3.2207 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=285 \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{9009 e^2 (d+e x)^7 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{1287 e^2 (d+e x)^8 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{143 e^2 (d+e x)^9 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e^2 (d+e x)^{10} (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(13*e^2*(2*c*d - b*
e)*(d + e*x)^10) - (2*(6*c*e*f + 20*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(7/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^9) - (8*c*(6*c*e*f + 20*c*
d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(1287*e^2*(2*c*d -
b*e)^3*(d + e*x)^8) - (16*c^2*(6*c*e*f + 20*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b
*e^2*x - c*e^2*x^2)^(7/2))/(9009*e^2*(2*c*d - b*e)^4*(d + e*x)^7)

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Rubi [A]  time = 1.00072, 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 \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{9009 e^2 (d+e x)^7 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{1287 e^2 (d+e x)^8 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{143 e^2 (d+e x)^9 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e^2 (d+e x)^{10} (2 c d-b e)} \]

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)^10,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(13*e^2*(2*c*d - b*
e)*(d + e*x)^10) - (2*(6*c*e*f + 20*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(7/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^9) - (8*c*(6*c*e*f + 20*c*
d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(1287*e^2*(2*c*d -
b*e)^3*(d + e*x)^8) - (16*c^2*(6*c*e*f + 20*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b
*e^2*x - c*e^2*x^2)^(7/2))/(9009*e^2*(2*c*d - b*e)^4*(d + e*x)^7)

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Rubi in Sympy [A]  time = 107.929, size = 274, normalized size = 0.96 \[ \frac{16 c^{2} \left (13 b e g - 20 c d g - 6 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{9009 e^{2} \left (d + e x\right )^{7} \left (b e - 2 c d\right )^{4}} - \frac{8 c \left (13 b e g - 20 c d g - 6 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{1287 e^{2} \left (d + e x\right )^{8} \left (b e - 2 c d\right )^{3}} + \frac{2 \left (13 b e g - 20 c d g - 6 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{143 e^{2} \left (d + e x\right )^{9} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{13 e^{2} \left (d + e x\right )^{10} \left (b e - 2 c d\right )} \]

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)**10,x)

[Out]

16*c**2*(13*b*e*g - 20*c*d*g - 6*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d
))**(7/2)/(9009*e**2*(d + e*x)**7*(b*e - 2*c*d)**4) - 8*c*(13*b*e*g - 20*c*d*g -
6*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(1287*e**2*(d + e*x)
**8*(b*e - 2*c*d)**3) + 2*(13*b*e*g - 20*c*d*g - 6*c*e*f)*(-b*e**2*x - c*e**2*x*
*2 + d*(-b*e + c*d))**(7/2)/(143*e**2*(d + e*x)**9*(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)/(13*e**2*(d + e*x)**10*(b
*e - 2*c*d))

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Mathematica [A]  time = 0.968299, size = 250, normalized size = 0.88 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (-63 b^3 e^3 (2 d g+11 e f+13 e g x)+14 b^2 c e^2 \left (53 d^2 g+4 d e (81 f+94 g x)+e^2 x (27 f+26 g x)\right )-4 b c^2 e \left (348 d^3 g+d^2 e (2499 f+2801 g x)+2 d e^2 x (231 f+200 g x)+2 e^3 x^2 (21 f+13 g x)\right )+8 c^3 \left (97 d^4 g+10 d^3 e (93 f+97 g x)+d^2 e^2 x (291 f+200 g x)+20 d e^3 x^2 (3 f+g x)+6 e^4 f x^3\right )\right )}{9009 e^2 (d+e x)^7 (b e-2 c d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^10,x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-63*b^3*e^3*
(11*e*f + 2*d*g + 13*e*g*x) + 14*b^2*c*e^2*(53*d^2*g + e^2*x*(27*f + 26*g*x) + 4
*d*e*(81*f + 94*g*x)) + 8*c^3*(97*d^4*g + 6*e^4*f*x^3 + 20*d*e^3*x^2*(3*f + g*x)
+ 10*d^3*e*(93*f + 97*g*x) + d^2*e^2*x*(291*f + 200*g*x)) - 4*b*c^2*e*(348*d^3*
g + 2*e^3*x^2*(21*f + 13*g*x) + 2*d*e^2*x*(231*f + 200*g*x) + d^2*e*(2499*f + 28
01*g*x))))/(9009*e^2*(-2*c*d + b*e)^4*(d + e*x)^7)

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Maple [A]  time = 0.018, size = 382, normalized size = 1.3 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 104\,b{c}^{2}{e}^{4}g{x}^{3}-160\,{c}^{3}d{e}^{3}g{x}^{3}-48\,{c}^{3}{e}^{4}f{x}^{3}-364\,{b}^{2}c{e}^{4}g{x}^{2}+1600\,b{c}^{2}d{e}^{3}g{x}^{2}+168\,b{c}^{2}{e}^{4}f{x}^{2}-1600\,{c}^{3}{d}^{2}{e}^{2}g{x}^{2}-480\,{c}^{3}d{e}^{3}f{x}^{2}+819\,{b}^{3}{e}^{4}gx-5264\,{b}^{2}cd{e}^{3}gx-378\,{b}^{2}c{e}^{4}fx+11204\,b{c}^{2}{d}^{2}{e}^{2}gx+1848\,b{c}^{2}d{e}^{3}fx-7760\,{c}^{3}{d}^{3}egx-2328\,{c}^{3}{d}^{2}{e}^{2}fx+126\,{b}^{3}d{e}^{3}g+693\,{b}^{3}{e}^{4}f-742\,{b}^{2}c{d}^{2}{e}^{2}g-4536\,{b}^{2}cd{e}^{3}f+1392\,b{c}^{2}{d}^{3}eg+9996\,b{c}^{2}{d}^{2}{e}^{2}f-776\,{c}^{3}{d}^{4}g-7440\,{c}^{3}{d}^{3}ef \right ) }{9009\, \left ( ex+d \right ) ^{9}{e}^{2} \left ({b}^{4}{e}^{4}-8\,{b}^{3}cd{e}^{3}+24\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}-32\,b{c}^{3}{d}^{3}e+16\,{c}^{4}{d}^{4} \right ) } \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^10,x)

[Out]

-2/9009*(c*e*x+b*e-c*d)*(104*b*c^2*e^4*g*x^3-160*c^3*d*e^3*g*x^3-48*c^3*e^4*f*x^
3-364*b^2*c*e^4*g*x^2+1600*b*c^2*d*e^3*g*x^2+168*b*c^2*e^4*f*x^2-1600*c^3*d^2*e^
2*g*x^2-480*c^3*d*e^3*f*x^2+819*b^3*e^4*g*x-5264*b^2*c*d*e^3*g*x-378*b^2*c*e^4*f
*x+11204*b*c^2*d^2*e^2*g*x+1848*b*c^2*d*e^3*f*x-7760*c^3*d^3*e*g*x-2328*c^3*d^2*
e^2*f*x+126*b^3*d*e^3*g+693*b^3*e^4*f-742*b^2*c*d^2*e^2*g-4536*b^2*c*d*e^3*f+139
2*b*c^2*d^3*e*g+9996*b*c^2*d^2*e^2*f-776*c^3*d^4*g-7440*c^3*d^3*e*f)*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^9/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)

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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)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 106.74, size = 1791, normalized size = 6.28 \[ \text{result too large to display} \]

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)^10,x, algorithm="fricas")

[Out]

2/9009*(8*(6*c^6*e^7*f + (20*c^6*d*e^6 - 13*b*c^5*e^7)*g)*x^6 + 4*(6*(14*c^6*d*e
^6 - b*c^5*e^7)*f + (280*c^6*d^2*e^5 - 202*b*c^5*d*e^6 + 13*b^2*c^4*e^7)*g)*x^5
+ (6*(172*c^6*d^2*e^5 - 32*b*c^5*d*e^6 + 3*b^2*c^4*e^7)*f + (3440*c^6*d^3*e^4 -
2876*b*c^5*d^2*e^5 + 476*b^2*c^4*d*e^6 - 39*b^3*c^3*e^7)*g)*x^4 + (3*(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 - (17864*c^6*d^4
*e^3 - 41684*b*c^5*d^3*e^4 + 33962*b^2*c^4*d^2*e^5 - 11611*b^3*c^3*d*e^6 + 1469*
b^4*c^2*e^7)*g)*x^3 - (3*(5272*c^6*d^4*e^3 - 11468*b*c^5*d^3*e^4 + 8778*b^2*c^4*
d^2*e^5 - 2953*b^3*c^3*d*e^6 + 371*b^4*c^2*e^7)*f - (19352*c^6*d^5*e^2 - 67268*b
*c^5*d^4*e^3 + 89930*b^2*c^4*d^3*e^4 - 57557*b^3*c^3*d^2*e^5 + 17636*b^4*c^2*d*e
^6 - 2093*b^5*c*e^7)*g)*x^2 - 3*(2480*c^6*d^6*e - 10772*b*c^5*d^5*e^2 + 18948*b^
2*c^4*d^4*e^3 - 17243*b^3*c^3*d^3*e^4 + 8561*b^4*c^2*d^2*e^5 - 2205*b^5*c*d*e^6
+ 231*b^6*e^7)*f - 2*(388*c^6*d^7 - 1860*b*c^5*d^6*e + 3623*b^2*c^4*d^5*e^2 - 36
52*b^3*c^3*d^4*e^3 + 1998*b^4*c^2*d^3*e^4 - 560*b^5*c*d^2*e^5 + 63*b^6*d*e^6)*g
+ (3*(6664*c^6*d^5*e^2 - 21932*b*c^5*d^4*e^3 + 27666*b^2*c^4*d^3*e^4 - 16759*b^3
*c^3*d^2*e^5 + 4928*b^4*c^2*d*e^6 - 567*b^5*c*e^7)*f - (5432*c^6*d^6*e - 25652*b
*c^5*d^5*e^2 + 49250*b^2*c^4*d^4*e^3 - 48977*b^3*c^3*d^3*e^4 + 26471*b^4*c^2*d^2
*e^5 - 7343*b^5*c*d*e^6 + 819*b^6*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 -
b*d*e)/(16*c^4*d^11*e^2 - 32*b*c^3*d^10*e^3 + 24*b^2*c^2*d^9*e^4 - 8*b^3*c*d^8*
e^5 + b^4*d^7*e^6 + (16*c^4*d^4*e^9 - 32*b*c^3*d^3*e^10 + 24*b^2*c^2*d^2*e^11 -
8*b^3*c*d*e^12 + b^4*e^13)*x^7 + 7*(16*c^4*d^5*e^8 - 32*b*c^3*d^4*e^9 + 24*b^2*c
^2*d^3*e^10 - 8*b^3*c*d^2*e^11 + b^4*d*e^12)*x^6 + 21*(16*c^4*d^6*e^7 - 32*b*c^3
*d^5*e^8 + 24*b^2*c^2*d^4*e^9 - 8*b^3*c*d^3*e^10 + b^4*d^2*e^11)*x^5 + 35*(16*c^
4*d^7*e^6 - 32*b*c^3*d^6*e^7 + 24*b^2*c^2*d^5*e^8 - 8*b^3*c*d^4*e^9 + b^4*d^3*e^
10)*x^4 + 35*(16*c^4*d^8*e^5 - 32*b*c^3*d^7*e^6 + 24*b^2*c^2*d^6*e^7 - 8*b^3*c*d
^5*e^8 + b^4*d^4*e^9)*x^3 + 21*(16*c^4*d^9*e^4 - 32*b*c^3*d^8*e^5 + 24*b^2*c^2*d
^7*e^6 - 8*b^3*c*d^6*e^7 + b^4*d^5*e^8)*x^2 + 7*(16*c^4*d^10*e^3 - 32*b*c^3*d^9*
e^4 + 24*b^2*c^2*d^8*e^5 - 8*b^3*c*d^7*e^6 + b^4*d^6*e^7)*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)**10,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)^10,x, algorithm="giac")

[Out]

Timed out