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

Optimal. Leaf size=153 \[ \frac{d+e x}{5 d^2 x \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 d^7 x}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^7}+\frac{8 d+5 e x}{5 d^6 x \sqrt{d^2-e^2 x^2}}+\frac{6 d+5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.402593, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{d+e x}{5 d^2 x \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 d^7 x}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^7}+\frac{8 d+5 e x}{5 d^6 x \sqrt{d^2-e^2 x^2}}+\frac{6 d+5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 52.7982, size = 126, normalized size = 0.82 \[ \frac{d + e x}{5 d^{2} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{6 d + 5 e x}{15 d^{4} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{24 d + 15 e x}{15 d^{6} x \sqrt{d^{2} - e^{2} x^{2}}} - \frac{e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{7}} - \frac{16 \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{7} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)/(5*d**2*x*(d**2 - e**2*x**2)**(5/2)) + (6*d + 5*e*x)/(15*d**4*x*(d**2
- e**2*x**2)**(3/2)) + (24*d + 15*e*x)/(15*d**6*x*sqrt(d**2 - e**2*x**2)) - e*at
anh(sqrt(d**2 - e**2*x**2)/d)/d**7 - 16*sqrt(d**2 - e**2*x**2)/(5*d**7*x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.11847, size = 123, normalized size = 0.8 \[ \frac{-15 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (15 d^5-38 d^4 e x-52 d^3 e^2 x^2+87 d^2 e^3 x^3+33 d e^4 x^4-48 e^5 x^5\right )}{x (e x-d)^3 (d+e x)^2}+15 e \log (x)}{15 d^7} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(15*d^5 - 38*d^4*e*x - 52*d^3*e^2*x^2 + 87*d^2*e^3*x^3 + 3
3*d*e^4*x^4 - 48*e^5*x^5))/(x*(-d + e*x)^3*(d + e*x)^2) + 15*e*Log[x] - 15*e*Log
[d + Sqrt[d^2 - e^2*x^2]])/(15*d^7)

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 195, normalized size = 1.3 \[ -{\frac{1}{dx} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{6\,{e}^{2}x}{5\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,{e}^{2}x}{5\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,{e}^{2}x}{5\,{d}^{7}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\frac{e}{5\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{e}{3\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{e}{{d}^{6}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{e}{{d}^{6}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.29642, size = 875, normalized size = 5.72 \[ \frac{23 \, e^{10} x^{10} + 217 \, d e^{9} x^{9} - 487 \, d^{2} e^{8} x^{8} - 1073 \, d^{3} e^{7} x^{7} + 1863 \, d^{4} e^{6} x^{6} + 1755 \, d^{5} e^{5} x^{5} - 2655 \, d^{6} e^{4} x^{4} - 1140 \, d^{7} e^{3} x^{3} + 1500 \, d^{8} e^{2} x^{2} + 240 \, d^{9} e x - 240 \, d^{10} + 15 \,{\left (e^{10} x^{10} - d e^{9} x^{9} - 14 \, d^{2} e^{8} x^{8} + 14 \, d^{3} e^{7} x^{7} + 41 \, d^{4} e^{6} x^{6} - 41 \, d^{5} e^{5} x^{5} - 44 \, d^{6} e^{4} x^{4} + 44 \, d^{7} e^{3} x^{3} + 16 \, d^{8} e^{2} x^{2} - 16 \, d^{9} e x +{\left (5 \, d e^{8} x^{8} - 5 \, d^{2} e^{7} x^{7} - 25 \, d^{3} e^{6} x^{6} + 25 \, d^{4} e^{5} x^{5} + 36 \, d^{5} e^{4} x^{4} - 36 \, d^{6} e^{3} x^{3} - 16 \, d^{7} e^{2} x^{2} + 16 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (48 \, e^{9} x^{9} - 148 \, d e^{8} x^{8} - 548 \, d^{2} e^{7} x^{7} + 1023 \, d^{3} e^{6} x^{6} + 1275 \, d^{4} e^{5} x^{5} - 1995 \, d^{5} e^{4} x^{4} - 1020 \, d^{6} e^{3} x^{3} + 1380 \, d^{7} e^{2} x^{2} + 240 \, d^{8} e x - 240 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{7} e^{9} x^{10} - d^{8} e^{8} x^{9} - 14 \, d^{9} e^{7} x^{8} + 14 \, d^{10} e^{6} x^{7} + 41 \, d^{11} e^{5} x^{6} - 41 \, d^{12} e^{4} x^{5} - 44 \, d^{13} e^{3} x^{4} + 44 \, d^{14} e^{2} x^{3} + 16 \, d^{15} e x^{2} - 16 \, d^{16} x +{\left (5 \, d^{8} e^{7} x^{8} - 5 \, d^{9} e^{6} x^{7} - 25 \, d^{10} e^{5} x^{6} + 25 \, d^{11} e^{4} x^{5} + 36 \, d^{12} e^{3} x^{4} - 36 \, d^{13} e^{2} x^{3} - 16 \, d^{14} e x^{2} + 16 \, d^{15} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(23*e^10*x^10 + 217*d*e^9*x^9 - 487*d^2*e^8*x^8 - 1073*d^3*e^7*x^7 + 1863*d
^4*e^6*x^6 + 1755*d^5*e^5*x^5 - 2655*d^6*e^4*x^4 - 1140*d^7*e^3*x^3 + 1500*d^8*e
^2*x^2 + 240*d^9*e*x - 240*d^10 + 15*(e^10*x^10 - d*e^9*x^9 - 14*d^2*e^8*x^8 + 1
4*d^3*e^7*x^7 + 41*d^4*e^6*x^6 - 41*d^5*e^5*x^5 - 44*d^6*e^4*x^4 + 44*d^7*e^3*x^
3 + 16*d^8*e^2*x^2 - 16*d^9*e*x + (5*d*e^8*x^8 - 5*d^2*e^7*x^7 - 25*d^3*e^6*x^6
+ 25*d^4*e^5*x^5 + 36*d^5*e^4*x^4 - 36*d^6*e^3*x^3 - 16*d^7*e^2*x^2 + 16*d^8*e*x
)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (48*e^9*x^9 - 148*d
*e^8*x^8 - 548*d^2*e^7*x^7 + 1023*d^3*e^6*x^6 + 1275*d^4*e^5*x^5 - 1995*d^5*e^4*
x^4 - 1020*d^6*e^3*x^3 + 1380*d^7*e^2*x^2 + 240*d^8*e*x - 240*d^9)*sqrt(-e^2*x^2
+ d^2))/(d^7*e^9*x^10 - d^8*e^8*x^9 - 14*d^9*e^7*x^8 + 14*d^10*e^6*x^7 + 41*d^1
1*e^5*x^6 - 41*d^12*e^4*x^5 - 44*d^13*e^3*x^4 + 44*d^14*e^2*x^3 + 16*d^15*e*x^2
- 16*d^16*x + (5*d^8*e^7*x^8 - 5*d^9*e^6*x^7 - 25*d^10*e^5*x^6 + 25*d^11*e^4*x^5
+ 36*d^12*e^3*x^4 - 36*d^13*e^2*x^3 - 16*d^14*e*x^2 + 16*d^15*x)*sqrt(-e^2*x^2
+ d^2))

_______________________________________________________________________________________

Sympy [A]  time = 43.2663, size = 2404, normalized size = 15.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*Piecewise((5*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*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)) + e*Piecewise((-46*I*d**6*sqrt(-1 + e**2*x**2/d**2)/(-30*
d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 15*d**6*lo
g(e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*
e**6*x**6) + 30*d**6*log(e*x/d)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x
**4 + 30*d**7*e**6*x**6) - 30*I*d**6*asin(d/(e*x))/(-30*d**13 + 90*d**11*e**2*x*
*2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 70*I*d**4*e**2*x**2*sqrt(-1 + e**2
*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x
**6) + 45*d**4*e**2*x**2*log(e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 9
0*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 90*d**4*e**2*x**2*log(e*x/d)/(-30*d**13
+ 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 90*I*d**4*e**2*x
**2*asin(d/(e*x))/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*
e**6*x**6) - 30*I*d**2*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)/(-30*d**13 + 90*d**11
*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 45*d**2*e**4*x**4*log(e**2
*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x
**6) + 90*d**2*e**4*x**4*log(e*x/d)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e*
*4*x**4 + 30*d**7*e**6*x**6) - 90*I*d**2*e**4*x**4*asin(d/(e*x))/(-30*d**13 + 90
*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 15*e**6*x**6*log(e**
2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*
x**6) - 30*e**6*x**6*log(e*x/d)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x
**4 + 30*d**7*e**6*x**6) + 30*I*e**6*x**6*asin(d/(e*x))/(-30*d**13 + 90*d**11*e*
*2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6), Abs(e**2*x**2/d**2) > 1), (-46
*d**6*sqrt(1 - e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x*
*4 + 30*d**7*e**6*x**6) - 15*d**6*log(e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2
*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 30*d**6*log(sqrt(1 - e**2*x**2/
d**2) + 1)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x*
*6) - 15*I*pi*d**6/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7
*e**6*x**6) + 70*d**4*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e
**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 45*d**4*e**2*x**2*log(e**2*x
**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**
6) - 90*d**4*e**2*x**2*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-30*d**13 + 90*d**11*e
**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 45*I*pi*d**4*e**2*x**2/(-30*
d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 30*d**2*e*
*4*x**4*sqrt(1 - e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*
x**4 + 30*d**7*e**6*x**6) - 45*d**2*e**4*x**4*log(e**2*x**2/d**2)/(-30*d**13 + 9
0*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 90*d**2*e**4*x**4*l
og(sqrt(1 - e**2*x**2/d**2) + 1)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*
x**4 + 30*d**7*e**6*x**6) - 45*I*pi*d**2*e**4*x**4/(-30*d**13 + 90*d**11*e**2*x*
*2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 15*e**6*x**6*log(e**2*x**2/d**2)/(
-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 30*e**
6*x**6*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d*
*9*e**4*x**4 + 30*d**7*e**6*x**6) + 15*I*pi*e**6*x**6/(-30*d**13 + 90*d**11*e**2
*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.300276, size = 255, normalized size = 1.67 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left (3 \,{\left (x{\left (\frac{11 \, x e^{6}}{d^{7}} + \frac{5 \, e^{5}}{d^{6}}\right )} - \frac{25 \, e^{4}}{d^{5}}\right )} x - \frac{35 \, e^{3}}{d^{4}}\right )} x + \frac{45 \, e^{2}}{d^{3}}\right )} x + \frac{23 \, e}{d^{2}}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{e{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d^{7}} + \frac{x e^{3}}{2 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{7}} - \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, d^{7} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*(((3*(x*(11*x*e^6/d^7 + 5*e^5/d^6) - 25*e^4/d^5)*x -
35*e^3/d^4)*x + 45*e^2/d^3)*x + 23*e/d^2)/(x^2*e^2 - d^2)^3 - e*ln(1/2*abs(-2*d*
e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^7 + 1/2*x*e^3/((d*e + sqrt(-x^2*e
^2 + d^2)*e)*d^7) - 1/2*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-1)/(d^7*x)