3.80 \(\int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac{55 e^9 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d}-\frac{55 e^7 \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6} \]

[Out]

(-55*e^7*Sqrt[d^2 - e^2*x^2])/(128*x^2) + (55*e^5*(d^2 - e^2*x^2)^(3/2))/(192*x^
4) - (11*e^3*(d^2 - e^2*x^2)^(5/2))/(48*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(9*x^9)
- (3*e*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (29*e^2*(d^2 - e^2*x^2)^(7/2))/(63*d*x^
7) + (55*e^9*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(128*d)

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Rubi [A]  time = 0.437093, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac{55 e^9 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d}-\frac{55 e^7 \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^10,x]

[Out]

(-55*e^7*Sqrt[d^2 - e^2*x^2])/(128*x^2) + (55*e^5*(d^2 - e^2*x^2)^(3/2))/(192*x^
4) - (11*e^3*(d^2 - e^2*x^2)^(5/2))/(48*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(9*x^9)
- (3*e*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (29*e^2*(d^2 - e^2*x^2)^(7/2))/(63*d*x^
7) + (55*e^9*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(128*d)

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Rubi in Sympy [A]  time = 122.167, size = 253, normalized size = 1.35 \[ - \frac{d^{7} \sqrt{d^{2} - e^{2} x^{2}}}{9 x^{9}} - \frac{3 d^{6} e \sqrt{d^{2} - e^{2} x^{2}}}{8 x^{8}} - \frac{8 d^{5} e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{63 x^{7}} + \frac{43 d^{4} e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{48 x^{6}} + \frac{22 d^{3} e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{21 x^{5}} - \frac{73 d^{2} e^{5} \sqrt{d^{2} - e^{2} x^{2}}}{192 x^{4}} - \frac{80 d e^{6} \sqrt{d^{2} - e^{2} x^{2}}}{63 x^{3}} - \frac{73 e^{7} \sqrt{d^{2} - e^{2} x^{2}}}{128 x^{2}} + \frac{55 e^{9} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{128 d} + \frac{29 e^{8} \sqrt{d^{2} - e^{2} x^{2}}}{63 d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**10,x)

[Out]

-d**7*sqrt(d**2 - e**2*x**2)/(9*x**9) - 3*d**6*e*sqrt(d**2 - e**2*x**2)/(8*x**8)
- 8*d**5*e**2*sqrt(d**2 - e**2*x**2)/(63*x**7) + 43*d**4*e**3*sqrt(d**2 - e**2*
x**2)/(48*x**6) + 22*d**3*e**4*sqrt(d**2 - e**2*x**2)/(21*x**5) - 73*d**2*e**5*s
qrt(d**2 - e**2*x**2)/(192*x**4) - 80*d*e**6*sqrt(d**2 - e**2*x**2)/(63*x**3) -
73*e**7*sqrt(d**2 - e**2*x**2)/(128*x**2) + 55*e**9*atanh(sqrt(d**2 - e**2*x**2)
/d)/(128*d) + 29*e**8*sqrt(d**2 - e**2*x**2)/(63*d*x)

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Mathematica [A]  time = 0.304955, size = 150, normalized size = 0.8 \[ -\frac{-3465 e^9 x^9 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (896 d^8+3024 d^7 e x+1024 d^6 e^2 x^2-7224 d^5 e^3 x^3-8448 d^4 e^4 x^4+3066 d^3 e^5 x^5+10240 d^2 e^6 x^6+4599 d e^7 x^7-3712 e^8 x^8\right )+3465 e^9 x^9 \log (x)}{8064 d x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^10,x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(896*d^8 + 3024*d^7*e*x + 1024*d^6*e^2*x^2 - 7224*d^5*e^3*
x^3 - 8448*d^4*e^4*x^4 + 3066*d^3*e^5*x^5 + 10240*d^2*e^6*x^6 + 4599*d*e^7*x^7 -
3712*e^8*x^8) + 3465*e^9*x^9*Log[x] - 3465*e^9*x^9*Log[d + Sqrt[d^2 - e^2*x^2]]
)/(8064*d*x^9)

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Maple [A]  time = 0.134, size = 250, normalized size = 1.3 \[ -{\frac{d}{9\,{x}^{9}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{29\,{e}^{2}}{63\,d{x}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{11\,{e}^{3}}{48\,{d}^{2}{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{11\,{e}^{5}}{192\,{d}^{4}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{11\,{e}^{7}}{128\,{d}^{6}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{11\,{e}^{9}}{128\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{55\,{e}^{9}}{384\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{55\,{e}^{9}}{128\,{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{55\,{e}^{9}}{128}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{3\,e}{8\,{x}^{8}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^10,x)

[Out]

-1/9*d*(-e^2*x^2+d^2)^(7/2)/x^9-29/63*e^2*(-e^2*x^2+d^2)^(7/2)/d/x^7-11/48*e^3/d
^2/x^6*(-e^2*x^2+d^2)^(7/2)+11/192*e^5/d^4/x^4*(-e^2*x^2+d^2)^(7/2)-11/128*e^7/d
^6/x^2*(-e^2*x^2+d^2)^(7/2)-11/128*e^9/d^6*(-e^2*x^2+d^2)^(5/2)-55/384*e^9/d^4*(
-e^2*x^2+d^2)^(3/2)-55/128*e^9/d^2*(-e^2*x^2+d^2)^(1/2)+55/128*e^9/(d^2)^(1/2)*l
n((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-3/8*e*(-e^2*x^2+d^2)^(7/2)/x^8

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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((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.579292, size = 887, normalized size = 4.74 \[ \frac{3712 \, e^{18} x^{18} - 4599 \, d e^{17} x^{17} - 162432 \, d^{2} e^{16} x^{16} + 185493 \, d^{3} e^{15} x^{15} + 1467648 \, d^{4} e^{14} x^{14} - 1154790 \, d^{5} e^{13} x^{13} - 5768448 \, d^{6} e^{12} x^{12} + 2006424 \, d^{7} e^{11} x^{11} + 12064896 \, d^{8} e^{10} x^{10} + 1018416 \, d^{9} e^{9} x^{9} - 14221440 \, d^{10} e^{8} x^{8} - 6797952 \, d^{11} e^{7} x^{7} + 9022464 \, d^{12} e^{6} x^{6} + 7951104 \, d^{13} e^{5} x^{5} - 2267136 \, d^{14} e^{4} x^{4} - 3978240 \, d^{15} e^{3} x^{3} - 368640 \, d^{16} e^{2} x^{2} + 774144 \, d^{17} e x + 229376 \, d^{18} - 3465 \,{\left (9 \, d e^{17} x^{17} - 120 \, d^{3} e^{15} x^{15} + 432 \, d^{5} e^{13} x^{13} - 576 \, d^{7} e^{11} x^{11} + 256 \, d^{9} e^{9} x^{9} -{\left (e^{17} x^{17} - 40 \, d^{2} e^{15} x^{15} + 240 \, d^{4} e^{13} x^{13} - 448 \, d^{6} e^{11} x^{11} + 256 \, d^{8} e^{9} x^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (33408 \, d e^{16} x^{16} - 41391 \, d^{2} e^{15} x^{15} - 537600 \, d^{3} e^{14} x^{14} + 524286 \, d^{4} e^{13} x^{13} + 2908416 \, d^{5} e^{12} x^{12} - 1553832 \, d^{6} e^{11} x^{11} - 7584768 \, d^{7} e^{10} x^{10} + 430416 \, d^{8} e^{9} x^{9} + 10612864 \, d^{9} e^{8} x^{8} + 4072320 \, d^{10} e^{7} x^{7} - 7822336 \, d^{11} e^{6} x^{6} - 6252288 \, d^{12} e^{5} x^{5} + 2365440 \, d^{13} e^{4} x^{4} + 3591168 \, d^{14} e^{3} x^{3} + 253952 \, d^{15} e^{2} x^{2} - 774144 \, d^{16} e x - 229376 \, d^{17}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{8064 \,{\left (9 \, d^{2} e^{8} x^{17} - 120 \, d^{4} e^{6} x^{15} + 432 \, d^{6} e^{4} x^{13} - 576 \, d^{8} e^{2} x^{11} + 256 \, d^{10} x^{9} -{\left (d e^{8} x^{17} - 40 \, d^{3} e^{6} x^{15} + 240 \, d^{5} e^{4} x^{13} - 448 \, d^{7} e^{2} x^{11} + 256 \, d^{9} x^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^10,x, algorithm="fricas")

[Out]

1/8064*(3712*e^18*x^18 - 4599*d*e^17*x^17 - 162432*d^2*e^16*x^16 + 185493*d^3*e^
15*x^15 + 1467648*d^4*e^14*x^14 - 1154790*d^5*e^13*x^13 - 5768448*d^6*e^12*x^12
+ 2006424*d^7*e^11*x^11 + 12064896*d^8*e^10*x^10 + 1018416*d^9*e^9*x^9 - 1422144
0*d^10*e^8*x^8 - 6797952*d^11*e^7*x^7 + 9022464*d^12*e^6*x^6 + 7951104*d^13*e^5*
x^5 - 2267136*d^14*e^4*x^4 - 3978240*d^15*e^3*x^3 - 368640*d^16*e^2*x^2 + 774144
*d^17*e*x + 229376*d^18 - 3465*(9*d*e^17*x^17 - 120*d^3*e^15*x^15 + 432*d^5*e^13
*x^13 - 576*d^7*e^11*x^11 + 256*d^9*e^9*x^9 - (e^17*x^17 - 40*d^2*e^15*x^15 + 24
0*d^4*e^13*x^13 - 448*d^6*e^11*x^11 + 256*d^8*e^9*x^9)*sqrt(-e^2*x^2 + d^2))*log
(-(d - sqrt(-e^2*x^2 + d^2))/x) + (33408*d*e^16*x^16 - 41391*d^2*e^15*x^15 - 537
600*d^3*e^14*x^14 + 524286*d^4*e^13*x^13 + 2908416*d^5*e^12*x^12 - 1553832*d^6*e
^11*x^11 - 7584768*d^7*e^10*x^10 + 430416*d^8*e^9*x^9 + 10612864*d^9*e^8*x^8 + 4
072320*d^10*e^7*x^7 - 7822336*d^11*e^6*x^6 - 6252288*d^12*e^5*x^5 + 2365440*d^13
*e^4*x^4 + 3591168*d^14*e^3*x^3 + 253952*d^15*e^2*x^2 - 774144*d^16*e*x - 229376
*d^17)*sqrt(-e^2*x^2 + d^2))/(9*d^2*e^8*x^17 - 120*d^4*e^6*x^15 + 432*d^6*e^4*x^
13 - 576*d^8*e^2*x^11 + 256*d^10*x^9 - (d*e^8*x^17 - 40*d^3*e^6*x^15 + 240*d^5*e
^4*x^13 - 448*d^7*e^2*x^11 + 256*d^9*x^9)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 90.8948, size = 1889, normalized size = 10.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**10,x)

[Out]

d**7*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(9*x**8) + e**3*sqrt(d**2/(e**2*x*
*2) - 1)/(63*d**2*x**6) + 2*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105*d**4*x**4) + 8*
e**7*sqrt(d**2/(e**2*x**2) - 1)/(315*d**6*x**2) + 16*e**9*sqrt(d**2/(e**2*x**2)
- 1)/(315*d**8), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(
9*x**8) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(63*d**2*x**6) + 2*I*e**5*sqrt(-d**
2/(e**2*x**2) + 1)/(105*d**4*x**4) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(315*d
**6*x**2) + 16*I*e**9*sqrt(-d**2/(e**2*x**2) + 1)/(315*d**8), True)) + 3*d**6*e*
Piecewise((-d**2/(8*e*x**9*sqrt(d**2/(e**2*x**2) - 1)) + 7*e/(48*x**7*sqrt(d**2/
(e**2*x**2) - 1)) + e**3/(192*d**2*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**5/(38
4*d**4*x**3*sqrt(d**2/(e**2*x**2) - 1)) - 5*e**7/(128*d**6*x*sqrt(d**2/(e**2*x**
2) - 1)) + 5*e**8*acosh(d/(e*x))/(128*d**7), Abs(d**2/(e**2*x**2)) > 1), (I*d**2
/(8*e*x**9*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e/(48*x**7*sqrt(-d**2/(e**2*x**2)
+ 1)) - I*e**3/(192*d**2*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e**5/(384*d**4*
x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 5*I*e**7/(128*d**6*x*sqrt(-d**2/(e**2*x**2)
+ 1)) - 5*I*e**8*asin(d/(e*x))/(128*d**7), True)) + d**5*e**2*Piecewise((-e*sqrt
(d**2/(e**2*x**2) - 1)/(7*x**6) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(35*d**2*x**4)
+ 4*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105*d**4*x**2) + 8*e**7*sqrt(d**2/(e**2*x*
*2) - 1)/(105*d**6), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) +
1)/(7*x**6) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(35*d**2*x**4) + 4*I*e**5*sqrt(
-d**2/(e**2*x**2) + 1)/(105*d**4*x**2) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(1
05*d**6), True)) - 5*d**4*e**3*Piecewise((-d**2/(6*e*x**7*sqrt(d**2/(e**2*x**2)
- 1)) + 5*e/(24*x**5*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(48*d**2*x**3*sqrt(d**2/
(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*acosh(d/(
e*x))/(16*d**5), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(6*e*x**7*sqrt(-d**2/(e**2*
x**2) + 1)) - 5*I*e/(24*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(48*d**2*x**3
*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-d**2/(e**2*x**2) + 1)) -
I*e**6*asin(d/(e*x))/(16*d**5), True)) - 5*d**3*e**4*Piecewise((3*I*d**3*sqrt(-
1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*I*d*e**2*x**2*sqrt(-1 + e
**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*I*e**6*x**6*sqrt(-1 + e**2*x**
2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e**4*x**4*sqrt(-1 + e**2*x**2/d*
*2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Abs(e**2*x**2/d**2) > 1), (3*d**3*sqrt(1 -
e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*d*e**2*x**2*sqrt(1 - e**2*x*
*2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*e**6*x**6*sqrt(1 - e**2*x**2/d**2)/(
-15*d**5*x**5 + 15*d**3*e**2*x**7) - e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-15*d**
3*x**5 + 15*d*e**2*x**7), True)) + d**2*e**5*Piecewise((-d**2/(4*e*x**5*sqrt(d**
2/(e**2*x**2) - 1)) + 3*e/(8*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**3/(8*d**2*x*s
qrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x))/(8*d**3), Abs(d**2/(e**2*x**2))
> 1), (I*d**2/(4*e*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8*x**3*sqrt(-d**2
/(e**2*x**2) + 1)) + I*e**3/(8*d**2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin
(d/(e*x))/(8*d**3), True)) + 3*d*e**6*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(
3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2), Abs(d**2/(e**2*x**2)) > 1),
(-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/
(3*d**2), True)) + e**7*Piecewise((-d**2/(2*e*x**3*sqrt(d**2/(e**2*x**2) - 1)) +
e/(2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**2*acosh(d/(e*x))/(2*d), Abs(d**2/(e**2*
x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(2*x) - I*e**2*asin(d/(e*x))/(2*d
), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.300282, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^10,x, algorithm="giac")

[Out]

Done