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

Optimal. Leaf size=184 \[ \frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(d + e*x)/(5*d^2*x^2*(d^2 - e^2*x^2)^(5/2)) + (7*d + 6*e*x)/(15*d^4*x^2*(d^2 - e
^2*x^2)^(3/2)) + (35*d + 24*e*x)/(15*d^6*x^2*Sqrt[d^2 - e^2*x^2]) - (7*Sqrt[d^2
- e^2*x^2])/(2*d^7*x^2) - (16*e*Sqrt[d^2 - e^2*x^2])/(5*d^8*x) - (7*e^2*ArcTanh[
Sqrt[d^2 - e^2*x^2]/d])/(2*d^8)

_______________________________________________________________________________________

Rubi [A]  time = 0.500497, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(d + e*x)/(5*d^2*x^2*(d^2 - e^2*x^2)^(5/2)) + (7*d + 6*e*x)/(15*d^4*x^2*(d^2 - e
^2*x^2)^(3/2)) + (35*d + 24*e*x)/(15*d^6*x^2*Sqrt[d^2 - e^2*x^2]) - (7*Sqrt[d^2
- e^2*x^2])/(2*d^7*x^2) - (16*e*Sqrt[d^2 - e^2*x^2])/(5*d^8*x) - (7*e^2*ArcTanh[
Sqrt[d^2 - e^2*x^2]/d])/(2*d^8)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 66.6412, size = 162, normalized size = 0.88 \[ \frac{d + e x}{5 d^{2} x^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{7 d + 6 e x}{15 d^{4} x^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{35 d + 24 e x}{15 d^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{7 \sqrt{d^{2} - e^{2} x^{2}}}{2 d^{7} x^{2}} - \frac{7 e^{2} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{8}} - \frac{16 e \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{8} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)/(5*d**2*x**2*(d**2 - e**2*x**2)**(5/2)) + (7*d + 6*e*x)/(15*d**4*x**2*
(d**2 - e**2*x**2)**(3/2)) + (35*d + 24*e*x)/(15*d**6*x**2*sqrt(d**2 - e**2*x**2
)) - 7*sqrt(d**2 - e**2*x**2)/(2*d**7*x**2) - 7*e**2*atanh(sqrt(d**2 - e**2*x**2
)/d)/(2*d**8) - 16*e*sqrt(d**2 - e**2*x**2)/(5*d**8*x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.132576, size = 138, normalized size = 0.75 \[ \frac{-105 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (15 d^6+15 d^5 e x-176 d^4 e^2 x^2-4 d^3 e^3 x^3+249 d^2 e^4 x^4-9 d e^5 x^5-96 e^6 x^6\right )}{x^2 (e x-d)^3 (d+e x)^2}+105 e^2 \log (x)}{30 d^8} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(15*d^6 + 15*d^5*e*x - 176*d^4*e^2*x^2 - 4*d^3*e^3*x^3 + 2
49*d^2*e^4*x^4 - 9*d*e^5*x^5 - 96*e^6*x^6))/(x^2*(-d + e*x)^3*(d + e*x)^2) + 105
*e^2*Log[x] - 105*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(30*d^8)

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 227, normalized size = 1.2 \[ -{\frac{1}{2\,d{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,{e}^{2}}{10\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,{e}^{2}}{6\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{e}^{2}}{2\,{d}^{7}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{7\,{e}^{2}}{2\,{d}^{7}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{e}{{d}^{2}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{6\,{e}^{3}x}{5\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,{e}^{3}x}{5\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,{e}^{3}x}{5\,{d}^{8}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.319681, size = 1013, normalized size = 5.51 \[ -\frac{96 \, e^{12} x^{12} - 687 \, d e^{11} x^{11} - 1281 \, d^{2} e^{10} x^{10} + 4946 \, d^{3} e^{9} x^{9} + 4162 \, d^{4} e^{8} x^{8} - 11487 \, d^{5} e^{7} x^{7} - 6375 \, d^{6} e^{6} x^{6} + 11310 \, d^{7} e^{5} x^{5} + 5550 \, d^{8} e^{4} x^{4} - 4560 \, d^{9} e^{3} x^{3} - 2640 \, d^{10} e^{2} x^{2} + 480 \, d^{11} e x + 480 \, d^{12} - 105 \,{\left (6 \, d e^{11} x^{11} - 6 \, d^{2} e^{10} x^{10} - 44 \, d^{3} e^{9} x^{9} + 44 \, d^{4} e^{8} x^{8} + 102 \, d^{5} e^{7} x^{7} - 102 \, d^{6} e^{6} x^{6} - 96 \, d^{7} e^{5} x^{5} + 96 \, d^{8} e^{4} x^{4} + 32 \, d^{9} e^{3} x^{3} - 32 \, d^{10} e^{2} x^{2} -{\left (e^{11} x^{11} - d e^{10} x^{10} - 19 \, d^{2} e^{9} x^{9} + 19 \, d^{3} e^{8} x^{8} + 66 \, d^{4} e^{7} x^{7} - 66 \, d^{5} e^{6} x^{6} - 80 \, d^{6} e^{5} x^{5} + 80 \, d^{7} e^{4} x^{4} + 32 \, d^{8} e^{3} x^{3} - 32 \, d^{9} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + 2 \,{\left (58 \, e^{11} x^{11} + 230 \, d e^{10} x^{10} - 1075 \, d^{2} e^{9} x^{9} - 1181 \, d^{3} e^{8} x^{8} + 3696 \, d^{4} e^{7} x^{7} + 2220 \, d^{5} e^{6} x^{6} - 4605 \, d^{6} e^{5} x^{5} - 2205 \, d^{7} e^{4} x^{4} + 2160 \, d^{8} e^{3} x^{3} + 1200 \, d^{9} e^{2} x^{2} - 240 \, d^{10} e x - 240 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (6 \, d^{9} e^{9} x^{11} - 6 \, d^{10} e^{8} x^{10} - 44 \, d^{11} e^{7} x^{9} + 44 \, d^{12} e^{6} x^{8} + 102 \, d^{13} e^{5} x^{7} - 102 \, d^{14} e^{4} x^{6} - 96 \, d^{15} e^{3} x^{5} + 96 \, d^{16} e^{2} x^{4} + 32 \, d^{17} e x^{3} - 32 \, d^{18} x^{2} -{\left (d^{8} e^{9} x^{11} - d^{9} e^{8} x^{10} - 19 \, d^{10} e^{7} x^{9} + 19 \, d^{11} e^{6} x^{8} + 66 \, d^{12} e^{5} x^{7} - 66 \, d^{13} e^{4} x^{6} - 80 \, d^{14} e^{3} x^{5} + 80 \, d^{15} e^{2} x^{4} + 32 \, d^{16} e x^{3} - 32 \, d^{17} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(96*e^12*x^12 - 687*d*e^11*x^11 - 1281*d^2*e^10*x^10 + 4946*d^3*e^9*x^9 +
4162*d^4*e^8*x^8 - 11487*d^5*e^7*x^7 - 6375*d^6*e^6*x^6 + 11310*d^7*e^5*x^5 + 55
50*d^8*e^4*x^4 - 4560*d^9*e^3*x^3 - 2640*d^10*e^2*x^2 + 480*d^11*e*x + 480*d^12
- 105*(6*d*e^11*x^11 - 6*d^2*e^10*x^10 - 44*d^3*e^9*x^9 + 44*d^4*e^8*x^8 + 102*d
^5*e^7*x^7 - 102*d^6*e^6*x^6 - 96*d^7*e^5*x^5 + 96*d^8*e^4*x^4 + 32*d^9*e^3*x^3
- 32*d^10*e^2*x^2 - (e^11*x^11 - d*e^10*x^10 - 19*d^2*e^9*x^9 + 19*d^3*e^8*x^8 +
66*d^4*e^7*x^7 - 66*d^5*e^6*x^6 - 80*d^6*e^5*x^5 + 80*d^7*e^4*x^4 + 32*d^8*e^3*
x^3 - 32*d^9*e^2*x^2)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) +
2*(58*e^11*x^11 + 230*d*e^10*x^10 - 1075*d^2*e^9*x^9 - 1181*d^3*e^8*x^8 + 3696*
d^4*e^7*x^7 + 2220*d^5*e^6*x^6 - 4605*d^6*e^5*x^5 - 2205*d^7*e^4*x^4 + 2160*d^8*
e^3*x^3 + 1200*d^9*e^2*x^2 - 240*d^10*e*x - 240*d^11)*sqrt(-e^2*x^2 + d^2))/(6*d
^9*e^9*x^11 - 6*d^10*e^8*x^10 - 44*d^11*e^7*x^9 + 44*d^12*e^6*x^8 + 102*d^13*e^5
*x^7 - 102*d^14*e^4*x^6 - 96*d^15*e^3*x^5 + 96*d^16*e^2*x^4 + 32*d^17*e*x^3 - 32
*d^18*x^2 - (d^8*e^9*x^11 - d^9*e^8*x^10 - 19*d^10*e^7*x^9 + 19*d^11*e^6*x^8 + 6
6*d^12*e^5*x^7 - 66*d^13*e^4*x^6 - 80*d^14*e^3*x^5 + 80*d^15*e^2*x^4 + 32*d^16*e
*x^3 - 32*d^17*x^2)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

Sympy [A]  time = 62.7193, size = 2691, normalized size = 14.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*Piecewise((30*I*d**8*sqrt(-1 + e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**
2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 322*I*d**6*e**2*x**2*sqrt(-1
+ e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 +
60*d**9*e**6*x**8) - 105*d**6*e**2*x**2*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 1
80*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 210*d**6*e**2*x*
*2*log(e*x/d)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d
**9*e**6*x**8) - 210*I*d**6*e**2*x**2*asin(d/(e*x))/(-60*d**15*x**2 + 180*d**13*
e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 490*I*d**4*e**4*x**4*sqrt
(-1 + e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**
6 + 60*d**9*e**6*x**8) + 315*d**4*e**4*x**4*log(e**2*x**2/d**2)/(-60*d**15*x**2
+ 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 630*d**4*e**4
*x**4*log(e*x/d)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 6
0*d**9*e**6*x**8) + 630*I*d**4*e**4*x**4*asin(d/(e*x))/(-60*d**15*x**2 + 180*d**
13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 210*I*d**2*e**6*x**6*s
qrt(-1 + e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*
x**6 + 60*d**9*e**6*x**8) - 315*d**2*e**6*x**6*log(e**2*x**2/d**2)/(-60*d**15*x*
*2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 630*d**2*e
**6*x**6*log(e*x/d)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6
+ 60*d**9*e**6*x**8) - 630*I*d**2*e**6*x**6*asin(d/(e*x))/(-60*d**15*x**2 + 180*
d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 105*e**8*x**8*log(e
**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*
d**9*e**6*x**8) - 210*e**8*x**8*log(e*x/d)/(-60*d**15*x**2 + 180*d**13*e**2*x**4
- 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 210*I*e**8*x**8*asin(d/(e*x))/(-60
*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8), Ab
s(e**2*x**2/d**2) > 1), (30*d**8*sqrt(1 - e**2*x**2/d**2)/(-60*d**15*x**2 + 180*
d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 322*d**6*e**2*x**2*
sqrt(1 - e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*
x**6 + 60*d**9*e**6*x**8) - 105*d**6*e**2*x**2*log(e**2*x**2/d**2)/(-60*d**15*x*
*2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 210*d**6*e
**2*x**2*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-60*d**15*x**2 + 180*d**13*e**2*x**4
- 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 105*I*pi*d**6*e**2*x**2/(-60*d**15
*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 490*d**
4*e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180
*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 315*d**4*e**4*x**4*log(e**2*x**2/d**2)/(
-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8)
- 630*d**4*e**4*x**4*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-60*d**15*x**2 + 180*d**
13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 315*I*pi*d**4*e**4*x**
4/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**
8) - 210*d**2*e**6*x**6*sqrt(1 - e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**
2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 315*d**2*e**6*x**6*log(e**2*
x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9
*e**6*x**8) + 630*d**2*e**6*x**6*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-60*d**15*x*
*2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 315*I*pi*d
**2*e**6*x**6/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d
**9*e**6*x**8) + 105*e**8*x**8*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e
**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 210*e**8*x**8*log(sqrt(1 -
e**2*x**2/d**2) + 1)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**
6 + 60*d**9*e**6*x**8) + 105*I*pi*e**8*x**8/(-60*d**15*x**2 + 180*d**13*e**2*x**
4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8), True)) + e*Piecewise((5*d**6*e*sqr
t(d**2/(e**2*x**2) - 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*
d**8*e**6*x**6) - 30*d**4*e**3*x**2*sqrt(d**2/(e**2*x**2) - 1)/(-5*d**14 + 15*d*
*12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6) + 40*d**2*e**5*x**4*sqrt(
d**2/(e**2*x**2) - 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d*
*8*e**6*x**6) - 16*e**7*x**6*sqrt(d**2/(e**2*x**2) - 1)/(-5*d**14 + 15*d**12*e**
2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6), Abs(d**2/(e**2*x**2)) > 1), (5*
I*d**6*e*sqrt(-d**2/(e**2*x**2) + 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e
**4*x**4 + 5*d**8*e**6*x**6) - 30*I*d**4*e**3*x**2*sqrt(-d**2/(e**2*x**2) + 1)/(
-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6) + 40*I*d*
*2*e**5*x**4*sqrt(-d**2/(e**2*x**2) + 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**
10*e**4*x**4 + 5*d**8*e**6*x**6) - 16*I*e**7*x**6*sqrt(-d**2/(e**2*x**2) + 1)/(-
5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.300302, size = 351, normalized size = 1.91 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left (3 \,{\left (x{\left (\frac{11 \, x e^{7}}{d^{8}} + \frac{15 \, e^{6}}{d^{7}}\right )} - \frac{25 \, e^{5}}{d^{6}}\right )} x - \frac{100 \, e^{4}}{d^{5}}\right )} x + \frac{45 \, e^{3}}{d^{4}}\right )} x + \frac{58 \, e^{2}}{d^{3}}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{7 \, e^{2}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{2 \, d^{8}} + \frac{x^{2}{\left (\frac{4 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{4}}{x} + e^{6}\right )}}{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{8}} - \frac{{\left (\frac{4 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{8} e^{8}}{x} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{8} e^{6}}{x^{2}}\right )} e^{\left (-8\right )}}{8 \, d^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*(((3*(x*(11*x*e^7/d^8 + 15*e^6/d^7) - 25*e^5/d^6)*x -
100*e^4/d^5)*x + 45*e^3/d^4)*x + 58*e^2/d^3)/(x^2*e^2 - d^2)^3 - 7/2*e^2*ln(1/2
*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^8 + 1/8*x^2*(4*(d*e + s
qrt(-x^2*e^2 + d^2)*e)*e^4/x + e^6)/((d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^8) - 1/8
*(4*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^8*e^8/x + (d*e + sqrt(-x^2*e^2 + d^2)*e)^2*
d^8*e^6/x^2)*e^(-8)/d^16