3.82 \(\int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=254 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{19 e^9 \sqrt{d^2-e^2 x^2}}{256 d^2 x^2}+\frac{19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac{19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}+\frac{19 e^{11} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{256 d^3}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7} \]

[Out]

(-19*e^9*Sqrt[d^2 - e^2*x^2])/(256*d^2*x^2) + (19*e^7*(d^2 - e^2*x^2)^(3/2))/(38
4*d^2*x^4) - (19*e^5*(d^2 - e^2*x^2)^(5/2))/(480*d^2*x^6) - (d*(d^2 - e^2*x^2)^(
7/2))/(11*x^11) - (3*e*(d^2 - e^2*x^2)^(7/2))/(10*x^10) - (37*e^2*(d^2 - e^2*x^2
)^(7/2))/(99*d*x^9) - (19*e^3*(d^2 - e^2*x^2)^(7/2))/(80*d^2*x^8) - (74*e^4*(d^2
- e^2*x^2)^(7/2))/(693*d^3*x^7) + (19*e^11*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(256
*d^3)

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Rubi [A]  time = 0.637824, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{19 e^9 \sqrt{d^2-e^2 x^2}}{256 d^2 x^2}+\frac{19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac{19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}+\frac{19 e^{11} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{256 d^3}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^12,x]

[Out]

(-19*e^9*Sqrt[d^2 - e^2*x^2])/(256*d^2*x^2) + (19*e^7*(d^2 - e^2*x^2)^(3/2))/(38
4*d^2*x^4) - (19*e^5*(d^2 - e^2*x^2)^(5/2))/(480*d^2*x^6) - (d*(d^2 - e^2*x^2)^(
7/2))/(11*x^11) - (3*e*(d^2 - e^2*x^2)^(7/2))/(10*x^10) - (37*e^2*(d^2 - e^2*x^2
)^(7/2))/(99*d*x^9) - (19*e^3*(d^2 - e^2*x^2)^(7/2))/(80*d^2*x^8) - (74*e^4*(d^2
- e^2*x^2)^(7/2))/(693*d^3*x^7) + (19*e^11*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(256
*d^3)

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Rubi in Sympy [A]  time = 164.825, size = 309, normalized size = 1.22 \[ - \frac{d^{7} \sqrt{d^{2} - e^{2} x^{2}}}{11 x^{11}} - \frac{3 d^{6} e \sqrt{d^{2} - e^{2} x^{2}}}{10 x^{10}} - \frac{10 d^{5} e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{99 x^{9}} + \frac{53 d^{4} e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{80 x^{8}} + \frac{514 d^{3} e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{693 x^{7}} - \frac{109 d^{2} e^{5} \sqrt{d^{2} - e^{2} x^{2}}}{480 x^{6}} - \frac{164 d e^{6} \sqrt{d^{2} - e^{2} x^{2}}}{231 x^{5}} - \frac{109 e^{7} \sqrt{d^{2} - e^{2} x^{2}}}{384 x^{4}} + \frac{37 e^{8} \sqrt{d^{2} - e^{2} x^{2}}}{693 d x^{3}} + \frac{19 e^{9} \sqrt{d^{2} - e^{2} x^{2}}}{256 d^{2} x^{2}} + \frac{19 e^{11} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{256 d^{3}} + \frac{74 e^{10} \sqrt{d^{2} - e^{2} x^{2}}}{693 d^{3} x} \]

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**12,x)

[Out]

-d**7*sqrt(d**2 - e**2*x**2)/(11*x**11) - 3*d**6*e*sqrt(d**2 - e**2*x**2)/(10*x*
*10) - 10*d**5*e**2*sqrt(d**2 - e**2*x**2)/(99*x**9) + 53*d**4*e**3*sqrt(d**2 -
e**2*x**2)/(80*x**8) + 514*d**3*e**4*sqrt(d**2 - e**2*x**2)/(693*x**7) - 109*d**
2*e**5*sqrt(d**2 - e**2*x**2)/(480*x**6) - 164*d*e**6*sqrt(d**2 - e**2*x**2)/(23
1*x**5) - 109*e**7*sqrt(d**2 - e**2*x**2)/(384*x**4) + 37*e**8*sqrt(d**2 - e**2*
x**2)/(693*d*x**3) + 19*e**9*sqrt(d**2 - e**2*x**2)/(256*d**2*x**2) + 19*e**11*a
tanh(sqrt(d**2 - e**2*x**2)/d)/(256*d**3) + 74*e**10*sqrt(d**2 - e**2*x**2)/(693
*d**3*x)

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Mathematica [A]  time = 0.251686, size = 172, normalized size = 0.68 \[ \frac{65835 e^{11} x^{11} \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (-80640 d^{10}-266112 d^9 e x-89600 d^8 e^2 x^2+587664 d^7 e^3 x^3+657920 d^6 e^4 x^4-201432 d^5 e^5 x^5-629760 d^4 e^6 x^6-251790 d^3 e^7 x^7+47360 d^2 e^8 x^8+65835 d e^9 x^9+94720 e^{10} x^{10}\right )-65835 e^{11} x^{11} \log (x)}{887040 d^3 x^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^12,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-80640*d^10 - 266112*d^9*e*x - 89600*d^8*e^2*x^2 + 587664*
d^7*e^3*x^3 + 657920*d^6*e^4*x^4 - 201432*d^5*e^5*x^5 - 629760*d^4*e^6*x^6 - 251
790*d^3*e^7*x^7 + 47360*d^2*e^8*x^8 + 65835*d*e^9*x^9 + 94720*e^10*x^10) - 65835
*e^11*x^11*Log[x] + 65835*e^11*x^11*Log[d + Sqrt[d^2 - e^2*x^2]])/(887040*d^3*x^
11)

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Maple [A]  time = 0.327, size = 303, normalized size = 1.2 \[ -{\frac{d}{11\,{x}^{11}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{37\,{e}^{2}}{99\,d{x}^{9}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{74\,{e}^{4}}{693\,{d}^{3}{x}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{19\,{e}^{3}}{80\,{d}^{2}{x}^{8}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{19\,{e}^{5}}{480\,{d}^{4}{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{19\,{e}^{7}}{1920\,{d}^{6}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{19\,{e}^{9}}{1280\,{d}^{8}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{19\,{e}^{11}}{1280\,{d}^{8}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{19\,{e}^{11}}{768\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{19\,{e}^{11}}{256\,{d}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{19\,{e}^{11}}{256\,{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{3\,e}{10\,{x}^{10}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

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^12,x)

[Out]

-1/11*d*(-e^2*x^2+d^2)^(7/2)/x^11-37/99*e^2*(-e^2*x^2+d^2)^(7/2)/d/x^9-74/693*e^
4*(-e^2*x^2+d^2)^(7/2)/d^3/x^7-19/80*e^3*(-e^2*x^2+d^2)^(7/2)/d^2/x^8-19/480*e^5
/d^4/x^6*(-e^2*x^2+d^2)^(7/2)+19/1920*e^7/d^6/x^4*(-e^2*x^2+d^2)^(7/2)-19/1280*e
^9/d^8/x^2*(-e^2*x^2+d^2)^(7/2)-19/1280*e^11/d^8*(-e^2*x^2+d^2)^(5/2)-19/768*e^1
1/d^6*(-e^2*x^2+d^2)^(3/2)-19/256*e^11/d^4*(-e^2*x^2+d^2)^(1/2)+19/256*e^11/d^2/
(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-3/10*e*(-e^2*x^2+d^
2)^(7/2)/x^10

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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^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.13114, size = 1068, normalized size = 4.2 \[ \text{result too large to display} \]

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^12,x, algorithm="fricas")

[Out]

1/887040*(94720*e^22*x^22 + 65835*d*e^21*x^21 - 5730560*d^2*e^20*x^20 - 4267725*
d^3*e^19*x^19 + 55207680*d^4*e^18*x^18 + 55975458*d^5*e^17*x^17 - 154344960*d^6*
e^16*x^16 - 298078704*d^7*e^15*x^15 - 154091520*d^8*e^14*x^14 + 700868784*d^9*e^
13*x^13 + 1773249280*d^10*e^12*x^12 - 396078144*d^11*e^11*x^11 - 4238178560*d^12
*e^10*x^10 - 1466868480*d^13*e^9*x^9 + 4999920640*d^14*e^8*x^8 + 3445499904*d^15
*e^7*x^7 - 3011768320*d^16*e^6*x^6 - 3252006912*d^17*e^5*x^5 + 641597440*d^18*e^
4*x^4 + 1487388672*d^19*e^3*x^3 + 176619520*d^20*e^2*x^2 - 272498688*d^21*e*x -
82575360*d^22 - 65835*(11*d*e^21*x^21 - 220*d^3*e^19*x^19 + 1232*d^5*e^17*x^17 -
2816*d^7*e^15*x^15 + 2816*d^9*e^13*x^13 - 1024*d^11*e^11*x^11 - (e^21*x^21 - 60
*d^2*e^19*x^19 + 560*d^4*e^17*x^17 - 1792*d^6*e^15*x^15 + 2304*d^8*e^13*x^13 - 1
024*d^10*e^11*x^11)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (
1041920*d*e^20*x^20 + 724185*d^2*e^19*x^19 - 20317440*d^3*e^18*x^18 - 17253390*d
^4*e^17*x^17 + 99348480*d^5*e^16*x^16 + 134286768*d^6*e^15*x^15 - 62599680*d^7*e
^14*x^14 - 444817296*d^8*e^13*x^13 - 788226560*d^9*e^12*x^12 + 514054464*d^10*e^
11*x^11 + 2639159040*d^11*e^10*x^10 + 573323520*d^12*e^9*x^9 - 3767249920*d^13*e
^8*x^8 - 2292111360*d^14*e^7*x^7 + 2650542080*d^15*e^6*x^6 + 2610499584*d^16*e^5
*x^5 - 698941440*d^17*e^4*x^4 - 1351139328*d^18*e^3*x^3 - 135331840*d^19*e^2*x^2
+ 272498688*d^20*e*x + 82575360*d^21)*sqrt(-e^2*x^2 + d^2))/(11*d^4*e^10*x^21 -
220*d^6*e^8*x^19 + 1232*d^8*e^6*x^17 - 2816*d^10*e^4*x^15 + 2816*d^12*e^2*x^13
- 1024*d^14*x^11 - (d^3*e^10*x^21 - 60*d^5*e^8*x^19 + 560*d^7*e^6*x^17 - 1792*d^
9*e^4*x^15 + 2304*d^11*e^2*x^13 - 1024*d^13*x^11)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 146.641, size = 2397, normalized size = 9.44 \[ \text{result too large to display} \]

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**12,x)

[Out]

d**7*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(11*x**10) + e**3*sqrt(d**2/(e**2*
x**2) - 1)/(99*d**2*x**8) + 8*e**5*sqrt(d**2/(e**2*x**2) - 1)/(693*d**4*x**6) +
16*e**7*sqrt(d**2/(e**2*x**2) - 1)/(1155*d**6*x**4) + 64*e**9*sqrt(d**2/(e**2*x*
*2) - 1)/(3465*d**8*x**2) + 128*e**11*sqrt(d**2/(e**2*x**2) - 1)/(3465*d**10), A
bs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(11*x**10) + I*e**3
*sqrt(-d**2/(e**2*x**2) + 1)/(99*d**2*x**8) + 8*I*e**5*sqrt(-d**2/(e**2*x**2) +
1)/(693*d**4*x**6) + 16*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(1155*d**6*x**4) + 64
*I*e**9*sqrt(-d**2/(e**2*x**2) + 1)/(3465*d**8*x**2) + 128*I*e**11*sqrt(-d**2/(e
**2*x**2) + 1)/(3465*d**10), True)) + 3*d**6*e*Piecewise((-d**2/(10*e*x**11*sqrt
(d**2/(e**2*x**2) - 1)) + 9*e/(80*x**9*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(480*d
**2*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**5/(1920*d**4*x**5*sqrt(d**2/(e**2*x*
*2) - 1)) + 7*e**7/(768*d**6*x**3*sqrt(d**2/(e**2*x**2) - 1)) - 7*e**9/(256*d**8
*x*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**10*acosh(d/(e*x))/(256*d**9), Abs(d**2/(e*
*2*x**2)) > 1), (I*d**2/(10*e*x**11*sqrt(-d**2/(e**2*x**2) + 1)) - 9*I*e/(80*x**
9*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(480*d**2*x**7*sqrt(-d**2/(e**2*x**2) +
1)) - 7*I*e**5/(1920*d**4*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e**7/(768*d**6
*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 7*I*e**9/(256*d**8*x*sqrt(-d**2/(e**2*x**2)
+ 1)) - 7*I*e**10*asin(d/(e*x))/(256*d**9), True)) + d**5*e**2*Piecewise((-e*sq
rt(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)/(1
05*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)) - 5*d**4*e**3*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/(384*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/(19
2*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*a
sin(d/(e*x))/(128*d**7), True)) - 5*d**3*e**4*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)/(105*d**6), True))
+ d**2*e**5*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), A
bs(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)) + 3*d*e**6*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)) + e**7*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*sqrt(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*s
qrt(-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))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.31805, size = 1, normalized size = 0. \[ \mathit{Done} \]

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^12,x, algorithm="giac")

[Out]

Done