3.14 \(\int \frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=172 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}+\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}-\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3} \]

[Out]

(e^5*Sqrt[d^2 - e^2*x^2])/(16*d^2*x^2) - (e^3*(d^2 - e^2*x^2)^(3/2))/(24*d^2*x^4
) - (d^2 - e^2*x^2)^(5/2)/(7*d*x^7) - (e*(d^2 - e^2*x^2)^(5/2))/(6*d^2*x^6) - (2
*e^2*(d^2 - e^2*x^2)^(5/2))/(35*d^3*x^5) - (e^7*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/
(16*d^3)

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Rubi [A]  time = 0.344622, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}+\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}-\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^8,x]

[Out]

(e^5*Sqrt[d^2 - e^2*x^2])/(16*d^2*x^2) - (e^3*(d^2 - e^2*x^2)^(3/2))/(24*d^2*x^4
) - (d^2 - e^2*x^2)^(5/2)/(7*d*x^7) - (e*(d^2 - e^2*x^2)^(5/2))/(6*d^2*x^6) - (2
*e^2*(d^2 - e^2*x^2)^(5/2))/(35*d^3*x^5) - (e^7*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/
(16*d^3)

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Rubi in Sympy [A]  time = 45.3999, size = 146, normalized size = 0.85 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{7 d x^{7}} + \frac{e^{5} \sqrt{d^{2} - e^{2} x^{2}}}{16 d^{2} x^{2}} - \frac{e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{24 d^{2} x^{4}} - \frac{e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{6 d^{2} x^{6}} - \frac{e^{7} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{16 d^{3}} - \frac{2 e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{35 d^{3} x^{5}} \]

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**8,x)

[Out]

-(d**2 - e**2*x**2)**(5/2)/(7*d*x**7) + e**5*sqrt(d**2 - e**2*x**2)/(16*d**2*x**
2) - e**3*(d**2 - e**2*x**2)**(3/2)/(24*d**2*x**4) - e*(d**2 - e**2*x**2)**(5/2)
/(6*d**2*x**6) - e**7*atanh(sqrt(d**2 - e**2*x**2)/d)/(16*d**3) - 2*e**2*(d**2 -
e**2*x**2)**(5/2)/(35*d**3*x**5)

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Mathematica [A]  time = 0.17817, size = 128, normalized size = 0.74 \[ -\frac{105 e^7 x^7 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (240 d^6+280 d^5 e x-384 d^4 e^2 x^2-490 d^3 e^3 x^3+48 d^2 e^4 x^4+105 d e^5 x^5+96 e^6 x^6\right )-105 e^7 x^7 \log (x)}{1680 d^3 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^8,x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(240*d^6 + 280*d^5*e*x - 384*d^4*e^2*x^2 - 490*d^3*e^3*x^3
+ 48*d^2*e^4*x^4 + 105*d*e^5*x^5 + 96*e^6*x^6) - 105*e^7*x^7*Log[x] + 105*e^7*x
^7*Log[d + Sqrt[d^2 - e^2*x^2]])/(1680*d^3*x^7)

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Maple [A]  time = 0.053, size = 211, normalized size = 1.2 \[ -{\frac{1}{7\,d{x}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{e}^{2}}{35\,{d}^{3}{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{e}{6\,{d}^{2}{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{3}}{24\,{d}^{4}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{5}}{48\,{d}^{6}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{7}}{48\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{7}}{16\,{d}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{e}^{7}}{16\,{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^8,x)

[Out]

-1/7*(-e^2*x^2+d^2)^(5/2)/d/x^7-2/35*e^2*(-e^2*x^2+d^2)^(5/2)/d^3/x^5-1/6*e*(-e^
2*x^2+d^2)^(5/2)/d^2/x^6-1/24*e^3/d^4/x^4*(-e^2*x^2+d^2)^(5/2)+1/48*e^5/d^6/x^2*
(-e^2*x^2+d^2)^(5/2)+1/48*e^7/d^6*(-e^2*x^2+d^2)^(3/2)+1/16*e^7/d^4*(-e^2*x^2+d^
2)^(1/2)-1/16*e^7/d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/
x)

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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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.37202, size = 711, normalized size = 4.13 \[ -\frac{96 \, e^{14} x^{14} + 105 \, d e^{13} x^{13} - 2352 \, d^{2} e^{12} x^{12} - 3115 \, d^{3} e^{11} x^{11} + 8400 \, d^{4} e^{10} x^{10} + 23450 \, d^{5} e^{9} x^{9} + 1008 \, d^{6} e^{8} x^{8} - 73080 \, d^{7} e^{7} x^{7} - 46704 \, d^{8} e^{6} x^{6} + 106400 \, d^{9} e^{5} x^{5} + 83328 \, d^{10} e^{4} x^{4} - 71680 \, d^{11} e^{3} x^{3} - 59136 \, d^{12} e^{2} x^{2} + 17920 \, d^{13} e x + 15360 \, d^{14} - 105 \,{\left (7 \, d e^{13} x^{13} - 56 \, d^{3} e^{11} x^{11} + 112 \, d^{5} e^{9} x^{9} - 64 \, d^{7} e^{7} x^{7} -{\left (e^{13} x^{13} - 24 \, d^{2} e^{11} x^{11} + 80 \, d^{4} e^{9} x^{9} - 64 \, d^{6} e^{7} x^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (672 \, d e^{12} x^{12} + 735 \, d^{2} e^{11} x^{11} - 5040 \, d^{3} e^{10} x^{10} - 9310 \, d^{4} e^{9} x^{9} + 5376 \, d^{5} e^{8} x^{8} + 41160 \, d^{6} e^{7} x^{7} + 22416 \, d^{7} e^{6} x^{6} - 77280 \, d^{8} e^{5} x^{5} - 59520 \, d^{9} e^{4} x^{4} + 62720 \, d^{10} e^{3} x^{3} + 51456 \, d^{11} e^{2} x^{2} - 17920 \, d^{12} e x - 15360 \, d^{13}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1680 \,{\left (7 \, d^{4} e^{6} x^{13} - 56 \, d^{6} e^{4} x^{11} + 112 \, d^{8} e^{2} x^{9} - 64 \, d^{10} x^{7} -{\left (d^{3} e^{6} x^{13} - 24 \, d^{5} e^{4} x^{11} + 80 \, d^{7} e^{2} x^{9} - 64 \, d^{9} x^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^8,x, algorithm="fricas")

[Out]

-1/1680*(96*e^14*x^14 + 105*d*e^13*x^13 - 2352*d^2*e^12*x^12 - 3115*d^3*e^11*x^1
1 + 8400*d^4*e^10*x^10 + 23450*d^5*e^9*x^9 + 1008*d^6*e^8*x^8 - 73080*d^7*e^7*x^
7 - 46704*d^8*e^6*x^6 + 106400*d^9*e^5*x^5 + 83328*d^10*e^4*x^4 - 71680*d^11*e^3
*x^3 - 59136*d^12*e^2*x^2 + 17920*d^13*e*x + 15360*d^14 - 105*(7*d*e^13*x^13 - 5
6*d^3*e^11*x^11 + 112*d^5*e^9*x^9 - 64*d^7*e^7*x^7 - (e^13*x^13 - 24*d^2*e^11*x^
11 + 80*d^4*e^9*x^9 - 64*d^6*e^7*x^7)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*
x^2 + d^2))/x) + (672*d*e^12*x^12 + 735*d^2*e^11*x^11 - 5040*d^3*e^10*x^10 - 931
0*d^4*e^9*x^9 + 5376*d^5*e^8*x^8 + 41160*d^6*e^7*x^7 + 22416*d^7*e^6*x^6 - 77280
*d^8*e^5*x^5 - 59520*d^9*e^4*x^4 + 62720*d^10*e^3*x^3 + 51456*d^11*e^2*x^2 - 179
20*d^12*e*x - 15360*d^13)*sqrt(-e^2*x^2 + d^2))/(7*d^4*e^6*x^13 - 56*d^6*e^4*x^1
1 + 112*d^8*e^2*x^9 - 64*d^10*x^7 - (d^3*e^6*x^13 - 24*d^5*e^4*x^11 + 80*d^7*e^2
*x^9 - 64*d^9*x^7)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 43.5264, size = 1037, normalized size = 6.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(-e**2*x**2+d**2)**(3/2)/x**8,x)

[Out]

d**3*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*sq
rt(-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*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)) - d*e**2*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*sqr
t(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**3*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*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8*x**3*sq
rt(-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))

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GIAC/XCAS [A]  time = 0.293522, 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)^(3/2)*(e*x + d)/x^8,x, algorithm="giac")

[Out]

Done