3.848 \(\int \frac{1}{(d+e x)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=238 \[ -\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{256 x}{2145 d^{11} \sqrt{d^2-e^2 x^2}}+\frac{128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(32*x)/(715*d^7*(d^2 - e^2*x^2)^(5/2)) - 1/(15*d*e*(d + e*x)^5*(d^2 - e^2*x^2)^(
5/2)) - 2/(39*d^2*e*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 6/(143*d^3*e*(d + e*x)^
3*(d^2 - e^2*x^2)^(5/2)) - 16/(429*d^4*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 16
/(429*d^5*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (128*x)/(2145*d^9*(d^2 - e^2*x^2)
^(3/2)) + (256*x)/(2145*d^11*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.312659, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{256 x}{2145 d^{11} \sqrt{d^2-e^2 x^2}}+\frac{128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^5*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(32*x)/(715*d^7*(d^2 - e^2*x^2)^(5/2)) - 1/(15*d*e*(d + e*x)^5*(d^2 - e^2*x^2)^(
5/2)) - 2/(39*d^2*e*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 6/(143*d^3*e*(d + e*x)^
3*(d^2 - e^2*x^2)^(5/2)) - 16/(429*d^4*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 16
/(429*d^5*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (128*x)/(2145*d^9*(d^2 - e^2*x^2)
^(3/2)) + (256*x)/(2145*d^11*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 36.812, size = 206, normalized size = 0.87 \[ - \frac{1}{15 d e \left (d + e x\right )^{5} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{2}{39 d^{2} e \left (d + e x\right )^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{6}{143 d^{3} e \left (d + e x\right )^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{16}{429 d^{4} e \left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{16}{429 d^{5} e \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{32 x}{715 d^{7} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{128 x}{2145 d^{9} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{256 x}{2145 d^{11} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**5/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-1/(15*d*e*(d + e*x)**5*(d**2 - e**2*x**2)**(5/2)) - 2/(39*d**2*e*(d + e*x)**4*(
d**2 - e**2*x**2)**(5/2)) - 6/(143*d**3*e*(d + e*x)**3*(d**2 - e**2*x**2)**(5/2)
) - 16/(429*d**4*e*(d + e*x)**2*(d**2 - e**2*x**2)**(5/2)) - 16/(429*d**5*e*(d +
e*x)*(d**2 - e**2*x**2)**(5/2)) + 32*x/(715*d**7*(d**2 - e**2*x**2)**(5/2)) + 1
28*x/(2145*d**9*(d**2 - e**2*x**2)**(3/2)) + 256*x/(2145*d**11*sqrt(d**2 - e**2*
x**2))

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Mathematica [A]  time = 0.135832, size = 148, normalized size = 0.62 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-503 d^{10}-370 d^9 e x+1590 d^8 e^2 x^2+3760 d^7 e^3 x^3+1520 d^6 e^4 x^4-3744 d^5 e^5 x^5-4640 d^4 e^6 x^6-640 d^3 e^7 x^7+1920 d^2 e^8 x^8+1280 d e^9 x^9+256 e^{10} x^{10}\right )}{2145 d^{11} e (d-e x)^3 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^5*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-503*d^10 - 370*d^9*e*x + 1590*d^8*e^2*x^2 + 3760*d^7*e^3*
x^3 + 1520*d^6*e^4*x^4 - 3744*d^5*e^5*x^5 - 4640*d^4*e^6*x^6 - 640*d^3*e^7*x^7 +
1920*d^2*e^8*x^8 + 1280*d*e^9*x^9 + 256*e^10*x^10))/(2145*d^11*e*(d - e*x)^3*(d
+ e*x)^8)

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Maple [A]  time = 0.016, size = 143, normalized size = 0.6 \[ -{\frac{ \left ( -ex+d \right ) \left ( -256\,{e}^{10}{x}^{10}-1280\,{e}^{9}{x}^{9}d-1920\,{e}^{8}{x}^{8}{d}^{2}+640\,{e}^{7}{x}^{7}{d}^{3}+4640\,{e}^{6}{x}^{6}{d}^{4}+3744\,{e}^{5}{x}^{5}{d}^{5}-1520\,{e}^{4}{x}^{4}{d}^{6}-3760\,{e}^{3}{x}^{3}{d}^{7}-1590\,{e}^{2}{x}^{2}{d}^{8}+370\,x{d}^{9}e+503\,{d}^{10} \right ) }{2145\,e{d}^{11} \left ( ex+d \right ) ^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/2145*(-e*x+d)*(-256*e^10*x^10-1280*d*e^9*x^9-1920*d^2*e^8*x^8+640*d^3*e^7*x^7
+4640*d^4*e^6*x^6+3744*d^5*e^5*x^5-1520*d^6*e^4*x^4-3760*d^7*e^3*x^3-1590*d^8*e^
2*x^2+370*d^9*e*x+503*d^10)/(e*x+d)^4/d^11/e/(-e^2*x^2+d^2)^(7/2)

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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(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.666315, size = 1166, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^5),x, algorithm="fricas")

[Out]

-1/2145*(256*e^19*x^20 + 6310*d*e^18*x^19 + 14270*d^2*e^17*x^18 - 109910*d^3*e^1
6*x^17 - 425790*d^4*e^15*x^16 + 204252*d^5*e^14*x^15 + 2736860*d^6*e^13*x^14 + 2
193380*d^7*e^12*x^13 - 6781190*d^8*e^11*x^12 - 11216660*d^9*e^10*x^11 + 5376371*
d^10*e^9*x^10 + 21680230*d^11*e^8*x^9 + 5954520*d^12*e^7*x^8 - 19608160*d^13*e^6
*x^7 - 14883440*d^14*e^5*x^6 + 6484192*d^15*e^4*x^5 + 10753600*d^16*e^3*x^4 + 14
64320*d^17*e^2*x^3 - 2745600*d^18*e*x^2 - 1098240*d^19*x - (503*e^18*x^19 - 45*d
*e^17*x^18 - 33926*d^2*e^16*x^17 - 103990*d^3*e^15*x^16 + 204158*d^4*e^14*x^15 +
1180526*d^5*e^13*x^14 + 473220*d^6*e^12*x^13 - 4246372*d^7*e^11*x^12 - 5573165*
d^8*e^10*x^11 + 5382091*d^9*e^9*x^10 + 14465022*d^10*e^8*x^9 + 1687400*d^11*e^7*
x^8 - 16160144*d^12*e^6*x^7 - 10536240*d^13*e^5*x^6 + 6804512*d^14*e^4*x^5 + 938
0800*d^15*e^3*x^4 + 915200*d^16*e^2*x^3 - 2745600*d^17*e*x^2 - 1098240*d^18*x)*s
qrt(-e^2*x^2 + d^2))/(10*d^12*e^19*x^19 + 50*d^13*e^18*x^18 - 90*d^14*e^17*x^17
- 850*d^15*e^16*x^16 - 668*d^16*e^15*x^15 + 4020*d^17*e^14*x^14 + 7340*d^18*e^13
*x^13 - 6020*d^19*e^12*x^12 - 23630*d^20*e^11*x^11 - 5318*d^21*e^10*x^10 + 34670
*d^22*e^9*x^9 + 27670*d^23*e^8*x^8 - 21440*d^24*e^7*x^7 - 34240*d^25*e^6*x^6 - 1
312*d^26*e^5*x^5 + 17760*d^27*e^4*x^4 + 7680*d^28*e^3*x^3 - 2560*d^29*e^2*x^2 -
2560*d^30*e*x - 512*d^31 - (d^11*e^19*x^19 + 5*d^12*e^18*x^18 - 42*d^13*e^17*x^1
7 - 250*d^14*e^16*x^16 - 14*d^15*e^15*x^15 + 1986*d^16*e^14*x^14 + 2780*d^17*e^1
3*x^13 - 4892*d^18*e^12*x^12 - 13275*d^19*e^11*x^11 + 401*d^20*e^10*x^10 + 25158
*d^21*e^9*x^9 + 16270*d^22*e^8*x^8 - 20016*d^23*e^7*x^7 - 26480*d^24*e^6*x^6 + 1
568*d^25*e^5*x^5 + 16288*d^26*e^4*x^4 + 6400*d^27*e^3*x^3 - 2816*d^28*e^2*x^2 -
2560*d^29*e*x - 512*d^30)*sqrt(-e^2*x^2 + d^2))

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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)**5/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.614621, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^5),x, algorithm="giac")

[Out]

sage0*x