3.15 \(\int \frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=201 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{3 e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d^4}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4} \]

[Out]

(3*e^6*Sqrt[d^2 - e^2*x^2])/(128*d^3*x^2) - (e^4*(d^2 - e^2*x^2)^(3/2))/(64*d^3*
x^4) - (d^2 - e^2*x^2)^(5/2)/(8*d*x^8) - (e*(d^2 - e^2*x^2)^(5/2))/(7*d^2*x^7) -
(e^2*(d^2 - e^2*x^2)^(5/2))/(16*d^3*x^6) - (2*e^3*(d^2 - e^2*x^2)^(5/2))/(35*d^
4*x^5) - (3*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(128*d^4)

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Rubi [A]  time = 0.442919, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{3 e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d^4}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^9,x]

[Out]

(3*e^6*Sqrt[d^2 - e^2*x^2])/(128*d^3*x^2) - (e^4*(d^2 - e^2*x^2)^(3/2))/(64*d^3*
x^4) - (d^2 - e^2*x^2)^(5/2)/(8*d*x^8) - (e*(d^2 - e^2*x^2)^(5/2))/(7*d^2*x^7) -
(e^2*(d^2 - e^2*x^2)^(5/2))/(16*d^3*x^6) - (2*e^3*(d^2 - e^2*x^2)^(5/2))/(35*d^
4*x^5) - (3*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(128*d^4)

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Rubi in Sympy [A]  time = 59.2166, size = 175, normalized size = 0.87 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{8 d x^{8}} - \frac{e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{7 d^{2} x^{7}} + \frac{3 e^{6} \sqrt{d^{2} - e^{2} x^{2}}}{128 d^{3} x^{2}} - \frac{e^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{64 d^{3} x^{4}} - \frac{e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{16 d^{3} x^{6}} - \frac{3 e^{8} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{128 d^{4}} - \frac{2 e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{35 d^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(-e**2*x**2+d**2)**(3/2)/x**9,x)

[Out]

-(d**2 - e**2*x**2)**(5/2)/(8*d*x**8) - e*(d**2 - e**2*x**2)**(5/2)/(7*d**2*x**7
) + 3*e**6*sqrt(d**2 - e**2*x**2)/(128*d**3*x**2) - e**4*(d**2 - e**2*x**2)**(3/
2)/(64*d**3*x**4) - e**2*(d**2 - e**2*x**2)**(5/2)/(16*d**3*x**6) - 3*e**8*atanh
(sqrt(d**2 - e**2*x**2)/d)/(128*d**4) - 2*e**3*(d**2 - e**2*x**2)**(5/2)/(35*d**
4*x**5)

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Mathematica [A]  time = 0.20593, size = 139, normalized size = 0.69 \[ -\frac{105 e^8 x^8 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (560 d^7+640 d^6 e x-840 d^5 e^2 x^2-1024 d^4 e^3 x^3+70 d^3 e^4 x^4+128 d^2 e^5 x^5+105 d e^6 x^6+256 e^7 x^7\right )-105 e^8 x^8 \log (x)}{4480 d^4 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^9,x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(560*d^7 + 640*d^6*e*x - 840*d^5*e^2*x^2 - 1024*d^4*e^3*x^
3 + 70*d^3*e^4*x^4 + 128*d^2*e^5*x^5 + 105*d*e^6*x^6 + 256*e^7*x^7) - 105*e^8*x^
8*Log[x] + 105*e^8*x^8*Log[d + Sqrt[d^2 - e^2*x^2]])/(4480*d^4*x^8)

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Maple [A]  time = 0.075, size = 236, normalized size = 1.2 \[ -{\frac{1}{8\,d{x}^{8}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{2}}{16\,{d}^{3}{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{4}}{64\,{d}^{5}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{6}}{128\,{d}^{7}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{8}}{128\,{d}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{8}}{128\,{d}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{8}}{128\,{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{e}{7\,{d}^{2}{x}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{e}^{3}}{35\,{d}^{4}{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^9,x)

[Out]

-1/8*(-e^2*x^2+d^2)^(5/2)/d/x^8-1/16*e^2*(-e^2*x^2+d^2)^(5/2)/d^3/x^6-1/64*e^4/d
^5/x^4*(-e^2*x^2+d^2)^(5/2)+1/128*e^6/d^7/x^2*(-e^2*x^2+d^2)^(5/2)+1/128*e^8/d^7
*(-e^2*x^2+d^2)^(3/2)+3/128*e^8/d^5*(-e^2*x^2+d^2)^(1/2)-3/128*e^8/d^3/(d^2)^(1/
2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-1/7*e*(-e^2*x^2+d^2)^(5/2)/d
^2/x^7-2/35*e^3*(-e^2*x^2+d^2)^(5/2)/d^4/x^5

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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)^(3/2)*(e*x + d)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.523407, size = 799, normalized size = 3.98 \[ \frac{2048 \, d e^{15} x^{15} + 840 \, d^{2} e^{14} x^{14} - 21504 \, d^{3} e^{13} x^{13} - 8680 \, d^{4} e^{12} x^{12} + 50176 \, d^{5} e^{11} x^{11} + 15680 \, d^{6} e^{10} x^{10} + 48128 \, d^{7} e^{9} x^{9} + 63840 \, d^{8} e^{8} x^{8} - 343040 \, d^{9} e^{7} x^{7} - 286720 \, d^{10} e^{6} x^{6} + 518144 \, d^{11} e^{5} x^{5} + 430080 \, d^{12} e^{4} x^{4} - 335872 \, d^{13} e^{3} x^{3} - 286720 \, d^{14} e^{2} x^{2} + 81920 \, d^{15} e x + 71680 \, d^{16} + 105 \,{\left (e^{16} x^{16} - 32 \, d^{2} e^{14} x^{14} + 160 \, d^{4} e^{12} x^{12} - 256 \, d^{6} e^{10} x^{10} + 128 \, d^{8} e^{8} x^{8} + 8 \,{\left (d e^{14} x^{14} - 10 \, d^{3} e^{12} x^{12} + 24 \, d^{5} e^{10} x^{10} - 16 \, d^{7} e^{8} x^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (256 \, e^{15} x^{15} + 105 \, d e^{14} x^{14} - 8064 \, d^{2} e^{13} x^{13} - 3290 \, d^{3} e^{12} x^{12} + 35840 \, d^{4} e^{11} x^{11} + 13720 \, d^{5} e^{10} x^{10} - 11648 \, d^{6} e^{9} x^{9} + 11760 \, d^{7} e^{8} x^{8} - 184320 \, d^{8} e^{7} x^{7} - 156800 \, d^{9} e^{6} x^{6} + 380928 \, d^{10} e^{5} x^{5} + 313600 \, d^{11} e^{4} x^{4} - 294912 \, d^{12} e^{3} x^{3} - 250880 \, d^{13} e^{2} x^{2} + 81920 \, d^{14} e x + 71680 \, d^{15}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{4480 \,{\left (d^{4} e^{8} x^{16} - 32 \, d^{6} e^{6} x^{14} + 160 \, d^{8} e^{4} x^{12} - 256 \, d^{10} e^{2} x^{10} + 128 \, d^{12} x^{8} + 8 \,{\left (d^{5} e^{6} x^{14} - 10 \, d^{7} e^{4} x^{12} + 24 \, d^{9} e^{2} x^{10} - 16 \, d^{11} x^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^9,x, algorithm="fricas")

[Out]

1/4480*(2048*d*e^15*x^15 + 840*d^2*e^14*x^14 - 21504*d^3*e^13*x^13 - 8680*d^4*e^
12*x^12 + 50176*d^5*e^11*x^11 + 15680*d^6*e^10*x^10 + 48128*d^7*e^9*x^9 + 63840*
d^8*e^8*x^8 - 343040*d^9*e^7*x^7 - 286720*d^10*e^6*x^6 + 518144*d^11*e^5*x^5 + 4
30080*d^12*e^4*x^4 - 335872*d^13*e^3*x^3 - 286720*d^14*e^2*x^2 + 81920*d^15*e*x
+ 71680*d^16 + 105*(e^16*x^16 - 32*d^2*e^14*x^14 + 160*d^4*e^12*x^12 - 256*d^6*e
^10*x^10 + 128*d^8*e^8*x^8 + 8*(d*e^14*x^14 - 10*d^3*e^12*x^12 + 24*d^5*e^10*x^1
0 - 16*d^7*e^8*x^8)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (
256*e^15*x^15 + 105*d*e^14*x^14 - 8064*d^2*e^13*x^13 - 3290*d^3*e^12*x^12 + 3584
0*d^4*e^11*x^11 + 13720*d^5*e^10*x^10 - 11648*d^6*e^9*x^9 + 11760*d^7*e^8*x^8 -
184320*d^8*e^7*x^7 - 156800*d^9*e^6*x^6 + 380928*d^10*e^5*x^5 + 313600*d^11*e^4*
x^4 - 294912*d^12*e^3*x^3 - 250880*d^13*e^2*x^2 + 81920*d^14*e*x + 71680*d^15)*s
qrt(-e^2*x^2 + d^2))/(d^4*e^8*x^16 - 32*d^6*e^6*x^14 + 160*d^8*e^4*x^12 - 256*d^
10*e^2*x^10 + 128*d^12*x^8 + 8*(d^5*e^6*x^14 - 10*d^7*e^4*x^12 + 24*d^9*e^2*x^10
- 16*d^11*x^8)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 63.5172, size = 1159, normalized size = 5.77 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(-e**2*x**2+d**2)**(3/2)/x**9,x)

[Out]

d**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/(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**2*e*Piecewise((-e*sq
rt(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*sqr
t(-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*e**2*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*sq
rt(-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)) - e**3*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 + 1
5*d*e**2*x**7), True))

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GIAC/XCAS [A]  time = 0.298833, 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)^(3/2)*(e*x + d)/x^9,x, algorithm="giac")

[Out]

Done