + 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(3/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^7) + (16*c*(13*b*e*g - 2*c
*(5*e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(1287*e^2*(2*c*d
- b*e)^3*(d + e*x)^6) - (32*c^2*(10*c*e*f + 16*c*d*g - 13*b*e*g)*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(3/2))/(3003*e^2*(2*c*d - b*e)^4*(d + e*x)^5) + (128*c^3*
(13*b*e*g - 2*c*(5*e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(1
5015*e^2*(2*c*d - b*e)^5*(d + e*x)^4) - (256*c^4*(10*c*e*f + 16*c*d*g - 13*b*e*g
)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(45045*e^2*(2*c*d - b*e)^6*(d + e
*x)^3)

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Rubi in Sympy [A]  time = 178.631, size = 423, normalized size = 0.96 \[ \frac{256 c^{4} \left (13 b e g - 16 c d g - 10 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{45045 e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )^{6}} - \frac{128 c^{3} \left (13 b e g - 16 c d g - 10 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{15015 e^{2} \left (d + e x\right )^{4} \left (b e - 2 c d\right )^{5}} + \frac{32 c^{2} \left (13 b e g - 16 c d g - 10 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3003 e^{2} \left (d + e x\right )^{5} \left (b e - 2 c d\right )^{4}} - \frac{16 c \left (13 b e g - 16 c d g - 10 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{1287 e^{2} \left (d + e x\right )^{6} \left (b e - 2 c d\right )^{3}} + \frac{2 \left (13 b e g - 16 c d g - 10 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{143 e^{2} \left (d + e x\right )^{7} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{13 e^{2} \left (d + e x\right )^{8} \left (b e - 2 c d\right )} \]

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)**(1/2)/(e*x+d)**8,x)

[Out]

256*c**4*(13*b*e*g - 16*c*d*g - 10*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c
*d))**(3/2)/(45045*e**2*(d + e*x)**3*(b*e - 2*c*d)**6) - 128*c**3*(13*b*e*g - 16
*c*d*g - 10*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(15015*e**2
*(d + e*x)**4*(b*e - 2*c*d)**5) + 32*c**2*(13*b*e*g - 16*c*d*g - 10*c*e*f)*(-b*e
**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(3003*e**2*(d + e*x)**5*(b*e - 2*c*
d)**4) - 16*c*(13*b*e*g - 16*c*d*g - 10*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*
e + c*d))**(3/2)/(1287*e**2*(d + e*x)**6*(b*e - 2*c*d)**3) + 2*(13*b*e*g - 16*c*
d*g - 10*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(143*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))**(3/2)/(13*e**2*(d + e*x)**8*(b*e - 2*c*d))

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Mathematica [A]  time = 0.739042, size = 280, normalized size = 0.64 \[ \frac{2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{128 c^5 (d+e x)^6 (2 c (8 d g+5 e f)-13 b e g)}{(b e-2 c d)^6}+\frac{64 c^4 (d+e x)^5 (2 c (8 d g+5 e f)-13 b e g)}{(2 c d-b e)^5}+\frac{48 c^3 (d+e x)^4 (2 c (8 d g+5 e f)-13 b e g)}{(b e-2 c d)^4}+\frac{40 c^2 (d+e x)^3 (2 c (8 d g+5 e f)-13 b e g)}{(2 c d-b e)^3}+\frac{35 c (d+e x)^2 (2 c (8 d g+5 e f)-13 b e g)}{(b e-2 c d)^2}-\frac{315 (d+e x) (13 b e g-27 c d g+c e f)}{b e-2 c d}+3465 (d g-e f)\right )}{45045 e^2 (d+e x)^7} \]

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)^8,x]

[Out]

(2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(3465*(-(e*f) + d*g) - (315*(c*e*f - 2
7*c*d*g + 13*b*e*g)*(d + e*x))/(-2*c*d + b*e) + (35*c*(-13*b*e*g + 2*c*(5*e*f +
8*d*g))*(d + e*x)^2)/(-2*c*d + b*e)^2 + (40*c^2*(-13*b*e*g + 2*c*(5*e*f + 8*d*g)
)*(d + e*x)^3)/(2*c*d - b*e)^3 + (48*c^3*(-13*b*e*g + 2*c*(5*e*f + 8*d*g))*(d +
e*x)^4)/(-2*c*d + b*e)^4 + (64*c^4*(-13*b*e*g + 2*c*(5*e*f + 8*d*g))*(d + e*x)^5
)/(2*c*d - b*e)^5 + (128*c^5*(-13*b*e*g + 2*c*(5*e*f + 8*d*g))*(d + e*x)^6)/(-2*
c*d + b*e)^6))/(45045*e^2*(d + e*x)^7)

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Maple [A]  time = 0.023, size = 782, normalized size = 1.8 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 1664\,b{c}^{4}{e}^{6}g{x}^{5}-2048\,{c}^{5}d{e}^{5}g{x}^{5}-1280\,{c}^{5}{e}^{6}f{x}^{5}-2496\,{b}^{2}{c}^{3}{e}^{6}g{x}^{4}+16384\,b{c}^{4}d{e}^{5}g{x}^{4}+1920\,b{c}^{4}{e}^{6}f{x}^{4}-16384\,{c}^{5}{d}^{2}{e}^{4}g{x}^{4}-10240\,{c}^{5}d{e}^{5}f{x}^{4}+3120\,{b}^{3}{c}^{2}{e}^{6}g{x}^{3}-26304\,{b}^{2}{c}^{3}d{e}^{5}g{x}^{3}-2400\,{b}^{2}{c}^{3}{e}^{6}f{x}^{3}+76736\,b{c}^{4}{d}^{2}{e}^{4}g{x}^{3}+17280\,b{c}^{4}d{e}^{5}f{x}^{3}-60416\,{c}^{5}{d}^{3}{e}^{3}g{x}^{3}-37760\,{c}^{5}{d}^{2}{e}^{4}f{x}^{3}-3640\,{b}^{4}c{e}^{6}g{x}^{2}+35680\,{b}^{3}{c}^{2}d{e}^{5}g{x}^{2}+2800\,{b}^{3}{c}^{2}{e}^{6}f{x}^{2}-134496\,{b}^{2}{c}^{3}{d}^{2}{e}^{4}g{x}^{2}-24000\,{b}^{2}{c}^{3}d{e}^{5}f{x}^{2}+231424\,b{c}^{4}{d}^{3}{e}^{3}g{x}^{2}+73920\,b{c}^{4}{d}^{2}{e}^{4}f{x}^{2}-139264\,{c}^{5}{d}^{4}{e}^{2}g{x}^{2}-87040\,{c}^{5}{d}^{3}{e}^{3}f{x}^{2}+4095\,{b}^{5}{e}^{6}gx-45080\,{b}^{4}cd{e}^{5}gx-3150\,{b}^{4}c{e}^{6}fx+200600\,{b}^{3}{c}^{2}{d}^{2}{e}^{4}gx+30800\,{b}^{3}{c}^{2}d{e}^{5}fx-452064\,{b}^{2}{c}^{3}{d}^{3}{e}^{3}gx-116400\,{b}^{2}{c}^{3}{d}^{2}{e}^{4}fx+516656\,b{c}^{4}{d}^{4}{e}^{2}gx+204480\,b{c}^{4}{d}^{3}{e}^{3}fx-233216\,{c}^{5}{d}^{5}egx-145760\,{c}^{5}{d}^{4}{e}^{2}fx+630\,{b}^{5}d{e}^{5}g+3465\,{b}^{5}{e}^{6}f-6790\,{b}^{4}c{d}^{2}{e}^{4}g-37800\,{b}^{4}cd{e}^{5}f+29440\,{b}^{3}{c}^{2}{d}^{3}{e}^{3}g+166600\,{b}^{3}{c}^{2}{d}^{2}{e}^{4}f-64176\,{b}^{2}{c}^{3}{d}^{4}{e}^{2}g-372000\,{b}^{2}{c}^{3}{d}^{3}{e}^{3}f+70048\,b{c}^{4}{d}^{5}eg+423120\,b{c}^{4}{d}^{4}{e}^{2}f-29152\,{c}^{5}{d}^{6}g-198400\,{c}^{5}{d}^{5}ef \right ) }{45045\, \left ( ex+d \right ) ^{7}{e}^{2} \left ({b}^{6}{e}^{6}-12\,{b}^{5}cd{e}^{5}+60\,{b}^{4}{c}^{2}{d}^{2}{e}^{4}-160\,{b}^{3}{c}^{3}{d}^{3}{e}^{3}+240\,{b}^{2}{c}^{4}{d}^{4}{e}^{2}-192\,b{c}^{5}{d}^{5}e+64\,{c}^{6}{d}^{6} \right ) }\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,x)

[Out]

-2/45045*(c*e*x+b*e-c*d)*(1664*b*c^4*e^6*g*x^5-2048*c^5*d*e^5*g*x^5-1280*c^5*e^6
*f*x^5-2496*b^2*c^3*e^6*g*x^4+16384*b*c^4*d*e^5*g*x^4+1920*b*c^4*e^6*f*x^4-16384
*c^5*d^2*e^4*g*x^4-10240*c^5*d*e^5*f*x^4+3120*b^3*c^2*e^6*g*x^3-26304*b^2*c^3*d*
e^5*g*x^3-2400*b^2*c^3*e^6*f*x^3+76736*b*c^4*d^2*e^4*g*x^3+17280*b*c^4*d*e^5*f*x
^3-60416*c^5*d^3*e^3*g*x^3-37760*c^5*d^2*e^4*f*x^3-3640*b^4*c*e^6*g*x^2+35680*b^
3*c^2*d*e^5*g*x^2+2800*b^3*c^2*e^6*f*x^2-134496*b^2*c^3*d^2*e^4*g*x^2-24000*b^2*
c^3*d*e^5*f*x^2+231424*b*c^4*d^3*e^3*g*x^2+73920*b*c^4*d^2*e^4*f*x^2-139264*c^5*
d^4*e^2*g*x^2-87040*c^5*d^3*e^3*f*x^2+4095*b^5*e^6*g*x-45080*b^4*c*d*e^5*g*x-315
0*b^4*c*e^6*f*x+200600*b^3*c^2*d^2*e^4*g*x+30800*b^3*c^2*d*e^5*f*x-452064*b^2*c^
3*d^3*e^3*g*x-116400*b^2*c^3*d^2*e^4*f*x+516656*b*c^4*d^4*e^2*g*x+204480*b*c^4*d
^3*e^3*f*x-233216*c^5*d^5*e*g*x-145760*c^5*d^4*e^2*f*x+630*b^5*d*e^5*g+3465*b^5*
e^6*f-6790*b^4*c*d^2*e^4*g-37800*b^4*c*d*e^5*f+29440*b^3*c^2*d^3*e^3*g+166600*b^
3*c^2*d^2*e^4*f-64176*b^2*c^3*d^4*e^2*g-372000*b^2*c^3*d^3*e^3*f+70048*b*c^4*d^5
*e*g+423120*b*c^4*d^4*e^2*f-29152*c^5*d^6*g-198400*c^5*d^5*e*f)*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^7/e^2/(b^6*e^6-12*b^5*c*d*e^5+60*b^4*c^2*d^2*e^4-
160*b^3*c^3*d^3*e^3+240*b^2*c^4*d^4*e^2-192*b*c^5*d^5*e+64*c^6*d^6)

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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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 96.673, size = 2093, normalized size = 4.77 \[ \text{result too large to display} \]

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)^8,x, algorithm="fricas")

[Out]

2/45045*(128*(10*c^6*e^7*f + (16*c^6*d*e^6 - 13*b*c^5*e^7)*g)*x^6 + 64*(10*(14*c
^6*d*e^6 - b*c^5*e^7)*f + (224*c^6*d^2*e^5 - 198*b*c^5*d*e^6 + 13*b^2*c^4*e^7)*g
)*x^5 + 16*(10*(172*c^6*d^2*e^5 - 32*b*c^5*d*e^6 + 3*b^2*c^4*e^7)*f + (2752*c^6*
d^3*e^4 - 2748*b*c^5*d^2*e^5 + 464*b^2*c^4*d*e^6 - 39*b^3*c^3*e^7)*g)*x^4 + 8*(1
0*(616*c^6*d^3*e^4 - 236*b*c^5*d^2*e^5 + 54*b^2*c^4*d*e^6 - 5*b^3*c^3*e^7)*f + (
9856*c^6*d^4*e^3 - 11784*b*c^5*d^3*e^4 + 3932*b^2*c^4*d^2*e^5 - 782*b^3*c^3*d*e^
6 + 65*b^4*c^2*e^7)*g)*x^3 + (10*(5872*c^6*d^4*e^3 - 4352*b*c^5*d^3*e^4 + 1848*b
^2*c^4*d^2*e^5 - 400*b^3*c^3*d*e^6 + 35*b^4*c^2*e^7)*f + (93952*c^6*d^5*e^2 - 14
5968*b*c^5*d^4*e^3 + 86144*b^2*c^4*d^3*e^4 - 30424*b^3*c^3*d^2*e^5 + 5760*b^4*c^
2*d*e^6 - 455*b^5*c*e^7)*g)*x^2 - 5*(39680*c^6*d^6*e - 124304*b*c^5*d^5*e^2 + 15
9024*b^2*c^4*d^4*e^3 - 107720*b^3*c^3*d^3*e^4 + 40880*b^4*c^2*d^2*e^5 - 8253*b^5
*c*d*e^6 + 693*b^6*e^7)*f - 2*(14576*c^6*d^7 - 49600*b*c^5*d^6*e + 67112*b^2*c^4
*d^5*e^2 - 46808*b^3*c^3*d^4*e^3 + 18115*b^4*c^2*d^3*e^4 - 3710*b^5*c*d^2*e^5 +
315*b^6*d*e^6)*g + (5*(10528*c^6*d^5*e^2 - 14576*b*c^5*d^4*e^3 + 10224*b^2*c^4*d
^3*e^4 - 3880*b^3*c^3*d^2*e^5 + 770*b^4*c^2*d*e^6 - 63*b^5*c*e^7)*f - (204064*c^
6*d^6*e - 679824*b*c^5*d^5*e^2 + 904544*b^2*c^4*d^4*e^3 - 623224*b^3*c^3*d^3*e^4
+ 238890*b^4*c^2*d^2*e^5 - 48545*b^5*c*d*e^6 + 4095*b^6*e^7)*g)*x)*sqrt(-c*e^2*
x^2 - b*e^2*x + c*d^2 - b*d*e)/(64*c^6*d^13*e^2 - 192*b*c^5*d^12*e^3 + 240*b^2*c
^4*d^11*e^4 - 160*b^3*c^3*d^10*e^5 + 60*b^4*c^2*d^9*e^6 - 12*b^5*c*d^8*e^7 + b^6
*d^7*e^8 + (64*c^6*d^6*e^9 - 192*b*c^5*d^5*e^10 + 240*b^2*c^4*d^4*e^11 - 160*b^3
*c^3*d^3*e^12 + 60*b^4*c^2*d^2*e^13 - 12*b^5*c*d*e^14 + b^6*e^15)*x^7 + 7*(64*c^
6*d^7*e^8 - 192*b*c^5*d^6*e^9 + 240*b^2*c^4*d^5*e^10 - 160*b^3*c^3*d^4*e^11 + 60
*b^4*c^2*d^3*e^12 - 12*b^5*c*d^2*e^13 + b^6*d*e^14)*x^6 + 21*(64*c^6*d^8*e^7 - 1
92*b*c^5*d^7*e^8 + 240*b^2*c^4*d^6*e^9 - 160*b^3*c^3*d^5*e^10 + 60*b^4*c^2*d^4*e
^11 - 12*b^5*c*d^3*e^12 + b^6*d^2*e^13)*x^5 + 35*(64*c^6*d^9*e^6 - 192*b*c^5*d^8
*e^7 + 240*b^2*c^4*d^7*e^8 - 160*b^3*c^3*d^6*e^9 + 60*b^4*c^2*d^5*e^10 - 12*b^5*
c*d^4*e^11 + b^6*d^3*e^12)*x^4 + 35*(64*c^6*d^10*e^5 - 192*b*c^5*d^9*e^6 + 240*b
^2*c^4*d^8*e^7 - 160*b^3*c^3*d^7*e^8 + 60*b^4*c^2*d^6*e^9 - 12*b^5*c*d^5*e^10 +
b^6*d^4*e^11)*x^3 + 21*(64*c^6*d^11*e^4 - 192*b*c^5*d^10*e^5 + 240*b^2*c^4*d^9*e
^6 - 160*b^3*c^3*d^8*e^7 + 60*b^4*c^2*d^7*e^8 - 12*b^5*c*d^6*e^9 + b^6*d^5*e^10)
*x^2 + 7*(64*c^6*d^12*e^3 - 192*b*c^5*d^11*e^4 + 240*b^2*c^4*d^10*e^5 - 160*b^3*
c^3*d^9*e^6 + 60*b^4*c^2*d^8*e^7 - 12*b^5*c*d^7*e^8 + b^6*d^6*e^9)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{8}}\, dx \]

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)**8,x)

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**8, x)

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GIAC/XCAS [A]  time = 2.32419, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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)^8,x, algorithm="giac")

[Out]

sage0*x