3.79 \(\int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=204 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}+e^8 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )+\frac{125}{128} e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )-\frac{e^6 (125 d+128 e x) \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4} \]

[Out]

-(e^6*(125*d + 128*e*x)*Sqrt[d^2 - e^2*x^2])/(128*x^2) + (e^4*(125*d + 64*e*x)*(
d^2 - e^2*x^2)^(3/2))/(192*x^4) - (e^2*(125*d + 48*e*x)*(d^2 - e^2*x^2)^(5/2))/(
240*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (3*e*(d^2 - e^2*x^2)^(7/2))/(7*x^
7) - e^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (125*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2
]/d])/128

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Rubi [A]  time = 0.605544, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}+e^8 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )+\frac{125}{128} e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )-\frac{e^6 (125 d+128 e x) \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^9,x]

[Out]

-(e^6*(125*d + 128*e*x)*Sqrt[d^2 - e^2*x^2])/(128*x^2) + (e^4*(125*d + 64*e*x)*(
d^2 - e^2*x^2)^(3/2))/(192*x^4) - (e^2*(125*d + 48*e*x)*(d^2 - e^2*x^2)^(5/2))/(
240*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (3*e*(d^2 - e^2*x^2)^(7/2))/(7*x^
7) - e^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (125*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2
]/d])/128

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Rubi in Sympy [A]  time = 109.416, size = 248, normalized size = 1.22 \[ - \frac{d^{7} \sqrt{d^{2} - e^{2} x^{2}}}{8 x^{8}} - \frac{3 d^{6} e \sqrt{d^{2} - e^{2} x^{2}}}{7 x^{7}} - \frac{7 d^{5} e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{48 x^{6}} + \frac{38 d^{4} e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{35 x^{5}} + \frac{253 d^{3} e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{192 x^{4}} - \frac{58 d^{2} e^{5} \sqrt{d^{2} - e^{2} x^{2}}}{105 x^{3}} - \frac{259 d e^{6} \sqrt{d^{2} - e^{2} x^{2}}}{128 x^{2}} - e^{8} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} + \frac{125 e^{8} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{128} - \frac{116 e^{7} \sqrt{d^{2} - e^{2} x^{2}}}{105 x} \]

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**9,x)

[Out]

-d**7*sqrt(d**2 - e**2*x**2)/(8*x**8) - 3*d**6*e*sqrt(d**2 - e**2*x**2)/(7*x**7)
- 7*d**5*e**2*sqrt(d**2 - e**2*x**2)/(48*x**6) + 38*d**4*e**3*sqrt(d**2 - e**2*
x**2)/(35*x**5) + 253*d**3*e**4*sqrt(d**2 - e**2*x**2)/(192*x**4) - 58*d**2*e**5
*sqrt(d**2 - e**2*x**2)/(105*x**3) - 259*d*e**6*sqrt(d**2 - e**2*x**2)/(128*x**2
) - e**8*atan(e*x/sqrt(d**2 - e**2*x**2)) + 125*e**8*atanh(sqrt(d**2 - e**2*x**2
)/d)/128 - 116*e**7*sqrt(d**2 - e**2*x**2)/(105*x)

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Mathematica [A]  time = 0.288094, size = 158, normalized size = 0.77 \[ \frac{125}{128} e^8 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+e^8 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )-\frac{\sqrt{d^2-e^2 x^2} \left (1680 d^7+5760 d^6 e x+1960 d^5 e^2 x^2-14592 d^4 e^3 x^3-17710 d^3 e^4 x^4+7424 d^2 e^5 x^5+27195 d e^6 x^6+14848 e^7 x^7\right )}{13440 x^8}-\frac{125}{128} e^8 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^9,x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(1680*d^7 + 5760*d^6*e*x + 1960*d^5*e^2*x^2 - 14592*d^4*e^
3*x^3 - 17710*d^3*e^4*x^4 + 7424*d^2*e^5*x^5 + 27195*d*e^6*x^6 + 14848*e^7*x^7))
/(13440*x^8) - e^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - (125*e^8*Log[x])/128 + (1
25*e^8*Log[d + Sqrt[d^2 - e^2*x^2]])/128

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Maple [B]  time = 0.094, size = 402, normalized size = 2. \[ -{\frac{d}{8\,{x}^{8}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{25\,{e}^{2}}{48\,d{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{25\,{e}^{4}}{192\,{d}^{3}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{25\,{e}^{6}}{128\,{d}^{5}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{25\,{e}^{8}}{128\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{125\,{e}^{8}}{384\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{125\,{e}^{8}}{128\,d}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{125\,d{e}^{8}}{128}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{{e}^{3}}{5\,{d}^{2}{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{2\,{e}^{5}}{15\,{d}^{4}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{e}^{7}}{15\,{d}^{6}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{e}^{9}x}{15\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{e}^{9}x}{3\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{9}x}{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{{e}^{9}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{3\,e}{7\,{x}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

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^9,x)

[Out]

-1/8*d*(-e^2*x^2+d^2)^(7/2)/x^8-25/48/d*e^2/x^6*(-e^2*x^2+d^2)^(7/2)+25/192/d^3*
e^4/x^4*(-e^2*x^2+d^2)^(7/2)-25/128/d^5*e^6/x^2*(-e^2*x^2+d^2)^(7/2)-25/128/d^5*
e^8*(-e^2*x^2+d^2)^(5/2)-125/384/d^3*e^8*(-e^2*x^2+d^2)^(3/2)-125/128/d*e^8*(-e^
2*x^2+d^2)^(1/2)+125/128*d*e^8/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2
)^(1/2))/x)-1/5*e^3/d^2/x^5*(-e^2*x^2+d^2)^(7/2)+2/15*e^5/d^4/x^3*(-e^2*x^2+d^2)
^(7/2)-8/15*e^7/d^6/x*(-e^2*x^2+d^2)^(7/2)-8/15*e^9/d^6*x*(-e^2*x^2+d^2)^(5/2)-2
/3*e^9/d^4*x*(-e^2*x^2+d^2)^(3/2)-e^9/d^2*x*(-e^2*x^2+d^2)^(1/2)-e^9/(e^2)^(1/2)
*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-3/7*e*(-e^2*x^2+d^2)^(7/2)/x^7

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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^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.553355, size = 980, normalized size = 4.8 \[ \text{result too large to display} \]

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^9,x, algorithm="fricas")

[Out]

1/13440*(118784*d*e^15*x^15 + 217560*d^2*e^14*x^14 - 1247232*d^3*e^13*x^13 - 253
4840*d^4*e^12*x^12 + 3268608*d^5*e^11*x^11 + 8971200*d^6*e^10*x^10 - 1401856*d^7
*e^9*x^9 - 13678560*d^8*e^8*x^8 - 4951040*d^9*e^7*x^7 + 9533440*d^10*e^6*x^6 + 7
186432*d^11*e^5*x^5 - 2437120*d^12*e^4*x^4 - 3710976*d^13*e^3*x^3 - 286720*d^14*
e^2*x^2 + 737280*d^15*e*x + 215040*d^16 + 26880*(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^10 - 16*d^7*e^8*x^8)*sqrt(-e^2*x^2 + d^2))*arctan(-(
d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 13125*(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^10 - 16*d^7*e^8*x^8)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-
e^2*x^2 + d^2))/x) - (14848*e^15*x^15 + 27195*d*e^14*x^14 - 467712*d^2*e^13*x^13
- 887950*d^3*e^12*x^12 + 2123520*d^4*e^11*x^11 + 4919880*d^5*e^10*x^10 - 214054
4*d^6*e^9*x^9 - 9856560*d^7*e^8*x^8 - 2519040*d^8*e^7*x^7 + 8274560*d^9*e^6*x^6
+ 5607424*d^10*e^5*x^5 - 2499840*d^11*e^4*x^4 - 3342336*d^12*e^3*x^3 - 179200*d^
13*e^2*x^2 + 737280*d^14*e*x + 215040*d^15)*sqrt(-e^2*x^2 + d^2))/(e^8*x^16 - 32
*d^2*e^6*x^14 + 160*d^4*e^4*x^12 - 256*d^6*e^2*x^10 + 128*d^8*x^8 + 8*(d*e^6*x^1
4 - 10*d^3*e^4*x^12 + 24*d^5*e^2*x^10 - 16*d^7*x^8)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 89.1017, size = 1719, normalized size = 8.43 \[ \text{result too large to display} \]

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**9,x)

[Out]

d**7*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)) + 3*d**6*e*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*s
qrt(-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**5*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*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**4*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 + 15*d*e**2*x**7), True)) - 5*d**3*e**4*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*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)) + d**2*e**5*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)) + 3*d*e**6*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)) + e**7*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*aco
sh(e*x/d) - I*e**2*x/(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-
d/(x*sqrt(1 - e**2*x**2/d**2)) - e*asin(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d*
*2)), True))

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GIAC/XCAS [A]  time = 0.305713, 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^9,x, algorithm="giac")

[Out]

Done