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

Optimal. Leaf size=161 \[ \frac{x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{(32 d+35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}-\frac{7 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^8}+\frac{x^2 (24 d+35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(x^6*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d + 7*e*x))/(15*e^4*(d^2
- e^2*x^2)^(3/2)) + (x^2*(24*d + 35*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + ((32*d
+ 35*e*x)*Sqrt[d^2 - e^2*x^2])/(10*e^8) - (7*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^
2]])/(2*e^8)

_______________________________________________________________________________________

Rubi [A]  time = 0.423463, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{(32 d+35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}-\frac{7 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^8}+\frac{x^2 (24 d+35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^6*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d + 7*e*x))/(15*e^4*(d^2
- e^2*x^2)^(3/2)) + (x^2*(24*d + 35*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + ((32*d
+ 35*e*x)*Sqrt[d^2 - e^2*x^2])/(10*e^8) - (7*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^
2]])/(2*e^8)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.4078, size = 146, normalized size = 0.91 \[ - \frac{7 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e^{8}} + \frac{x^{6} \left (2 d + 2 e x\right )}{10 e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{x^{4} \left (24 d + 28 e x\right )}{60 e^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{x^{2} \left (192 d + 280 e x\right )}{120 e^{6} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{\left (768 d + 840 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{240 e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*d**2*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e**8) + x**6*(2*d + 2*e*x)/(10*e**2*
(d**2 - e**2*x**2)**(5/2)) - x**4*(24*d + 28*e*x)/(60*e**4*(d**2 - e**2*x**2)**(
3/2)) + x**2*(192*d + 280*e*x)/(120*e**6*sqrt(d**2 - e**2*x**2)) + (768*d + 840*
e*x)*sqrt(d**2 - e**2*x**2)/(240*e**8)

_______________________________________________________________________________________

Mathematica [A]  time = 0.193679, size = 128, normalized size = 0.8 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (96 d^6+9 d^5 e x-249 d^4 e^2 x^2+4 d^3 e^3 x^3+176 d^2 e^4 x^4-15 d e^5 x^5-15 e^6 x^6\right )}{(d-e x)^3 (d+e x)^2}-105 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{30 e^8} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(96*d^6 + 9*d^5*e*x - 249*d^4*e^2*x^2 + 4*d^3*e^3*x^3 + 17
6*d^2*e^4*x^4 - 15*d*e^5*x^5 - 15*e^6*x^6))/((d - e*x)^3*(d + e*x)^2) - 105*d^2*
ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^8)

_______________________________________________________________________________________

Maple [A]  time = 0.102, size = 227, normalized size = 1.4 \[ -{\frac{d{x}^{6}}{{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+6\,{\frac{{d}^{3}{x}^{4}}{{e}^{4} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-8\,{\frac{{d}^{5}{x}^{2}}{{e}^{6} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}+{\frac{16\,{d}^{7}}{5\,{e}^{8}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{{x}^{7}}{2\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,{d}^{2}{x}^{5}}{10\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{7\,{d}^{2}{x}^{3}}{6\,{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{d}^{2}x}{2\,{e}^{7}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{7\,{d}^{2}}{2\,{e}^{7}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*x^6/e^2/(-e^2*x^2+d^2)^(5/2)+6*d^3/e^4*x^4/(-e^2*x^2+d^2)^(5/2)-8*d^5/e^6*x^2
/(-e^2*x^2+d^2)^(5/2)+16/5*d^7/e^8/(-e^2*x^2+d^2)^(5/2)-1/2*x^7/e/(-e^2*x^2+d^2)
^(5/2)+7/10*d^2/e^3*x^5/(-e^2*x^2+d^2)^(5/2)-7/6*d^2/e^5*x^3/(-e^2*x^2+d^2)^(3/2
)+7/2*d^2/e^7*x/(-e^2*x^2+d^2)^(1/2)-7/2*d^2/e^7/(e^2)^(1/2)*arctan((e^2)^(1/2)*
x/(-e^2*x^2+d^2)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.804259, size = 439, normalized size = 2.73 \[ -\frac{x^{7}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{7 \, d^{2} x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )}}{30 \, e} - \frac{d x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{7 \, d^{2} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )}}{6 \, e^{3}} + \frac{6 \, d^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} - \frac{8 \, d^{5} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}} + \frac{16 \, d^{7}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{8}} + \frac{14 \, d^{4} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{7}} - \frac{49 \, d^{2} x}{30 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{7}} - \frac{7 \, d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x^7/((-e^2*x^2 + d^2)^(5/2)*e) + 7/30*d^2*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)
*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*
e^6))/e - d*x^6/((-e^2*x^2 + d^2)^(5/2)*e^2) - 7/6*d^2*x*(3*x^2/((-e^2*x^2 + d^2
)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4))/e^3 + 6*d^3*x^4/((-e^2*x^2 +
d^2)^(5/2)*e^4) - 8*d^5*x^2/((-e^2*x^2 + d^2)^(5/2)*e^6) + 16/5*d^7/((-e^2*x^2 +
d^2)^(5/2)*e^8) + 14/15*d^4*x/((-e^2*x^2 + d^2)^(3/2)*e^7) - 49/30*d^2*x/(sqrt(
-e^2*x^2 + d^2)*e^7) - 7/2*d^2*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^7)

_______________________________________________________________________________________

Fricas [A]  time = 0.312542, size = 965, normalized size = 5.99 \[ \frac{15 \, e^{12} x^{12} + 15 \, d e^{11} x^{11} - 446 \, d^{2} e^{10} x^{10} + 302 \, d^{3} e^{9} x^{9} + 3561 \, d^{4} e^{8} x^{8} - 3441 \, d^{5} e^{7} x^{7} - 9282 \, d^{6} e^{6} x^{6} + 9282 \, d^{7} e^{5} x^{5} + 9520 \, d^{8} e^{4} x^{4} - 9520 \, d^{9} e^{3} x^{3} - 3360 \, d^{10} e^{2} x^{2} + 3360 \, d^{11} e x + 210 \,{\left (6 \, d^{3} e^{9} x^{9} - 6 \, d^{4} e^{8} x^{8} - 44 \, d^{5} e^{7} x^{7} + 44 \, d^{6} e^{6} x^{6} + 102 \, d^{7} e^{5} x^{5} - 102 \, d^{8} e^{4} x^{4} - 96 \, d^{9} e^{3} x^{3} + 96 \, d^{10} e^{2} x^{2} + 32 \, d^{11} e x - 32 \, d^{12} -{\left (d^{2} e^{9} x^{9} - d^{3} e^{8} x^{8} - 19 \, d^{4} e^{7} x^{7} + 19 \, d^{5} e^{6} x^{6} + 66 \, d^{6} e^{5} x^{5} - 66 \, d^{7} e^{4} x^{4} - 80 \, d^{8} e^{3} x^{3} + 80 \, d^{9} e^{2} x^{2} + 32 \, d^{10} e x - 32 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 2 \,{\left (45 \, d e^{10} x^{10} - 3 \, d^{2} e^{9} x^{9} - 720 \, d^{3} e^{8} x^{8} + 660 \, d^{4} e^{7} x^{7} + 2891 \, d^{5} e^{6} x^{6} - 2891 \, d^{6} e^{5} x^{5} - 3920 \, d^{7} e^{4} x^{4} + 3920 \, d^{8} e^{3} x^{3} + 1680 \, d^{9} e^{2} x^{2} - 1680 \, d^{10} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (6 \, d e^{17} x^{9} - 6 \, d^{2} e^{16} x^{8} - 44 \, d^{3} e^{15} x^{7} + 44 \, d^{4} e^{14} x^{6} + 102 \, d^{5} e^{13} x^{5} - 102 \, d^{6} e^{12} x^{4} - 96 \, d^{7} e^{11} x^{3} + 96 \, d^{8} e^{10} x^{2} + 32 \, d^{9} e^{9} x - 32 \, d^{10} e^{8} -{\left (e^{17} x^{9} - d e^{16} x^{8} - 19 \, d^{2} e^{15} x^{7} + 19 \, d^{3} e^{14} x^{6} + 66 \, d^{4} e^{13} x^{5} - 66 \, d^{5} e^{12} x^{4} - 80 \, d^{6} e^{11} x^{3} + 80 \, d^{7} e^{10} x^{2} + 32 \, d^{8} e^{9} x - 32 \, d^{9} e^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(15*e^12*x^12 + 15*d*e^11*x^11 - 446*d^2*e^10*x^10 + 302*d^3*e^9*x^9 + 3561
*d^4*e^8*x^8 - 3441*d^5*e^7*x^7 - 9282*d^6*e^6*x^6 + 9282*d^7*e^5*x^5 + 9520*d^8
*e^4*x^4 - 9520*d^9*e^3*x^3 - 3360*d^10*e^2*x^2 + 3360*d^11*e*x + 210*(6*d^3*e^9
*x^9 - 6*d^4*e^8*x^8 - 44*d^5*e^7*x^7 + 44*d^6*e^6*x^6 + 102*d^7*e^5*x^5 - 102*d
^8*e^4*x^4 - 96*d^9*e^3*x^3 + 96*d^10*e^2*x^2 + 32*d^11*e*x - 32*d^12 - (d^2*e^9
*x^9 - d^3*e^8*x^8 - 19*d^4*e^7*x^7 + 19*d^5*e^6*x^6 + 66*d^6*e^5*x^5 - 66*d^7*e
^4*x^4 - 80*d^8*e^3*x^3 + 80*d^9*e^2*x^2 + 32*d^10*e*x - 32*d^11)*sqrt(-e^2*x^2
+ d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 2*(45*d*e^10*x^10 - 3*d^2*e^
9*x^9 - 720*d^3*e^8*x^8 + 660*d^4*e^7*x^7 + 2891*d^5*e^6*x^6 - 2891*d^6*e^5*x^5
- 3920*d^7*e^4*x^4 + 3920*d^8*e^3*x^3 + 1680*d^9*e^2*x^2 - 1680*d^10*e*x)*sqrt(-
e^2*x^2 + d^2))/(6*d*e^17*x^9 - 6*d^2*e^16*x^8 - 44*d^3*e^15*x^7 + 44*d^4*e^14*x
^6 + 102*d^5*e^13*x^5 - 102*d^6*e^12*x^4 - 96*d^7*e^11*x^3 + 96*d^8*e^10*x^2 + 3
2*d^9*e^9*x - 32*d^10*e^8 - (e^17*x^9 - d*e^16*x^8 - 19*d^2*e^15*x^7 + 19*d^3*e^
14*x^6 + 66*d^4*e^13*x^5 - 66*d^5*e^12*x^4 - 80*d^6*e^11*x^3 + 80*d^7*e^10*x^2 +
32*d^8*e^9*x - 32*d^9*e^8)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

Sympy [A]  time = 56.1707, size = 2004, normalized size = 12.45 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((16*d**6/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sq
rt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) - 40*d**4*e**2*x**2/
(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2)
+ 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) + 30*d**2*e**4*x**4/(5*d**4*e**8*sqrt(d**
2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d
**2 - e**2*x**2)) - 5*e**6*x**6/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e*
*10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)), Ne(e, 0)
), (x**8/(8*(d**2)**(7/2)), True)) + e*Piecewise((210*I*d**7*sqrt(-1 + e**2*x**2
/d**2)*acosh(e*x/d)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**
2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 105*p
i*d**7*sqrt(-1 + e**2*x**2/d**2)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d
**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d
**2)) - 210*I*d**6*e*x/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*
x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 42
0*I*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(60*d**5*e**9*sqrt(-1
+ e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x
**4*sqrt(-1 + e**2*x**2/d**2)) + 210*pi*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)
/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x*
*2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 490*I*d**4*e**3*x**3/(60
*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d
**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 210*I*d**3*e**4*x**4*sqrt(-1
+ e**2*x**2/d**2)*acosh(e*x/d)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d*
*3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d*
*2)) - 105*pi*d**3*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)/(60*d**5*e**9*sqrt(-1 + e
**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4
*sqrt(-1 + e**2*x**2/d**2)) - 322*I*d**2*e**5*x**5/(60*d**5*e**9*sqrt(-1 + e**2*
x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqr
t(-1 + e**2*x**2/d**2)) + 30*I*e**7*x**7/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2)
- 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**
2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-105*d**7*sqrt(1 - e**2*x**2/d**2)*asi
n(e*x/d)/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e*
*2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) + 105*d**6*e*x/(30*d**
5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) +
30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) + 210*d**5*e**2*x**2*sqrt(1 - e**2*x**
2/d**2)*asin(e*x/d)/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*
sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) - 245*d**4*
e**3*x**3/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e
**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) - 105*d**3*e**4*x**4*
sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60
*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d
**2)) + 161*d**2*e**5*x**5/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**1
1*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) - 15
*e**7*x**7/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 -
e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.298067, size = 162, normalized size = 1.01 \[ -\frac{7}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-8\right )}{\rm sign}\left (d\right ) - \frac{{\left (96 \, d^{7} e^{\left (-8\right )} +{\left (105 \, d^{6} e^{\left (-7\right )} -{\left (240 \, d^{5} e^{\left (-6\right )} +{\left (245 \, d^{4} e^{\left (-5\right )} -{\left (180 \, d^{3} e^{\left (-4\right )} +{\left (161 \, d^{2} e^{\left (-3\right )} - 15 \,{\left (x e^{\left (-1\right )} + 2 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{30 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/2*d^2*arcsin(x*e/d)*e^(-8)*sign(d) - 1/30*(96*d^7*e^(-8) + (105*d^6*e^(-7) -
(240*d^5*e^(-6) + (245*d^4*e^(-5) - (180*d^3*e^(-4) + (161*d^2*e^(-3) - 15*(x*e^
(-1) + 2*d*e^(-2))*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3