3.2206 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=210 \[ -\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+18 c d g+4 c e f)}{693 e^2 (d+e x)^7 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+18 c d g+4 c e f)}{99 e^2 (d+e x)^8 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e^2 (d+e x)^9 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(11*e^2*(2*c*d - b*
e)*(d + e*x)^9) - (2*(4*c*e*f + 18*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(7/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^8) - (4*c*(4*c*e*f + 18*c*d*
g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(693*e^2*(2*c*d - b*e
)^3*(d + e*x)^7)

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Rubi [A]  time = 0.764411, 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 \[ -\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+18 c d g+4 c e f)}{693 e^2 (d+e x)^7 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+18 c d g+4 c e f)}{99 e^2 (d+e x)^8 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e^2 (d+e x)^9 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^9,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(11*e^2*(2*c*d - b*
e)*(d + e*x)^9) - (2*(4*c*e*f + 18*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(7/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^8) - (4*c*(4*c*e*f + 18*c*d*
g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(693*e^2*(2*c*d - b*e
)^3*(d + e*x)^7)

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Rubi in Sympy [A]  time = 78.7255, size = 199, normalized size = 0.95 \[ - \frac{4 c \left (11 b e g - 18 c d g - 4 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{693 e^{2} \left (d + e x\right )^{7} \left (b e - 2 c d\right )^{3}} + \frac{2 \left (11 b e g - 18 c d g - 4 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{99 e^{2} \left (d + e x\right )^{8} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{11 e^{2} \left (d + e x\right )^{9} \left (b e - 2 c d\right )} \]

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)**(5/2)/(e*x+d)**9,x)

[Out]

-4*c*(11*b*e*g - 18*c*d*g - 4*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))*
*(7/2)/(693*e**2*(d + e*x)**7*(b*e - 2*c*d)**3) + 2*(11*b*e*g - 18*c*d*g - 4*c*e
*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(99*e**2*(d + e*x)**8*(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))**(7/2)/
(11*e**2*(d + e*x)**9*(b*e - 2*c*d))

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Mathematica [A]  time = 0.573682, size = 169, normalized size = 0.8 \[ -\frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (7 b^2 e^2 (2 d g+9 e f+11 e g x)-2 b c e \left (25 d^2 g+2 d e (70 f+81 g x)+e^2 x (14 f+11 g x)\right )+4 c^2 \left (9 d^3 g+d^2 e (79 f+81 g x)+9 d e^2 x (2 f+g x)+2 e^3 f x^2\right )\right )}{693 e^2 (d+e x)^6 (b e-2 c d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^9,x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(7*b^2*e^2*(
9*e*f + 2*d*g + 11*e*g*x) - 2*b*c*e*(25*d^2*g + e^2*x*(14*f + 11*g*x) + 2*d*e*(7
0*f + 81*g*x)) + 4*c^2*(9*d^3*g + 2*e^3*f*x^2 + 9*d*e^2*x*(2*f + g*x) + d^2*e*(7
9*f + 81*g*x))))/(693*e^2*(-2*c*d + b*e)^3*(d + e*x)^6)

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Maple [A]  time = 0.014, size = 236, normalized size = 1.1 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -22\,bc{e}^{3}g{x}^{2}+36\,{c}^{2}d{e}^{2}g{x}^{2}+8\,{c}^{2}{e}^{3}f{x}^{2}+77\,{b}^{2}{e}^{3}gx-324\,bcd{e}^{2}gx-28\,bc{e}^{3}fx+324\,{c}^{2}{d}^{2}egx+72\,{c}^{2}d{e}^{2}fx+14\,{b}^{2}d{e}^{2}g+63\,{b}^{2}{e}^{3}f-50\,bc{d}^{2}eg-280\,bcd{e}^{2}f+36\,{c}^{2}{d}^{3}g+316\,{c}^{2}{d}^{2}ef \right ) }{693\, \left ( ex+d \right ) ^{8} \left ({b}^{3}{e}^{3}-6\,{b}^{2}cd{e}^{2}+12\,b{c}^{2}{d}^{2}e-8\,{c}^{3}{d}^{3} \right ){e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^9,x)

[Out]

-2/693*(c*e*x+b*e-c*d)*(-22*b*c*e^3*g*x^2+36*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x^2+77*
b^2*e^3*g*x-324*b*c*d*e^2*g*x-28*b*c*e^3*f*x+324*c^2*d^2*e*g*x+72*c^2*d*e^2*f*x+
14*b^2*d*e^2*g+63*b^2*e^3*f-50*b*c*d^2*e*g-280*b*c*d*e^2*f+36*c^2*d^3*g+316*c^2*
d^2*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^8/(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

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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((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 36.7871, size = 1303, normalized size = 6.2 \[ \text{result too large to display} \]

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)^(5/2)*(g*x + f)/(e*x + d)^9,x, algorithm="fricas")

[Out]

2/693*(2*(4*c^5*e^6*f + (18*c^5*d*e^5 - 11*b*c^4*e^6)*g)*x^5 + (4*(12*c^5*d*e^5
- b*c^4*e^6)*f + (216*c^5*d^2*e^4 - 150*b*c^4*d*e^5 + 11*b^2*c^3*e^6)*g)*x^4 + (
(124*c^5*d^2*e^4 - 28*b*c^4*d*e^5 + 3*b^2*c^3*e^6)*f - (828*c^5*d^3*e^3 - 1612*b
*c^4*d^2*e^4 + 949*b^2*c^3*d*e^5 - 165*b^3*c^2*e^6)*g)*x^3 - ((740*c^5*d^3*e^3 -
1296*b*c^4*d^2*e^4 + 669*b^2*c^3*d*e^5 - 113*b^3*c^2*e^6)*f - (828*c^5*d^4*e^2
- 2528*b*c^4*d^3*e^3 + 2781*b^2*c^3*d^2*e^4 - 1290*b^3*c^2*d*e^5 + 209*b^4*c*e^6
)*g)*x^2 - (316*c^5*d^5*e - 1228*b*c^4*d^4*e^2 + 1851*b^2*c^3*d^3*e^3 - 1345*b^3
*c^2*d^2*e^4 + 469*b^4*c*d*e^5 - 63*b^5*e^6)*f - 2*(18*c^5*d^6 - 79*b*c^4*d^5*e
+ 136*b^2*c^3*d^4*e^2 - 114*b^3*c^2*d^3*e^3 + 46*b^4*c*d^2*e^4 - 7*b^5*d*e^5)*g
+ ((876*c^5*d^4*e^2 - 2492*b*c^4*d^3*e^3 + 2517*b^2*c^3*d^2*e^4 - 1062*b^3*c^2*d
*e^5 + 161*b^4*c*e^6)*f - (216*c^5*d^5*e - 930*b*c^4*d^4*e^2 + 1571*b^2*c^3*d^3*
e^3 - 1293*b^3*c^2*d^2*e^4 + 513*b^4*c*d*e^5 - 77*b^5*e^6)*g)*x)*sqrt(-c*e^2*x^2
- b*e^2*x + c*d^2 - b*d*e)/(8*c^3*d^9*e^2 - 12*b*c^2*d^8*e^3 + 6*b^2*c*d^7*e^4
- b^3*d^6*e^5 + (8*c^3*d^3*e^8 - 12*b*c^2*d^2*e^9 + 6*b^2*c*d*e^10 - b^3*e^11)*x
^6 + 6*(8*c^3*d^4*e^7 - 12*b*c^2*d^3*e^8 + 6*b^2*c*d^2*e^9 - b^3*d*e^10)*x^5 + 1
5*(8*c^3*d^5*e^6 - 12*b*c^2*d^4*e^7 + 6*b^2*c*d^3*e^8 - b^3*d^2*e^9)*x^4 + 20*(8
*c^3*d^6*e^5 - 12*b*c^2*d^5*e^6 + 6*b^2*c*d^4*e^7 - b^3*d^3*e^8)*x^3 + 15*(8*c^3
*d^7*e^4 - 12*b*c^2*d^6*e^5 + 6*b^2*c*d^5*e^6 - b^3*d^4*e^7)*x^2 + 6*(8*c^3*d^8*
e^3 - 12*b*c^2*d^7*e^4 + 6*b^2*c*d^6*e^5 - b^3*d^5*e^6)*x)

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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((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**9,x)

[Out]

Timed out

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

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)^(5/2)*(g*x + f)/(e*x + d)^9,x, algorithm="giac")

[Out]

Timed out