3.215 \(\int \frac{1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=205 \[ -\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{128 x}{715 d^{10} \sqrt{d^2-e^2 x^2}}+\frac{64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(48*x)/(715*d^6*(d^2 - e^2*x^2)^(5/2)) - 1/(13*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(
5/2)) - 9/(143*d^2*e*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^3*e*(d + e*x)
^2*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^4*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (64*
x)/(715*d^8*(d^2 - e^2*x^2)^(3/2)) + (128*x)/(715*d^10*Sqrt[d^2 - e^2*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.217793, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{128 x}{715 d^{10} \sqrt{d^2-e^2 x^2}}+\frac{64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(48*x)/(715*d^6*(d^2 - e^2*x^2)^(5/2)) - 1/(13*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(
5/2)) - 9/(143*d^2*e*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^3*e*(d + e*x)
^2*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^4*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (64*
x)/(715*d^8*(d^2 - e^2*x^2)^(3/2)) + (128*x)/(715*d^10*Sqrt[d^2 - e^2*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.2195, size = 177, normalized size = 0.86 \[ - \frac{1}{13 d e \left (d + e x\right )^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{9}{143 d^{2} e \left (d + e x\right )^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{8}{143 d^{3} e \left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{8}{143 d^{4} e \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{48 x}{715 d^{6} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{64 x}{715 d^{8} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{128 x}{715 d^{10} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-1/(13*d*e*(d + e*x)**4*(d**2 - e**2*x**2)**(5/2)) - 9/(143*d**2*e*(d + e*x)**3*
(d**2 - e**2*x**2)**(5/2)) - 8/(143*d**3*e*(d + e*x)**2*(d**2 - e**2*x**2)**(5/2
)) - 8/(143*d**4*e*(d + e*x)*(d**2 - e**2*x**2)**(5/2)) + 48*x/(715*d**6*(d**2 -
e**2*x**2)**(5/2)) + 64*x/(715*d**8*(d**2 - e**2*x**2)**(3/2)) + 128*x/(715*d**
10*sqrt(d**2 - e**2*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0992181, size = 137, normalized size = 0.67 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-180 d^9-5 d^8 e x+800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7+512 d e^8 x^8+128 e^9 x^9\right )}{715 d^{10} e (d-e x)^3 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-180*d^9 - 5*d^8*e*x + 800*d^7*e^2*x^2 + 1080*d^6*e^3*x^3
- 320*d^5*e^4*x^4 - 1552*d^4*e^5*x^5 - 768*d^3*e^6*x^6 + 448*d^2*e^7*x^7 + 512*d
*e^8*x^8 + 128*e^9*x^9))/(715*d^10*e*(d - e*x)^3*(d + e*x)^7)

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 132, normalized size = 0.6 \[ -{\frac{ \left ( -ex+d \right ) \left ( -128\,{e}^{9}{x}^{9}-512\,{e}^{8}{x}^{8}d-448\,{e}^{7}{x}^{7}{d}^{2}+768\,{e}^{6}{x}^{6}{d}^{3}+1552\,{e}^{5}{x}^{5}{d}^{4}+320\,{e}^{4}{x}^{4}{d}^{5}-1080\,{e}^{3}{x}^{3}{d}^{6}-800\,{e}^{2}{x}^{2}{d}^{7}+5\,x{d}^{8}e+180\,{d}^{9} \right ) }{715\,e{d}^{10} \left ( ex+d \right ) ^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/715*(-e*x+d)*(-128*e^9*x^9-512*d*e^8*x^8-448*d^2*e^7*x^7+768*d^3*e^6*x^6+1552
*d^4*e^5*x^5+320*d^5*e^4*x^4-1080*d^6*e^3*x^3-800*d^7*e^2*x^2+5*d^8*e*x+180*d^9)
/(e*x+d)^3/d^10/e/(-e^2*x^2+d^2)^(7/2)

_______________________________________________________________________________________

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(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.518105, size = 1045, normalized size = 5.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4),x, algorithm="fricas")

[Out]

-1/715*(180*e^17*x^18 - 432*d*e^16*x^17 - 11268*d^2*e^15*x^16 - 18912*d^3*e^14*x
^15 + 87432*d^4*e^13*x^14 + 242832*d^5*e^12*x^13 - 158184*d^6*e^11*x^12 - 982488
*d^7*e^10*x^11 - 320892*d^8*e^9*x^10 + 1796509*d^9*e^8*x^9 + 1555840*d^10*e^7*x^
8 - 1470040*d^11*e^6*x^7 - 2251392*d^12*e^5*x^6 + 203632*d^13*e^4*x^5 + 1464320*
d^14*e^3*x^4 + 411840*d^15*e^2*x^3 - 366080*d^16*e*x^2 - 183040*d^17*x + (128*e^
16*x^17 + 2132*d*e^15*x^16 + 1808*d^2*e^14*x^15 - 36368*d^3*e^13*x^14 - 81632*d^
4*e^12*x^13 + 128440*d^5*e^11*x^12 + 504296*d^6*e^10*x^11 + 29744*d^7*e^9*x^10 -
1216501*d^8*e^8*x^9 - 864864*d^9*e^7*x^8 + 1270984*d^10*e^6*x^7 + 1656512*d^11*
e^5*x^6 - 340912*d^12*e^4*x^5 - 1281280*d^13*e^3*x^4 - 320320*d^14*e^2*x^3 + 366
080*d^15*e*x^2 + 183040*d^16*x)*sqrt(-e^2*x^2 + d^2))/(d^10*e^18*x^18 + 4*d^11*e
^17*x^17 - 37*d^12*e^16*x^16 - 168*d^13*e^15*x^15 + 106*d^14*e^14*x^14 + 1280*d^
15*e^13*x^13 + 846*d^16*e^12*x^12 - 3704*d^17*e^11*x^11 - 5011*d^18*e^10*x^10 +
4284*d^19*e^9*x^9 + 10519*d^20*e^8*x^8 + 32*d^21*e^7*x^7 - 10536*d^22*e^6*x^6 -
4544*d^23*e^5*x^5 + 4688*d^24*e^4*x^4 + 3840*d^25*e^3*x^3 - 320*d^26*e^2*x^2 - 1
024*d^27*e*x - 256*d^28 + (9*d^11*e^16*x^16 + 36*d^12*e^15*x^15 - 84*d^13*e^14*x
^14 - 516*d^14*e^13*x^13 - 138*d^15*e^12*x^12 + 2172*d^16*e^11*x^11 + 2388*d^17*
e^10*x^10 - 3516*d^18*e^9*x^9 - 6839*d^19*e^8*x^8 + 1120*d^20*e^7*x^7 + 8392*d^2
1*e^6*x^6 + 3008*d^22*e^5*x^5 - 4432*d^23*e^4*x^4 - 3328*d^24*e^3*x^3 + 448*d^25
*e^2*x^2 + 1024*d^26*e*x + 256*d^27)*sqrt(-e^2*x^2 + d^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(1/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 1]