3.587 \(\int \frac{(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=398 \[ \frac{e^2 g^3 \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{2 (e f-d g)^2 (d g+e f)^5 \sqrt{e^2 f^2-d^2 g^2}}+\frac{3 e g^4 \sqrt{d^2-e^2 x^2} (3 e f-2 d g)}{2 (f+g x) (e f-d g)^2 (d g+e f)^5}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}-\frac{e^2 (5 d (e f-5 d g)-e x (31 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}+\frac{4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}+\frac{e^2 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^5} \]

[Out]

(4*d*e^2*(d + e*x))/(5*(e*f + d*g)^3*(d^2 - e^2*x^2)^(5/2)) - (e^2*(5*d*(e*f - 5
*d*g) - e*(e*f + 31*d*g)*x))/(15*d*(e*f + d*g)^4*(d^2 - e^2*x^2)^(3/2)) + (e^2*(
90*d^3*g^2 + e*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*g^2)*x))/(15*d^3*(e*f + d*g)^5*
Sqrt[d^2 - e^2*x^2]) + (g^4*Sqrt[d^2 - e^2*x^2])/(2*(e*f - d*g)*(e*f + d*g)^4*(f
+ g*x)^2) + (3*e*g^4*(3*e*f - 2*d*g)*Sqrt[d^2 - e^2*x^2])/(2*(e*f - d*g)^2*(e*f
+ d*g)^5*(f + g*x)) + (e^2*g^3*(20*e^2*f^2 - 30*d*e*f*g + 13*d^2*g^2)*ArcTan[(d
^2*g + e^2*f*x)/(Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2*x^2])])/(2*(e*f - d*g)^2
*(e*f + d*g)^5*Sqrt[e^2*f^2 - d^2*g^2])

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Rubi [A]  time = 5.42682, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{e^2 g^3 \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{2 (e f-d g)^2 (d g+e f)^5 \sqrt{e^2 f^2-d^2 g^2}}+\frac{3 e g^4 \sqrt{d^2-e^2 x^2} (3 e f-2 d g)}{2 (f+g x) (e f-d g)^2 (d g+e f)^5}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}-\frac{e^2 (5 d (e f-5 d g)-e x (31 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}+\frac{4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}+\frac{e^2 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^5} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/((f + g*x)^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(4*d*e^2*(d + e*x))/(5*(e*f + d*g)^3*(d^2 - e^2*x^2)^(5/2)) - (e^2*(5*d*(e*f - 5
*d*g) - e*(e*f + 31*d*g)*x))/(15*d*(e*f + d*g)^4*(d^2 - e^2*x^2)^(3/2)) + (e^2*(
90*d^3*g^2 + e*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*g^2)*x))/(15*d^3*(e*f + d*g)^5*
Sqrt[d^2 - e^2*x^2]) + (g^4*Sqrt[d^2 - e^2*x^2])/(2*(e*f - d*g)*(e*f + d*g)^4*(f
+ g*x)^2) + (3*e*g^4*(3*e*f - 2*d*g)*Sqrt[d^2 - e^2*x^2])/(2*(e*f - d*g)^2*(e*f
+ d*g)^5*(f + g*x)) + (e^2*g^3*(20*e^2*f^2 - 30*d*e*f*g + 13*d^2*g^2)*ArcTan[(d
^2*g + e^2*f*x)/(Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2*x^2])])/(2*(e*f - d*g)^2
*(e*f + d*g)^5*Sqrt[e^2*f^2 - d^2*g^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(g*x+f)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

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Mathematica [C]  time = 1.65175, size = 387, normalized size = 0.97 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\frac{2 e^2 (d g+e f) (17 d g+2 e f)}{d^2 (d-e x)^2}+\frac{2 e^2 \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )}{d^3 (d-e x)}+\frac{6 e^2 (d g+e f)^2}{d (d-e x)^3}+\frac{45 e g^4 (3 e f-2 d g)}{(f+g x) (e f-d g)^2}+\frac{15 g^4 (d g+e f)}{(f+g x)^2 (e f-d g)}\right )-\frac{15 i e^2 g^3 \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \log \left (\frac{4 (e f-d g)^2 (d g+e f)^5 \left (\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}+i d^2 g+i e^2 f x\right )}{e^2 g^2 (f+g x) \sqrt{e^2 f^2-d^2 g^2} \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right )}\right )}{(e f-d g)^2 \sqrt{e^2 f^2-d^2 g^2}}}{30 (d g+e f)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/((f + g*x)^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*((6*e^2*(e*f + d*g)^2)/(d*(d - e*x)^3) + (2*e^2*(e*f + d*g)
*(2*e*f + 17*d*g))/(d^2*(d - e*x)^2) + (2*e^2*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*
g^2))/(d^3*(d - e*x)) + (15*g^4*(e*f + d*g))/((e*f - d*g)*(f + g*x)^2) + (45*e*g
^4*(3*e*f - 2*d*g))/((e*f - d*g)^2*(f + g*x))) - ((15*I)*e^2*g^3*(20*e^2*f^2 - 3
0*d*e*f*g + 13*d^2*g^2)*Log[(4*(e*f - d*g)^2*(e*f + d*g)^5*(I*d^2*g + I*e^2*f*x
+ Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2*x^2]))/(e^2*g^2*Sqrt[e^2*f^2 - d^2*g^2]
*(20*e^2*f^2 - 30*d*e*f*g + 13*d^2*g^2)*(f + g*x))])/((e*f - d*g)^2*Sqrt[e^2*f^2
- d^2*g^2]))/(30*(e*f + d*g)^5)

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Maple [B]  time = 0.046, size = 9593, normalized size = 24.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x)

[Out]

result too large to display

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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((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*(g*x + f)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.905889, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*(g*x + f)^3),x, algorithm="fricas")

[Out]

[1/30*(15*(320*d^10*e^4*f^6*g^3 - 480*d^11*e^3*f^5*g^4 + 208*d^12*e^2*f^4*g^5 -
(20*d^3*e^11*f^4*g^5 - 30*d^4*e^10*f^3*g^6 + 13*d^5*e^9*f^2*g^7)*x^9 - (40*d^3*e
^11*f^5*g^4 - 200*d^4*e^10*f^4*g^5 + 236*d^5*e^9*f^3*g^6 - 91*d^6*e^8*f^2*g^7)*x
^8 - (20*d^3*e^11*f^6*g^3 - 310*d^4*e^10*f^5*g^4 + 493*d^5*e^9*f^4*g^5 - 272*d^6
*e^8*f^3*g^6 + 39*d^7*e^7*f^2*g^7)*x^7 + (140*d^4*e^10*f^6*g^3 - 330*d^5*e^9*f^5
*g^4 - 349*d^6*e^8*f^4*g^5 + 852*d^7*e^7*f^3*g^6 - 403*d^8*e^6*f^2*g^7)*x^6 - (6
0*d^5*e^9*f^6*g^3 + 1150*d^6*e^8*f^5*g^4 - 2621*d^7*e^7*f^4*g^5 + 2006*d^8*e^6*f
^3*g^6 - 520*d^9*e^5*f^2*g^7)*x^5 - (620*d^6*e^8*f^6*g^3 - 2530*d^7*e^7*f^5*g^4
+ 2563*d^8*e^6*f^4*g^5 - 680*d^9*e^5*f^3*g^6 - 156*d^10*e^4*f^2*g^7)*x^4 + 8*(10
0*d^7*e^7*f^6*g^3 - 90*d^8*e^6*f^5*g^4 - 125*d^9*e^5*f^4*g^5 + 189*d^10*e^4*f^3*
g^6 - 65*d^11*e^3*f^2*g^7)*x^3 + 4*(60*d^8*e^6*f^6*g^3 - 490*d^9*e^5*f^5*g^4 + 7
19*d^10*e^4*f^4*g^5 - 380*d^11*e^3*f^3*g^6 + 52*d^12*e^2*f^2*g^7)*x^2 - 8*(100*d
^9*e^5*f^6*g^3 - 230*d^10*e^4*f^5*g^4 + 185*d^11*e^3*f^4*g^5 - 52*d^12*e^2*f^3*g
^6)*x - (320*d^9*e^4*f^6*g^3 - 480*d^10*e^3*f^5*g^4 + 208*d^11*e^2*f^4*g^5 + (20
*d^3*e^10*f^4*g^5 - 30*d^4*e^9*f^3*g^6 + 13*d^5*e^8*f^2*g^7)*x^8 + 2*(20*d^3*e^1
0*f^5*g^4 - 10*d^4*e^9*f^4*g^5 - 17*d^5*e^8*f^3*g^6 + 13*d^6*e^7*f^2*g^7)*x^7 +
(20*d^3*e^10*f^6*g^3 + 50*d^4*e^9*f^5*g^4 - 487*d^5*e^8*f^4*g^5 + 622*d^6*e^7*f^
3*g^6 - 247*d^7*e^6*f^2*g^7)*x^6 + 2*(20*d^4*e^9*f^6*g^3 - 410*d^5*e^8*f^5*g^4 +
783*d^6*e^7*f^4*g^5 - 547*d^7*e^6*f^3*g^6 + 130*d^8*e^5*f^2*g^7)*x^5 - (380*d^5
*e^8*f^6*g^3 - 1370*d^6*e^7*f^5*g^4 + 1047*d^7*e^6*f^4*g^5 + 80*d^8*e^5*f^3*g^6
- 260*d^9*e^4*f^2*g^7)*x^4 + 20*(20*d^6*e^7*f^6*g^3 + 10*d^7*e^6*f^5*g^4 - 87*d^
8*e^5*f^4*g^5 + 86*d^9*e^4*f^3*g^6 - 26*d^10*e^3*f^2*g^7)*x^3 + 4*(100*d^7*e^6*f
^6*g^3 - 550*d^8*e^5*f^5*g^4 + 745*d^9*e^4*f^4*g^5 - 380*d^10*e^3*f^3*g^6 + 52*d
^11*e^2*f^2*g^7)*x^2 - 8*(100*d^8*e^5*f^6*g^3 - 230*d^9*e^4*f^5*g^4 + 185*d^10*e
^3*f^4*g^5 - 52*d^11*e^2*f^3*g^6)*x)*sqrt(-e^2*x^2 + d^2))*log(((d*e^2*f^2*g - d
^3*g^3)*x^2 - (e^2*f^3 - d^2*f*g^2)*sqrt(-e^2*x^2 + d^2)*x + (d*e^2*f^3 - d^3*f*
g^2)*x + (d^2*f*g*x + d^2*f^2 - (e^2*f^2 - d^2*g^2)*x^2 - (d*f*g*x + d*f^2)*sqrt
(-e^2*x^2 + d^2))*sqrt(-e^2*f^2 + d^2*g^2))/(d*g*x + d*f - sqrt(-e^2*x^2 + d^2)*
(g*x + f))) - (3*(6*e^13*f^6*g^2 + 30*d*e^12*f^5*g^3 + 78*d^2*e^11*f^4*g^4 - 315
*d^3*e^10*f^3*g^5 + 236*d^4*e^9*f^2*g^6 - 30*d^5*e^8*f*g^7 - 5*d^6*e^7*g^8)*x^9
+ 3*(12*e^13*f^7*g + 30*d*e^12*f^6*g^2 + 36*d^2*e^11*f^5*g^3 - 650*d^3*e^10*f^4*
g^4 + 952*d^4*e^9*f^3*g^5 - 735*d^5*e^8*f^2*g^6 + 200*d^6*e^7*f*g^7 + 35*d^7*e^6
*g^8)*x^8 + 3*(6*e^13*f^8 - 30*d*e^12*f^7*g - 174*d^2*e^11*f^6*g^2 - 540*d^3*e^1
0*f^5*g^3 + 640*d^4*e^9*f^4*g^4 + 1280*d^5*e^8*f^3*g^5 - 1027*d^6*e^7*f^2*g^6 -
20*d^7*e^6*f*g^7 - 15*d^8*e^5*g^8)*x^7 - (90*d*e^12*f^8 + 432*d^2*e^11*f^7*g + 5
60*d^3*e^10*f^6*g^2 - 84*d^4*e^9*f^5*g^3 - 14200*d^5*e^8*f^4*g^4 + 22512*d^6*e^7
*f^3*g^5 - 14095*d^7*e^6*f^2*g^6 + 2880*d^8*e^5*f*g^7 + 465*d^9*e^4*g^8)*x^6 - (
36*d^2*e^11*f^8 - 590*d^3*e^10*f^7*g - 2162*d^4*e^9*f^6*g^2 - 9530*d^5*e^8*f^5*g
^3 + 31216*d^6*e^7*f^4*g^4 - 23345*d^7*e^6*f^3*g^5 + 9085*d^8*e^5*f^2*g^6 - 2670
*d^9*e^4*f*g^7 - 600*d^10*e^3*g^8)*x^5 + 20*(26*d^3*e^10*f^8 + 74*d^4*e^9*f^7*g
+ 14*d^5*e^8*f^6*g^2 - 634*d^6*e^7*f^5*g^3 + 320*d^7*e^6*f^4*g^4 + 401*d^8*e^5*f
^3*g^5 - 414*d^9*e^4*f^2*g^6 + 114*d^10*e^3*f*g^7 + 9*d^11*e^2*g^8)*x^4 - 20*(20
*d^4*e^9*f^8 + 90*d^5*e^8*f^7*g + 110*d^6*e^7*f^6*g^2 + 150*d^7*e^6*f^5*g^3 - 14
80*d^8*e^5*f^4*g^4 + 1443*d^9*e^4*f^3*g^5 - 615*d^10*e^3*f^2*g^6 + 162*d^11*e^2*
f*g^7 + 30*d^12*e*g^8)*x^3 - 120*(4*d^5*e^8*f^8 + 10*d^6*e^7*f^7*g - 2*d^7*e^6*f
^6*g^2 - 122*d^8*e^5*f^5*g^3 + 200*d^9*e^4*f^4*g^4 - 141*d^10*e^3*f^3*g^5 + 49*d
^11*e^2*f^2*g^6 - 2*d^12*e*f*g^7 - 2*d^13*g^8)*x^2 + 240*(2*d^6*e^7*f^8 + 6*d^7*
e^6*f^7*g + 2*d^8*e^5*f^6*g^2 - 30*d^9*e^4*f^5*g^3 + 20*d^10*e^3*f^4*g^4 - 11*d^
11*e^2*f^3*g^5 + 6*d^12*e*f^2*g^6 + 2*d^13*f*g^7)*x + (5*(2*e^12*f^6*g^2 + 6*d*e
^11*f^5*g^3 - 10*d^2*e^10*f^4*g^4 + 21*d^3*e^9*f^3*g^5 + 20*d^4*e^8*f^2*g^6 - 18
*d^5*e^7*f*g^7 - 3*d^6*e^6*g^8)*x^8 + (20*e^12*f^7*g + 116*d*e^11*f^6*g^2 + 230*
d^2*e^10*f^5*g^3 + 1088*d^3*e^9*f^4*g^4 - 3865*d^4*e^8*f^3*g^5 + 2561*d^5*e^7*f^
2*g^6 - 210*d^6*e^6*f*g^7 - 30*d^7*e^5*g^8)*x^7 + (10*e^12*f^8 + 142*d*e^11*f^7*
g + 330*d^2*e^10*f^6*g^2 + 766*d^3*e^9*f^5*g^3 - 8400*d^4*e^8*f^4*g^4 + 12157*d^
5*e^7*f^3*g^5 - 7750*d^6*e^6*f^2*g^6 + 1650*d^7*e^5*f*g^7 + 285*d^8*e^4*g^8)*x^6
+ (56*d*e^11*f^8 - 230*d^2*e^10*f^7*g - 1242*d^3*e^9*f^6*g^2 - 5330*d^4*e^8*f^5
*g^3 + 14616*d^5*e^7*f^4*g^4 - 7925*d^6*e^6*f^3*g^5 + 2395*d^7*e^5*f^2*g^6 - 123
0*d^8*e^4*f*g^7 - 300*d^9*e^3*g^8)*x^5 - 20*(14*d^2*e^10*f^8 + 44*d^3*e^9*f^7*g
+ 20*d^4*e^8*f^6*g^2 - 268*d^5*e^7*f^5*g^3 - 280*d^6*e^6*f^4*g^4 + 824*d^7*e^5*f
^3*g^5 - 561*d^8*e^4*f^2*g^6 + 120*d^9*e^3*f*g^7 + 15*d^10*e^2*g^8)*x^4 + 20*(8*
d^3*e^9*f^8 + 54*d^4*e^8*f^7*g + 98*d^5*e^7*f^6*g^2 + 330*d^6*e^6*f^5*g^3 - 1600
*d^7*e^5*f^4*g^4 + 1509*d^8*e^4*f^3*g^5 - 651*d^9*e^3*f^2*g^6 + 150*d^10*e^2*f*g
^7 + 30*d^11*e*g^8)*x^3 + 120*(4*d^4*e^8*f^8 + 10*d^5*e^7*f^7*g - 2*d^6*e^6*f^6*
g^2 - 122*d^7*e^5*f^5*g^3 + 200*d^8*e^4*f^4*g^4 - 141*d^9*e^3*f^3*g^5 + 49*d^10*
e^2*f^2*g^6 - 2*d^11*e*f*g^7 - 2*d^12*g^8)*x^2 - 240*(2*d^5*e^7*f^8 + 6*d^6*e^6*
f^7*g + 2*d^7*e^5*f^6*g^2 - 30*d^8*e^4*f^5*g^3 + 20*d^9*e^3*f^4*g^4 - 11*d^10*e^
2*f^3*g^5 + 6*d^11*e*f^2*g^6 + 2*d^12*f*g^7)*x)*sqrt(-e^2*x^2 + d^2))*sqrt(-e^2*
f^2 + d^2*g^2))/((16*d^10*e^7*f^11 + 48*d^11*e^6*f^10*g + 16*d^12*e^5*f^9*g^2 -
80*d^13*e^4*f^8*g^3 - 80*d^14*e^3*f^7*g^4 + 16*d^15*e^2*f^6*g^5 + 48*d^16*e*f^5*
g^6 + 16*d^17*f^4*g^7 - (d^3*e^14*f^9*g^2 + 3*d^4*e^13*f^8*g^3 + d^5*e^12*f^7*g^
4 - 5*d^6*e^11*f^6*g^5 - 5*d^7*e^10*f^5*g^6 + d^8*e^9*f^4*g^7 + 3*d^9*e^8*f^3*g^
8 + d^10*e^7*f^2*g^9)*x^9 - (2*d^3*e^14*f^10*g - d^4*e^13*f^9*g^2 - 19*d^5*e^12*
f^8*g^3 - 17*d^6*e^11*f^7*g^4 + 25*d^7*e^10*f^6*g^5 + 37*d^8*e^9*f^5*g^6 - d^9*e
^8*f^4*g^7 - 19*d^10*e^7*f^3*g^8 - 7*d^11*e^6*f^2*g^9)*x^8 - (d^3*e^14*f^11 - 11
*d^4*e^13*f^10*g - 38*d^5*e^12*f^9*g^2 - 10*d^6*e^11*f^8*g^3 + 68*d^7*e^10*f^7*g
^4 + 56*d^8*e^9*f^6*g^5 - 26*d^9*e^8*f^5*g^6 - 38*d^10*e^7*f^4*g^7 - 5*d^11*e^6*
f^3*g^8 + 3*d^12*e^5*f^2*g^9)*x^7 + (7*d^4*e^13*f^11 + 15*d^5*e^12*f^10*g - 42*d
^6*e^11*f^9*g^2 - 134*d^7*e^10*f^8*g^3 - 36*d^8*e^9*f^7*g^4 + 192*d^9*e^8*f^6*g^
5 + 170*d^10*e^7*f^5*g^6 - 42*d^11*e^6*f^4*g^7 - 99*d^12*e^5*f^3*g^8 - 31*d^13*e
^4*f^2*g^9)*x^6 - (3*d^5*e^12*f^11 + 71*d^6*e^11*f^10*g + 149*d^7*e^10*f^9*g^2 -
73*d^8*e^9*f^8*g^3 - 365*d^9*e^8*f^7*g^4 - 107*d^10*e^7*f^6*g^5 + 271*d^11*e^6*
f^5*g^6 + 149*d^12*e^5*f^4*g^7 - 58*d^13*e^4*f^3*g^8 - 40*d^14*e^3*f^2*g^9)*x^5
- (31*d^6*e^11*f^11 + 13*d^7*e^10*f^10*g - 221*d^8*e^9*f^9*g^2 - 271*d^9*e^8*f^8
*g^3 + 233*d^10*e^7*f^7*g^4 + 491*d^11*e^6*f^6*g^5 + 73*d^12*e^5*f^5*g^6 - 221*d
^13*e^4*f^4*g^7 - 116*d^14*e^3*f^3*g^8 - 12*d^15*e^2*f^2*g^9)*x^4 + 8*(5*d^7*e^1
0*f^11 + 18*d^8*e^9*f^10*g + 9*d^9*e^8*f^9*g^2 - 37*d^10*e^7*f^8*g^3 - 45*d^11*e
^6*f^7*g^4 + 15*d^12*e^5*f^6*g^5 + 43*d^13*e^4*f^5*g^6 + 9*d^14*e^3*f^4*g^7 - 12
*d^15*e^2*f^3*g^8 - 5*d^16*e*f^2*g^9)*x^3 + 4*(3*d^8*e^9*f^11 - 11*d^9*e^8*f^10*
g - 53*d^10*e^7*f^9*g^2 - 23*d^11*e^6*f^8*g^3 + 89*d^12*e^5*f^7*g^4 + 83*d^13*e^
4*f^6*g^5 - 31*d^14*e^3*f^5*g^6 - 53*d^15*e^2*f^4*g^7 - 8*d^16*e*f^3*g^8 + 4*d^1
7*f^2*g^9)*x^2 - 8*(5*d^9*e^8*f^11 + 11*d^10*e^7*f^10*g - 7*d^11*e^6*f^9*g^2 - 2
9*d^12*e^5*f^8*g^3 - 5*d^13*e^4*f^7*g^4 + 25*d^14*e^3*f^6*g^5 + 11*d^15*e^2*f^5*
g^6 - 7*d^16*e*f^4*g^7 - 4*d^17*f^3*g^8)*x - (16*d^9*e^7*f^11 + 48*d^10*e^6*f^10
*g + 16*d^11*e^5*f^9*g^2 - 80*d^12*e^4*f^8*g^3 - 80*d^13*e^3*f^7*g^4 + 16*d^14*e
^2*f^6*g^5 + 48*d^15*e*f^5*g^6 + 16*d^16*f^4*g^7 + (d^3*e^13*f^9*g^2 + 3*d^4*e^1
2*f^8*g^3 + d^5*e^11*f^7*g^4 - 5*d^6*e^10*f^6*g^5 - 5*d^7*e^9*f^5*g^6 + d^8*e^8*
f^4*g^7 + 3*d^9*e^7*f^3*g^8 + d^10*e^6*f^2*g^9)*x^8 + 2*(d^3*e^13*f^10*g + 4*d^4
*e^12*f^9*g^2 + 4*d^5*e^11*f^8*g^3 - 4*d^6*e^10*f^7*g^4 - 10*d^7*e^9*f^6*g^5 - 4
*d^8*e^8*f^5*g^6 + 4*d^9*e^7*f^4*g^7 + 4*d^10*e^6*f^3*g^8 + d^11*e^5*f^2*g^9)*x^
7 + (d^3*e^13*f^11 + 7*d^4*e^12*f^10*g - 6*d^5*e^11*f^9*g^2 - 58*d^6*e^10*f^8*g^
3 - 44*d^7*e^9*f^7*g^4 + 76*d^8*e^8*f^6*g^5 + 102*d^9*e^7*f^5*g^6 - 6*d^10*e^6*f
^4*g^7 - 53*d^11*e^5*f^3*g^8 - 19*d^12*e^4*f^2*g^9)*x^6 + 2*(d^4*e^12*f^11 - 16*
d^5*e^11*f^10*g - 46*d^6*e^10*f^9*g^2 + 6*d^7*e^9*f^8*g^3 + 100*d^8*e^8*f^7*g^4
+ 46*d^9*e^7*f^6*g^5 - 66*d^10*e^6*f^5*g^6 - 46*d^11*e^5*f^4*g^7 + 11*d^12*e^4*f
^3*g^8 + 10*d^13*e^3*f^2*g^9)*x^5 - (19*d^5*e^11*f^11 + 17*d^6*e^10*f^10*g - 121
*d^7*e^9*f^9*g^2 - 195*d^8*e^8*f^8*g^3 + 85*d^9*e^7*f^7*g^4 + 319*d^10*e^6*f^6*g
^5 + 117*d^11*e^5*f^5*g^6 - 121*d^12*e^4*f^4*g^7 - 100*d^13*e^3*f^3*g^8 - 20*d^1
4*e^2*f^2*g^9)*x^4 + 20*(d^6*e^10*f^11 + 5*d^7*e^9*f^10*g + 5*d^8*e^8*f^9*g^2 -
9*d^9*e^7*f^8*g^3 - 17*d^10*e^6*f^7*g^4 + d^11*e^5*f^6*g^5 + 15*d^12*e^4*f^5*g^6
+ 5*d^13*e^3*f^4*g^7 - 4*d^14*e^2*f^3*g^8 - 2*d^15*e*f^2*g^9)*x^3 + 4*(5*d^7*e^
9*f^11 - 5*d^8*e^8*f^10*g - 51*d^9*e^7*f^9*g^2 - 33*d^10*e^6*f^8*g^3 + 79*d^11*e
^5*f^7*g^4 + 85*d^12*e^4*f^6*g^5 - 25*d^13*e^3*f^5*g^6 - 51*d^14*e^2*f^4*g^7 - 8
*d^15*e*f^3*g^8 + 4*d^16*f^2*g^9)*x^2 - 8*(5*d^8*e^8*f^11 + 11*d^9*e^7*f^10*g -
7*d^10*e^6*f^9*g^2 - 29*d^11*e^5*f^8*g^3 - 5*d^12*e^4*f^7*g^4 + 25*d^13*e^3*f^6*
g^5 + 11*d^14*e^2*f^5*g^6 - 7*d^15*e*f^4*g^7 - 4*d^16*f^3*g^8)*x)*sqrt(-e^2*x^2
+ d^2))*sqrt(-e^2*f^2 + d^2*g^2)), 1/30*(30*(320*d^10*e^4*f^6*g^3 - 480*d^11*e^3
*f^5*g^4 + 208*d^12*e^2*f^4*g^5 - (20*d^3*e^11*f^4*g^5 - 30*d^4*e^10*f^3*g^6 + 1
3*d^5*e^9*f^2*g^7)*x^9 - (40*d^3*e^11*f^5*g^4 - 200*d^4*e^10*f^4*g^5 + 236*d^5*e
^9*f^3*g^6 - 91*d^6*e^8*f^2*g^7)*x^8 - (20*d^3*e^11*f^6*g^3 - 310*d^4*e^10*f^5*g
^4 + 493*d^5*e^9*f^4*g^5 - 272*d^6*e^8*f^3*g^6 + 39*d^7*e^7*f^2*g^7)*x^7 + (140*
d^4*e^10*f^6*g^3 - 330*d^5*e^9*f^5*g^4 - 349*d^6*e^8*f^4*g^5 + 852*d^7*e^7*f^3*g
^6 - 403*d^8*e^6*f^2*g^7)*x^6 - (60*d^5*e^9*f^6*g^3 + 1150*d^6*e^8*f^5*g^4 - 262
1*d^7*e^7*f^4*g^5 + 2006*d^8*e^6*f^3*g^6 - 520*d^9*e^5*f^2*g^7)*x^5 - (620*d^6*e
^8*f^6*g^3 - 2530*d^7*e^7*f^5*g^4 + 2563*d^8*e^6*f^4*g^5 - 680*d^9*e^5*f^3*g^6 -
156*d^10*e^4*f^2*g^7)*x^4 + 8*(100*d^7*e^7*f^6*g^3 - 90*d^8*e^6*f^5*g^4 - 125*d
^9*e^5*f^4*g^5 + 189*d^10*e^4*f^3*g^6 - 65*d^11*e^3*f^2*g^7)*x^3 + 4*(60*d^8*e^6
*f^6*g^3 - 490*d^9*e^5*f^5*g^4 + 719*d^10*e^4*f^4*g^5 - 380*d^11*e^3*f^3*g^6 + 5
2*d^12*e^2*f^2*g^7)*x^2 - 8*(100*d^9*e^5*f^6*g^3 - 230*d^10*e^4*f^5*g^4 + 185*d^
11*e^3*f^4*g^5 - 52*d^12*e^2*f^3*g^6)*x - (320*d^9*e^4*f^6*g^3 - 480*d^10*e^3*f^
5*g^4 + 208*d^11*e^2*f^4*g^5 + (20*d^3*e^10*f^4*g^5 - 30*d^4*e^9*f^3*g^6 + 13*d^
5*e^8*f^2*g^7)*x^8 + 2*(20*d^3*e^10*f^5*g^4 - 10*d^4*e^9*f^4*g^5 - 17*d^5*e^8*f^
3*g^6 + 13*d^6*e^7*f^2*g^7)*x^7 + (20*d^3*e^10*f^6*g^3 + 50*d^4*e^9*f^5*g^4 - 48
7*d^5*e^8*f^4*g^5 + 622*d^6*e^7*f^3*g^6 - 247*d^7*e^6*f^2*g^7)*x^6 + 2*(20*d^4*e
^9*f^6*g^3 - 410*d^5*e^8*f^5*g^4 + 783*d^6*e^7*f^4*g^5 - 547*d^7*e^6*f^3*g^6 + 1
30*d^8*e^5*f^2*g^7)*x^5 - (380*d^5*e^8*f^6*g^3 - 1370*d^6*e^7*f^5*g^4 + 1047*d^7
*e^6*f^4*g^5 + 80*d^8*e^5*f^3*g^6 - 260*d^9*e^4*f^2*g^7)*x^4 + 20*(20*d^6*e^7*f^
6*g^3 + 10*d^7*e^6*f^5*g^4 - 87*d^8*e^5*f^4*g^5 + 86*d^9*e^4*f^3*g^6 - 26*d^10*e
^3*f^2*g^7)*x^3 + 4*(100*d^7*e^6*f^6*g^3 - 550*d^8*e^5*f^5*g^4 + 745*d^9*e^4*f^4
*g^5 - 380*d^10*e^3*f^3*g^6 + 52*d^11*e^2*f^2*g^7)*x^2 - 8*(100*d^8*e^5*f^6*g^3
- 230*d^9*e^4*f^5*g^4 + 185*d^10*e^3*f^4*g^5 - 52*d^11*e^2*f^3*g^6)*x)*sqrt(-e^2
*x^2 + d^2))*arctan((d*g*x + d*f - sqrt(-e^2*x^2 + d^2)*f)/(sqrt(e^2*f^2 - d^2*g
^2)*x)) - (3*(6*e^13*f^6*g^2 + 30*d*e^12*f^5*g^3 + 78*d^2*e^11*f^4*g^4 - 315*d^3
*e^10*f^3*g^5 + 236*d^4*e^9*f^2*g^6 - 30*d^5*e^8*f*g^7 - 5*d^6*e^7*g^8)*x^9 + 3*
(12*e^13*f^7*g + 30*d*e^12*f^6*g^2 + 36*d^2*e^11*f^5*g^3 - 650*d^3*e^10*f^4*g^4
+ 952*d^4*e^9*f^3*g^5 - 735*d^5*e^8*f^2*g^6 + 200*d^6*e^7*f*g^7 + 35*d^7*e^6*g^8
)*x^8 + 3*(6*e^13*f^8 - 30*d*e^12*f^7*g - 174*d^2*e^11*f^6*g^2 - 540*d^3*e^10*f^
5*g^3 + 640*d^4*e^9*f^4*g^4 + 1280*d^5*e^8*f^3*g^5 - 1027*d^6*e^7*f^2*g^6 - 20*d
^7*e^6*f*g^7 - 15*d^8*e^5*g^8)*x^7 - (90*d*e^12*f^8 + 432*d^2*e^11*f^7*g + 560*d
^3*e^10*f^6*g^2 - 84*d^4*e^9*f^5*g^3 - 14200*d^5*e^8*f^4*g^4 + 22512*d^6*e^7*f^3
*g^5 - 14095*d^7*e^6*f^2*g^6 + 2880*d^8*e^5*f*g^7 + 465*d^9*e^4*g^8)*x^6 - (36*d
^2*e^11*f^8 - 590*d^3*e^10*f^7*g - 2162*d^4*e^9*f^6*g^2 - 9530*d^5*e^8*f^5*g^3 +
31216*d^6*e^7*f^4*g^4 - 23345*d^7*e^6*f^3*g^5 + 9085*d^8*e^5*f^2*g^6 - 2670*d^9
*e^4*f*g^7 - 600*d^10*e^3*g^8)*x^5 + 20*(26*d^3*e^10*f^8 + 74*d^4*e^9*f^7*g + 14
*d^5*e^8*f^6*g^2 - 634*d^6*e^7*f^5*g^3 + 320*d^7*e^6*f^4*g^4 + 401*d^8*e^5*f^3*g
^5 - 414*d^9*e^4*f^2*g^6 + 114*d^10*e^3*f*g^7 + 9*d^11*e^2*g^8)*x^4 - 20*(20*d^4
*e^9*f^8 + 90*d^5*e^8*f^7*g + 110*d^6*e^7*f^6*g^2 + 150*d^7*e^6*f^5*g^3 - 1480*d
^8*e^5*f^4*g^4 + 1443*d^9*e^4*f^3*g^5 - 615*d^10*e^3*f^2*g^6 + 162*d^11*e^2*f*g^
7 + 30*d^12*e*g^8)*x^3 - 120*(4*d^5*e^8*f^8 + 10*d^6*e^7*f^7*g - 2*d^7*e^6*f^6*g
^2 - 122*d^8*e^5*f^5*g^3 + 200*d^9*e^4*f^4*g^4 - 141*d^10*e^3*f^3*g^5 + 49*d^11*
e^2*f^2*g^6 - 2*d^12*e*f*g^7 - 2*d^13*g^8)*x^2 + 240*(2*d^6*e^7*f^8 + 6*d^7*e^6*
f^7*g + 2*d^8*e^5*f^6*g^2 - 30*d^9*e^4*f^5*g^3 + 20*d^10*e^3*f^4*g^4 - 11*d^11*e
^2*f^3*g^5 + 6*d^12*e*f^2*g^6 + 2*d^13*f*g^7)*x + (5*(2*e^12*f^6*g^2 + 6*d*e^11*
f^5*g^3 - 10*d^2*e^10*f^4*g^4 + 21*d^3*e^9*f^3*g^5 + 20*d^4*e^8*f^2*g^6 - 18*d^5
*e^7*f*g^7 - 3*d^6*e^6*g^8)*x^8 + (20*e^12*f^7*g + 116*d*e^11*f^6*g^2 + 230*d^2*
e^10*f^5*g^3 + 1088*d^3*e^9*f^4*g^4 - 3865*d^4*e^8*f^3*g^5 + 2561*d^5*e^7*f^2*g^
6 - 210*d^6*e^6*f*g^7 - 30*d^7*e^5*g^8)*x^7 + (10*e^12*f^8 + 142*d*e^11*f^7*g +
330*d^2*e^10*f^6*g^2 + 766*d^3*e^9*f^5*g^3 - 8400*d^4*e^8*f^4*g^4 + 12157*d^5*e^
7*f^3*g^5 - 7750*d^6*e^6*f^2*g^6 + 1650*d^7*e^5*f*g^7 + 285*d^8*e^4*g^8)*x^6 + (
56*d*e^11*f^8 - 230*d^2*e^10*f^7*g - 1242*d^3*e^9*f^6*g^2 - 5330*d^4*e^8*f^5*g^3
+ 14616*d^5*e^7*f^4*g^4 - 7925*d^6*e^6*f^3*g^5 + 2395*d^7*e^5*f^2*g^6 - 1230*d^
8*e^4*f*g^7 - 300*d^9*e^3*g^8)*x^5 - 20*(14*d^2*e^10*f^8 + 44*d^3*e^9*f^7*g + 20
*d^4*e^8*f^6*g^2 - 268*d^5*e^7*f^5*g^3 - 280*d^6*e^6*f^4*g^4 + 824*d^7*e^5*f^3*g
^5 - 561*d^8*e^4*f^2*g^6 + 120*d^9*e^3*f*g^7 + 15*d^10*e^2*g^8)*x^4 + 20*(8*d^3*
e^9*f^8 + 54*d^4*e^8*f^7*g + 98*d^5*e^7*f^6*g^2 + 330*d^6*e^6*f^5*g^3 - 1600*d^7
*e^5*f^4*g^4 + 1509*d^8*e^4*f^3*g^5 - 651*d^9*e^3*f^2*g^6 + 150*d^10*e^2*f*g^7 +
30*d^11*e*g^8)*x^3 + 120*(4*d^4*e^8*f^8 + 10*d^5*e^7*f^7*g - 2*d^6*e^6*f^6*g^2
- 122*d^7*e^5*f^5*g^3 + 200*d^8*e^4*f^4*g^4 - 141*d^9*e^3*f^3*g^5 + 49*d^10*e^2*
f^2*g^6 - 2*d^11*e*f*g^7 - 2*d^12*g^8)*x^2 - 240*(2*d^5*e^7*f^8 + 6*d^6*e^6*f^7*
g + 2*d^7*e^5*f^6*g^2 - 30*d^8*e^4*f^5*g^3 + 20*d^9*e^3*f^4*g^4 - 11*d^10*e^2*f^
3*g^5 + 6*d^11*e*f^2*g^6 + 2*d^12*f*g^7)*x)*sqrt(-e^2*x^2 + d^2))*sqrt(e^2*f^2 -
d^2*g^2))/((16*d^10*e^7*f^11 + 48*d^11*e^6*f^10*g + 16*d^12*e^5*f^9*g^2 - 80*d^
13*e^4*f^8*g^3 - 80*d^14*e^3*f^7*g^4 + 16*d^15*e^2*f^6*g^5 + 48*d^16*e*f^5*g^6 +
16*d^17*f^4*g^7 - (d^3*e^14*f^9*g^2 + 3*d^4*e^13*f^8*g^3 + d^5*e^12*f^7*g^4 - 5
*d^6*e^11*f^6*g^5 - 5*d^7*e^10*f^5*g^6 + d^8*e^9*f^4*g^7 + 3*d^9*e^8*f^3*g^8 + d
^10*e^7*f^2*g^9)*x^9 - (2*d^3*e^14*f^10*g - d^4*e^13*f^9*g^2 - 19*d^5*e^12*f^8*g
^3 - 17*d^6*e^11*f^7*g^4 + 25*d^7*e^10*f^6*g^5 + 37*d^8*e^9*f^5*g^6 - d^9*e^8*f^
4*g^7 - 19*d^10*e^7*f^3*g^8 - 7*d^11*e^6*f^2*g^9)*x^8 - (d^3*e^14*f^11 - 11*d^4*
e^13*f^10*g - 38*d^5*e^12*f^9*g^2 - 10*d^6*e^11*f^8*g^3 + 68*d^7*e^10*f^7*g^4 +
56*d^8*e^9*f^6*g^5 - 26*d^9*e^8*f^5*g^6 - 38*d^10*e^7*f^4*g^7 - 5*d^11*e^6*f^3*g
^8 + 3*d^12*e^5*f^2*g^9)*x^7 + (7*d^4*e^13*f^11 + 15*d^5*e^12*f^10*g - 42*d^6*e^
11*f^9*g^2 - 134*d^7*e^10*f^8*g^3 - 36*d^8*e^9*f^7*g^4 + 192*d^9*e^8*f^6*g^5 + 1
70*d^10*e^7*f^5*g^6 - 42*d^11*e^6*f^4*g^7 - 99*d^12*e^5*f^3*g^8 - 31*d^13*e^4*f^
2*g^9)*x^6 - (3*d^5*e^12*f^11 + 71*d^6*e^11*f^10*g + 149*d^7*e^10*f^9*g^2 - 73*d
^8*e^9*f^8*g^3 - 365*d^9*e^8*f^7*g^4 - 107*d^10*e^7*f^6*g^5 + 271*d^11*e^6*f^5*g
^6 + 149*d^12*e^5*f^4*g^7 - 58*d^13*e^4*f^3*g^8 - 40*d^14*e^3*f^2*g^9)*x^5 - (31
*d^6*e^11*f^11 + 13*d^7*e^10*f^10*g - 221*d^8*e^9*f^9*g^2 - 271*d^9*e^8*f^8*g^3
+ 233*d^10*e^7*f^7*g^4 + 491*d^11*e^6*f^6*g^5 + 73*d^12*e^5*f^5*g^6 - 221*d^13*e
^4*f^4*g^7 - 116*d^14*e^3*f^3*g^8 - 12*d^15*e^2*f^2*g^9)*x^4 + 8*(5*d^7*e^10*f^1
1 + 18*d^8*e^9*f^10*g + 9*d^9*e^8*f^9*g^2 - 37*d^10*e^7*f^8*g^3 - 45*d^11*e^6*f^
7*g^4 + 15*d^12*e^5*f^6*g^5 + 43*d^13*e^4*f^5*g^6 + 9*d^14*e^3*f^4*g^7 - 12*d^15
*e^2*f^3*g^8 - 5*d^16*e*f^2*g^9)*x^3 + 4*(3*d^8*e^9*f^11 - 11*d^9*e^8*f^10*g - 5
3*d^10*e^7*f^9*g^2 - 23*d^11*e^6*f^8*g^3 + 89*d^12*e^5*f^7*g^4 + 83*d^13*e^4*f^6
*g^5 - 31*d^14*e^3*f^5*g^6 - 53*d^15*e^2*f^4*g^7 - 8*d^16*e*f^3*g^8 + 4*d^17*f^2
*g^9)*x^2 - 8*(5*d^9*e^8*f^11 + 11*d^10*e^7*f^10*g - 7*d^11*e^6*f^9*g^2 - 29*d^1
2*e^5*f^8*g^3 - 5*d^13*e^4*f^7*g^4 + 25*d^14*e^3*f^6*g^5 + 11*d^15*e^2*f^5*g^6 -
7*d^16*e*f^4*g^7 - 4*d^17*f^3*g^8)*x - (16*d^9*e^7*f^11 + 48*d^10*e^6*f^10*g +
16*d^11*e^5*f^9*g^2 - 80*d^12*e^4*f^8*g^3 - 80*d^13*e^3*f^7*g^4 + 16*d^14*e^2*f^
6*g^5 + 48*d^15*e*f^5*g^6 + 16*d^16*f^4*g^7 + (d^3*e^13*f^9*g^2 + 3*d^4*e^12*f^8
*g^3 + d^5*e^11*f^7*g^4 - 5*d^6*e^10*f^6*g^5 - 5*d^7*e^9*f^5*g^6 + d^8*e^8*f^4*g
^7 + 3*d^9*e^7*f^3*g^8 + d^10*e^6*f^2*g^9)*x^8 + 2*(d^3*e^13*f^10*g + 4*d^4*e^12
*f^9*g^2 + 4*d^5*e^11*f^8*g^3 - 4*d^6*e^10*f^7*g^4 - 10*d^7*e^9*f^6*g^5 - 4*d^8*
e^8*f^5*g^6 + 4*d^9*e^7*f^4*g^7 + 4*d^10*e^6*f^3*g^8 + d^11*e^5*f^2*g^9)*x^7 + (
d^3*e^13*f^11 + 7*d^4*e^12*f^10*g - 6*d^5*e^11*f^9*g^2 - 58*d^6*e^10*f^8*g^3 - 4
4*d^7*e^9*f^7*g^4 + 76*d^8*e^8*f^6*g^5 + 102*d^9*e^7*f^5*g^6 - 6*d^10*e^6*f^4*g^
7 - 53*d^11*e^5*f^3*g^8 - 19*d^12*e^4*f^2*g^9)*x^6 + 2*(d^4*e^12*f^11 - 16*d^5*e
^11*f^10*g - 46*d^6*e^10*f^9*g^2 + 6*d^7*e^9*f^8*g^3 + 100*d^8*e^8*f^7*g^4 + 46*
d^9*e^7*f^6*g^5 - 66*d^10*e^6*f^5*g^6 - 46*d^11*e^5*f^4*g^7 + 11*d^12*e^4*f^3*g^
8 + 10*d^13*e^3*f^2*g^9)*x^5 - (19*d^5*e^11*f^11 + 17*d^6*e^10*f^10*g - 121*d^7*
e^9*f^9*g^2 - 195*d^8*e^8*f^8*g^3 + 85*d^9*e^7*f^7*g^4 + 319*d^10*e^6*f^6*g^5 +
117*d^11*e^5*f^5*g^6 - 121*d^12*e^4*f^4*g^7 - 100*d^13*e^3*f^3*g^8 - 20*d^14*e^2
*f^2*g^9)*x^4 + 20*(d^6*e^10*f^11 + 5*d^7*e^9*f^10*g + 5*d^8*e^8*f^9*g^2 - 9*d^9
*e^7*f^8*g^3 - 17*d^10*e^6*f^7*g^4 + d^11*e^5*f^6*g^5 + 15*d^12*e^4*f^5*g^6 + 5*
d^13*e^3*f^4*g^7 - 4*d^14*e^2*f^3*g^8 - 2*d^15*e*f^2*g^9)*x^3 + 4*(5*d^7*e^9*f^1
1 - 5*d^8*e^8*f^10*g - 51*d^9*e^7*f^9*g^2 - 33*d^10*e^6*f^8*g^3 + 79*d^11*e^5*f^
7*g^4 + 85*d^12*e^4*f^6*g^5 - 25*d^13*e^3*f^5*g^6 - 51*d^14*e^2*f^4*g^7 - 8*d^15
*e*f^3*g^8 + 4*d^16*f^2*g^9)*x^2 - 8*(5*d^8*e^8*f^11 + 11*d^9*e^7*f^10*g - 7*d^1
0*e^6*f^9*g^2 - 29*d^11*e^5*f^8*g^3 - 5*d^12*e^4*f^7*g^4 + 25*d^13*e^3*f^6*g^5 +
11*d^14*e^2*f^5*g^6 - 7*d^15*e*f^4*g^7 - 4*d^16*f^3*g^8)*x)*sqrt(-e^2*x^2 + d^2
))*sqrt(e^2*f^2 - d^2*g^2))]

_______________________________________________________________________________________

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((e*x+d)**3/(g*x+f)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 1.15921, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*(g*x + f)^3),x, algorithm="giac")

[Out]

Done