3.2181 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=360 \[ -\frac{32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{3465 e^2 (d+e x)^3 (2 c d-b e)^5}-\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{1155 e^2 (d+e x)^4 (2 c d-b e)^4}-\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{231 e^2 (d+e x)^5 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{99 e^2 (d+e x)^6 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 e^2 (d+e x)^7 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(11*e^2*(2*c*d - b*
e)*(d + e*x)^7) - (2*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(3/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^6) - (4*c*(8*c*e*f + 14*c*d*
g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(231*e^2*(2*c*d - b*e
)^3*(d + e*x)^5) - (16*c^2*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^
2*x - c*e^2*x^2)^(3/2))/(1155*e^2*(2*c*d - b*e)^4*(d + e*x)^4) - (32*c^3*(8*c*e*
f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3465*e^2*
(2*c*d - b*e)^5*(d + e*x)^3)

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Rubi [A]  time = 1.25093, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068 \[ -\frac{32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{3465 e^2 (d+e x)^3 (2 c d-b e)^5}-\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{1155 e^2 (d+e x)^4 (2 c d-b e)^4}-\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{231 e^2 (d+e x)^5 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{99 e^2 (d+e x)^6 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 e^2 (d+e x)^7 (2 c d-b e)} \]

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)^7,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(11*e^2*(2*c*d - b*
e)*(d + e*x)^7) - (2*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(3/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^6) - (4*c*(8*c*e*f + 14*c*d*
g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(231*e^2*(2*c*d - b*e
)^3*(d + e*x)^5) - (16*c^2*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^
2*x - c*e^2*x^2)^(3/2))/(1155*e^2*(2*c*d - b*e)^4*(d + e*x)^4) - (32*c^3*(8*c*e*
f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3465*e^2*
(2*c*d - b*e)^5*(d + e*x)^3)

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Rubi in Sympy [A]  time = 142.695, size = 348, normalized size = 0.97 \[ - \frac{32 c^{3} \left (11 b e g - 14 c d g - 8 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3465 e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )^{5}} + \frac{16 c^{2} \left (11 b e g - 14 c d g - 8 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{1155 e^{2} \left (d + e x\right )^{4} \left (b e - 2 c d\right )^{4}} - \frac{4 c \left (11 b e g - 14 c d g - 8 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{231 e^{2} \left (d + e x\right )^{5} \left (b e - 2 c d\right )^{3}} + \frac{2 \left (11 b e g - 14 c d g - 8 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{99 e^{2} \left (d + e x\right )^{6} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{11 e^{2} \left (d + e x\right )^{7} \left (b e - 2 c d\right )} \]

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)**7,x)

[Out]

-32*c**3*(11*b*e*g - 14*c*d*g - 8*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*
d))**(3/2)/(3465*e**2*(d + e*x)**3*(b*e - 2*c*d)**5) + 16*c**2*(11*b*e*g - 14*c*
d*g - 8*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(1155*e**2*(d +
e*x)**4*(b*e - 2*c*d)**4) - 4*c*(11*b*e*g - 14*c*d*g - 8*c*e*f)*(-b*e**2*x - c*
e**2*x**2 + d*(-b*e + c*d))**(3/2)/(231*e**2*(d + e*x)**5*(b*e - 2*c*d)**3) + 2*
(11*b*e*g - 14*c*d*g - 8*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2
)/(99*e**2*(d + e*x)**6*(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)/(11*e**2*(d + e*x)**7*(b*e - 2*c*d))

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Mathematica [A]  time = 0.488385, size = 232, normalized size = 0.64 \[ \frac{2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{16 c^4 (d+e x)^5 (-11 b e g+14 c d g+8 c e f)}{(2 c d-b e)^5}+\frac{8 c^3 (d+e x)^4 (-11 b e g+14 c d g+8 c e f)}{(b e-2 c d)^4}+\frac{6 c^2 (d+e x)^3 (-11 b e g+14 c d g+8 c e f)}{(2 c d-b e)^3}+\frac{5 c (d+e x)^2 (-11 b e g+14 c d g+8 c e f)}{(b e-2 c d)^2}-\frac{35 (d+e x) (11 b e g-23 c d g+c e f)}{b e-2 c d}+315 (d g-e f)\right )}{3465 e^2 (d+e x)^6} \]

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)^7,x]

[Out]

(2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(315*(-(e*f) + d*g) - (35*(c*e*f - 23*
c*d*g + 11*b*e*g)*(d + e*x))/(-2*c*d + b*e) + (5*c*(8*c*e*f + 14*c*d*g - 11*b*e*
g)*(d + e*x)^2)/(-2*c*d + b*e)^2 + (6*c^2*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d + e
*x)^3)/(2*c*d - b*e)^3 + (8*c^3*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d + e*x)^4)/(-2
*c*d + b*e)^4 + (16*c^4*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d + e*x)^5)/(2*c*d - b*
e)^5))/(3465*e^2*(d + e*x)^6)

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Maple [A]  time = 0.02, size = 564, normalized size = 1.6 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -176\,b{c}^{3}{e}^{5}g{x}^{4}+224\,{c}^{4}d{e}^{4}g{x}^{4}+128\,{c}^{4}{e}^{5}f{x}^{4}+264\,{b}^{2}{c}^{2}{e}^{5}g{x}^{3}-1568\,b{c}^{3}d{e}^{4}g{x}^{3}-192\,b{c}^{3}{e}^{5}f{x}^{3}+1568\,{c}^{4}{d}^{2}{e}^{3}g{x}^{3}+896\,{c}^{4}d{e}^{4}f{x}^{3}-330\,{b}^{3}c{e}^{5}g{x}^{2}+2532\,{b}^{2}{c}^{2}d{e}^{4}g{x}^{2}+240\,{b}^{2}{c}^{2}{e}^{5}f{x}^{2}-6648\,b{c}^{3}{d}^{2}{e}^{3}g{x}^{2}-1536\,b{c}^{3}d{e}^{4}f{x}^{2}+5040\,{c}^{4}{d}^{3}{e}^{2}g{x}^{2}+2880\,{c}^{4}{d}^{2}{e}^{3}f{x}^{2}+385\,{b}^{4}{e}^{5}gx-3460\,{b}^{3}cd{e}^{4}gx-280\,{b}^{3}c{e}^{5}fx+11832\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}gx+2160\,{b}^{2}{c}^{2}d{e}^{4}fx-18256\,b{c}^{3}{d}^{3}{e}^{2}gx-5856\,b{c}^{3}{d}^{2}{e}^{3}fx+10192\,{c}^{4}{d}^{4}egx+5824\,{c}^{4}{d}^{3}{e}^{2}fx+70\,{b}^{4}d{e}^{4}g+315\,{b}^{4}{e}^{5}f-610\,{b}^{3}c{d}^{2}{e}^{3}g-2800\,{b}^{3}cd{e}^{4}f+2004\,{b}^{2}{c}^{2}{d}^{3}{e}^{2}g+9480\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}f-2920\,b{c}^{3}{d}^{4}eg-14592\,b{c}^{3}{d}^{3}{e}^{2}f+1456\,{c}^{4}{d}^{5}g+8752\,{c}^{4}{d}^{4}ef \right ) }{3465\, \left ( ex+d \right ) ^{6}{e}^{2} \left ({b}^{5}{e}^{5}-10\,{b}^{4}cd{e}^{4}+40\,{b}^{3}{c}^{2}{d}^{2}{e}^{3}-80\,{b}^{2}{c}^{3}{d}^{3}{e}^{2}+80\,b{c}^{4}{d}^{4}e-32\,{c}^{5}{d}^{5} \right ) }\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}} \]

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)^7,x)

[Out]

-2/3465*(c*e*x+b*e-c*d)*(-176*b*c^3*e^5*g*x^4+224*c^4*d*e^4*g*x^4+128*c^4*e^5*f*
x^4+264*b^2*c^2*e^5*g*x^3-1568*b*c^3*d*e^4*g*x^3-192*b*c^3*e^5*f*x^3+1568*c^4*d^
2*e^3*g*x^3+896*c^4*d*e^4*f*x^3-330*b^3*c*e^5*g*x^2+2532*b^2*c^2*d*e^4*g*x^2+240
*b^2*c^2*e^5*f*x^2-6648*b*c^3*d^2*e^3*g*x^2-1536*b*c^3*d*e^4*f*x^2+5040*c^4*d^3*
e^2*g*x^2+2880*c^4*d^2*e^3*f*x^2+385*b^4*e^5*g*x-3460*b^3*c*d*e^4*g*x-280*b^3*c*
e^5*f*x+11832*b^2*c^2*d^2*e^3*g*x+2160*b^2*c^2*d*e^4*f*x-18256*b*c^3*d^3*e^2*g*x
-5856*b*c^3*d^2*e^3*f*x+10192*c^4*d^4*e*g*x+5824*c^4*d^3*e^2*f*x+70*b^4*d*e^4*g+
315*b^4*e^5*f-610*b^3*c*d^2*e^3*g-2800*b^3*c*d*e^4*f+2004*b^2*c^2*d^3*e^2*g+9480
*b^2*c^2*d^2*e^3*f-2920*b*c^3*d^4*e*g-14592*b*c^3*d^3*e^2*f+1456*c^4*d^5*g+8752*
c^4*d^4*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6/e^2/(b^5*e^5-10*b^
4*c*d*e^4+40*b^3*c^2*d^2*e^3-80*b^2*c^3*d^3*e^2+80*b*c^4*d^4*e-32*c^5*d^5)

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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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 37.2833, size = 1569, normalized size = 4.36 \[ \text{result too large to display} \]

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)^7,x, algorithm="fricas")

[Out]

2/3465*(16*(8*c^5*e^6*f + (14*c^5*d*e^5 - 11*b*c^4*e^6)*g)*x^5 + 8*(8*(12*c^5*d*
e^5 - b*c^4*e^6)*f + (168*c^5*d^2*e^4 - 146*b*c^4*d*e^5 + 11*b^2*c^3*e^6)*g)*x^4
+ 2*(8*(124*c^5*d^2*e^4 - 28*b*c^4*d*e^5 + 3*b^2*c^3*e^6)*f + (1736*c^5*d^3*e^3
- 1756*b*c^4*d^2*e^4 + 350*b^2*c^3*d*e^5 - 33*b^3*c^2*e^6)*g)*x^3 + (8*(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 + (5152*c^5*d
^4*e^2 - 6568*b*c^4*d^3*e^3 + 2652*b^2*c^3*d^2*e^4 - 598*b^3*c^2*d*e^5 + 55*b^4*
c*e^6)*g)*x^2 - (8752*c^5*d^5*e - 23344*b*c^4*d^4*e^2 + 24072*b^2*c^3*d^3*e^3 -
12280*b^3*c^2*d^2*e^4 + 3115*b^4*c*d*e^5 - 315*b^5*e^6)*f - 2*(728*c^5*d^6 - 218
8*b*c^4*d^5*e + 2462*b^2*c^3*d^4*e^2 - 1307*b^3*c^2*d^3*e^3 + 340*b^4*c*d^2*e^4
- 35*b^5*d*e^5)*g + ((2928*c^5*d^4*e^2 - 2912*b*c^4*d^3*e^3 + 1464*b^2*c^3*d^2*e
^4 - 360*b^3*c^2*d*e^5 + 35*b^4*c*e^6)*f - (8736*c^5*d^5*e - 25528*b*c^4*d^4*e^2
+ 28084*b^2*c^3*d^3*e^3 - 14682*b^3*c^2*d^2*e^4 + 3775*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)/(32*c^5*d^11*e^2 - 80*b*c^4*
d^10*e^3 + 80*b^2*c^3*d^9*e^4 - 40*b^3*c^2*d^8*e^5 + 10*b^4*c*d^7*e^6 - b^5*d^6*
e^7 + (32*c^5*d^5*e^8 - 80*b*c^4*d^4*e^9 + 80*b^2*c^3*d^3*e^10 - 40*b^3*c^2*d^2*
e^11 + 10*b^4*c*d*e^12 - b^5*e^13)*x^6 + 6*(32*c^5*d^6*e^7 - 80*b*c^4*d^5*e^8 +
80*b^2*c^3*d^4*e^9 - 40*b^3*c^2*d^3*e^10 + 10*b^4*c*d^2*e^11 - b^5*d*e^12)*x^5 +
15*(32*c^5*d^7*e^6 - 80*b*c^4*d^6*e^7 + 80*b^2*c^3*d^5*e^8 - 40*b^3*c^2*d^4*e^9
+ 10*b^4*c*d^3*e^10 - b^5*d^2*e^11)*x^4 + 20*(32*c^5*d^8*e^5 - 80*b*c^4*d^7*e^6
+ 80*b^2*c^3*d^6*e^7 - 40*b^3*c^2*d^5*e^8 + 10*b^4*c*d^4*e^9 - b^5*d^3*e^10)*x^
3 + 15*(32*c^5*d^9*e^4 - 80*b*c^4*d^8*e^5 + 80*b^2*c^3*d^7*e^6 - 40*b^3*c^2*d^6*
e^7 + 10*b^4*c*d^5*e^8 - b^5*d^4*e^9)*x^2 + 6*(32*c^5*d^10*e^3 - 80*b*c^4*d^9*e^
4 + 80*b^2*c^3*d^8*e^5 - 40*b^3*c^2*d^7*e^6 + 10*b^4*c*d^6*e^7 - b^5*d^5*e^8)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{7}}\, dx \]

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)**7,x)

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**7, x)

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GIAC/XCAS [A]  time = 1.78625, 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)^7,x, algorithm="giac")

[Out]

sage0*x