3.76 \(\int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=216 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac{e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}+\frac{d^2 e^4 (52 d+25 e x) \sqrt{d^2-e^2 x^2}}{8 x}+\frac{d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}+\frac{13}{2} d^3 e^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{25}{8} d^3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

(d^2*e^4*(52*d + 25*e*x)*Sqrt[d^2 - e^2*x^2])/(8*x) + (d*e^3*(25*d - 52*e*x)*(d^
2 - e^2*x^2)^(3/2))/(24*x^2) - (e^2*(52*d + 25*e*x)*(d^2 - e^2*x^2)^(5/2))/(60*x
^3) - (d*(d^2 - e^2*x^2)^(7/2))/(5*x^5) - (3*e*(d^2 - e^2*x^2)^(7/2))/(4*x^4) +
(13*d^3*e^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 - (25*d^3*e^5*ArcTanh[Sqrt[d^2
- e^2*x^2]/d])/8

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Rubi [A]  time = 0.618965, antiderivative size = 216, 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 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac{e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}+\frac{d^2 e^4 (52 d+25 e x) \sqrt{d^2-e^2 x^2}}{8 x}+\frac{d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}+\frac{13}{2} d^3 e^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{25}{8} d^3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^6,x]

[Out]

(d^2*e^4*(52*d + 25*e*x)*Sqrt[d^2 - e^2*x^2])/(8*x) + (d*e^3*(25*d - 52*e*x)*(d^
2 - e^2*x^2)^(3/2))/(24*x^2) - (e^2*(52*d + 25*e*x)*(d^2 - e^2*x^2)^(5/2))/(60*x
^3) - (d*(d^2 - e^2*x^2)^(7/2))/(5*x^5) - (3*e*(d^2 - e^2*x^2)^(7/2))/(4*x^4) +
(13*d^3*e^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 - (25*d^3*e^5*ArcTanh[Sqrt[d^2
- e^2*x^2]/d])/8

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Rubi in Sympy [A]  time = 90.7145, size = 245, normalized size = 1.13 \[ - \frac{d^{7} \sqrt{d^{2} - e^{2} x^{2}}}{5 x^{5}} - \frac{3 d^{6} e \sqrt{d^{2} - e^{2} x^{2}}}{4 x^{4}} - \frac{4 d^{5} e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 x^{3}} + \frac{23 d^{4} e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{8 x^{2}} + \frac{13 d^{3} e^{5} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2} - \frac{25 d^{3} e^{5} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8} + \frac{82 d^{3} e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{15 x} + d^{2} e^{5} \sqrt{d^{2} - e^{2} x^{2}} + \frac{3 d e^{6} x \sqrt{d^{2} - e^{2} x^{2}}}{2} - \frac{e^{5} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3} \]

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**6,x)

[Out]

-d**7*sqrt(d**2 - e**2*x**2)/(5*x**5) - 3*d**6*e*sqrt(d**2 - e**2*x**2)/(4*x**4)
- 4*d**5*e**2*sqrt(d**2 - e**2*x**2)/(15*x**3) + 23*d**4*e**3*sqrt(d**2 - e**2*
x**2)/(8*x**2) + 13*d**3*e**5*atan(e*x/sqrt(d**2 - e**2*x**2))/2 - 25*d**3*e**5*
atanh(sqrt(d**2 - e**2*x**2)/d)/8 + 82*d**3*e**4*sqrt(d**2 - e**2*x**2)/(15*x) +
d**2*e**5*sqrt(d**2 - e**2*x**2) + 3*d*e**6*x*sqrt(d**2 - e**2*x**2)/2 - e**5*(
d**2 - e**2*x**2)**(3/2)/3

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Mathematica [A]  time = 0.486642, size = 167, normalized size = 0.77 \[ \frac{1}{8} \left (25 d^3 e^5 \log (x)-25 d^3 e^5 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+52 d^3 e^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (-24 d^7-90 d^6 e x-32 d^5 e^2 x^2+345 d^4 e^3 x^3+656 d^3 e^4 x^4+80 d^2 e^5 x^5+180 d e^6 x^6+40 e^7 x^7\right )}{15 x^5}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^6,x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(-24*d^7 - 90*d^6*e*x - 32*d^5*e^2*x^2 + 345*d^4*e^3*x^3 +
656*d^3*e^4*x^4 + 80*d^2*e^5*x^5 + 180*d*e^6*x^6 + 40*e^7*x^7))/(15*x^5) + 52*d
^3*e^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + 25*d^3*e^5*Log[x] - 25*d^3*e^5*Log[d
+ Sqrt[d^2 - e^2*x^2]])/8

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Maple [A]  time = 0.035, size = 327, normalized size = 1.5 \[ -{\frac{d}{5\,{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{13\,{e}^{2}}{15\,d{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{52\,{e}^{4}}{15\,{d}^{3}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{52\,{e}^{6}x}{15\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{13\,{e}^{6}x}{3\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{13\,d{e}^{6}x}{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{13\,{d}^{3}{e}^{6}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{5\,{e}^{3}}{8\,{d}^{2}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{5\,{e}^{5}}{8\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{25\,{e}^{5}}{24} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{25\,{e}^{5}{d}^{2}}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{25\,{e}^{5}{d}^{4}}{8}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{3\,e}{4\,{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

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^6,x)

[Out]

-1/5*d*(-e^2*x^2+d^2)^(7/2)/x^5-13/15/d*e^2/x^3*(-e^2*x^2+d^2)^(7/2)+52/15/d^3*e
^4/x*(-e^2*x^2+d^2)^(7/2)+52/15/d^3*e^6*x*(-e^2*x^2+d^2)^(5/2)+13/3/d*e^6*x*(-e^
2*x^2+d^2)^(3/2)+13/2*d*e^6*x*(-e^2*x^2+d^2)^(1/2)+13/2*d^3*e^6/(e^2)^(1/2)*arct
an((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+5/8*e^3/d^2/x^2*(-e^2*x^2+d^2)^(7/2)+5/8*
e^5/d^2*(-e^2*x^2+d^2)^(5/2)+25/24*e^5*(-e^2*x^2+d^2)^(3/2)+25/8*e^5*d^2*(-e^2*x
^2+d^2)^(1/2)-25/8*e^5*d^4/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1
/2))/x)-3/4*e*(-e^2*x^2+d^2)^(7/2)/x^4

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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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303481, size = 994, normalized size = 4.6 \[ -\frac{320 \, d e^{15} x^{15} + 1440 \, d^{2} e^{14} x^{14} - 2960 \, d^{3} e^{13} x^{13} - 10592 \, d^{4} e^{12} x^{12} + 9160 \, d^{5} e^{11} x^{11} - 9024 \, d^{6} e^{10} x^{10} - 34920 \, d^{7} e^{9} x^{9} + 123456 \, d^{8} e^{8} x^{8} + 101760 \, d^{9} e^{7} x^{7} - 193472 \, d^{10} e^{6} x^{6} - 134880 \, d^{11} e^{5} x^{5} + 87680 \, d^{12} e^{4} x^{4} + 72960 \, d^{13} e^{3} x^{3} + 3584 \, d^{14} e^{2} x^{2} - 11520 \, d^{15} e x - 3072 \, d^{16} + 1560 \,{\left (d^{3} e^{13} x^{13} - 32 \, d^{5} e^{11} x^{11} + 160 \, d^{7} e^{9} x^{9} - 256 \, d^{9} e^{7} x^{7} + 128 \, d^{11} e^{5} x^{5} + 8 \,{\left (d^{4} e^{11} x^{11} - 10 \, d^{6} e^{9} x^{9} + 24 \, d^{8} e^{7} x^{7} - 16 \, d^{10} e^{5} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 375 \,{\left (d^{3} e^{13} x^{13} - 32 \, d^{5} e^{11} x^{11} + 160 \, d^{7} e^{9} x^{9} - 256 \, d^{9} e^{7} x^{7} + 128 \, d^{11} e^{5} x^{5} + 8 \,{\left (d^{4} e^{11} x^{11} - 10 \, d^{6} e^{9} x^{9} + 24 \, d^{8} e^{7} x^{7} - 16 \, d^{10} e^{5} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (40 \, e^{15} x^{15} + 180 \, d e^{14} x^{14} - 1200 \, d^{2} e^{13} x^{13} - 5104 \, d^{3} e^{12} x^{12} + 4825 \, d^{4} e^{11} x^{11} + 7776 \, d^{5} e^{10} x^{10} - 14970 \, d^{6} e^{9} x^{9} + 59880 \, d^{7} e^{8} x^{8} + 58080 \, d^{8} e^{7} x^{7} - 149248 \, d^{9} e^{6} x^{6} - 102720 \, d^{10} e^{5} x^{5} + 88320 \, d^{11} e^{4} x^{4} + 67200 \, d^{12} e^{3} x^{3} + 2048 \, d^{13} e^{2} x^{2} - 11520 \, d^{14} e x - 3072 \, d^{15}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{120 \,{\left (e^{8} x^{13} - 32 \, d^{2} e^{6} x^{11} + 160 \, d^{4} e^{4} x^{9} - 256 \, d^{6} e^{2} x^{7} + 128 \, d^{8} x^{5} + 8 \,{\left (d e^{6} x^{11} - 10 \, d^{3} e^{4} x^{9} + 24 \, d^{5} e^{2} x^{7} - 16 \, d^{7} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

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^6,x, algorithm="fricas")

[Out]

-1/120*(320*d*e^15*x^15 + 1440*d^2*e^14*x^14 - 2960*d^3*e^13*x^13 - 10592*d^4*e^
12*x^12 + 9160*d^5*e^11*x^11 - 9024*d^6*e^10*x^10 - 34920*d^7*e^9*x^9 + 123456*d
^8*e^8*x^8 + 101760*d^9*e^7*x^7 - 193472*d^10*e^6*x^6 - 134880*d^11*e^5*x^5 + 87
680*d^12*e^4*x^4 + 72960*d^13*e^3*x^3 + 3584*d^14*e^2*x^2 - 11520*d^15*e*x - 307
2*d^16 + 1560*(d^3*e^13*x^13 - 32*d^5*e^11*x^11 + 160*d^7*e^9*x^9 - 256*d^9*e^7*
x^7 + 128*d^11*e^5*x^5 + 8*(d^4*e^11*x^11 - 10*d^6*e^9*x^9 + 24*d^8*e^7*x^7 - 16
*d^10*e^5*x^5)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) -
375*(d^3*e^13*x^13 - 32*d^5*e^11*x^11 + 160*d^7*e^9*x^9 - 256*d^9*e^7*x^7 + 128
*d^11*e^5*x^5 + 8*(d^4*e^11*x^11 - 10*d^6*e^9*x^9 + 24*d^8*e^7*x^7 - 16*d^10*e^5
*x^5)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (40*e^15*x^15 +
180*d*e^14*x^14 - 1200*d^2*e^13*x^13 - 5104*d^3*e^12*x^12 + 4825*d^4*e^11*x^11
+ 7776*d^5*e^10*x^10 - 14970*d^6*e^9*x^9 + 59880*d^7*e^8*x^8 + 58080*d^8*e^7*x^7
- 149248*d^9*e^6*x^6 - 102720*d^10*e^5*x^5 + 88320*d^11*e^4*x^4 + 67200*d^12*e^
3*x^3 + 2048*d^13*e^2*x^2 - 11520*d^14*e*x - 3072*d^15)*sqrt(-e^2*x^2 + d^2))/(e
^8*x^13 - 32*d^2*e^6*x^11 + 160*d^4*e^4*x^9 - 256*d^6*e^2*x^7 + 128*d^8*x^5 + 8*
(d*e^6*x^11 - 10*d^3*e^4*x^9 + 24*d^5*e^2*x^7 - 16*d^7*x^5)*sqrt(-e^2*x^2 + d^2)
)

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Sympy [A]  time = 43.5764, size = 1178, normalized size = 5.45 \[ \text{result too large to display} \]

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**6,x)

[Out]

d**7*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*sq
rt(1 - e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), True)) + 3*d**6*e*Piece
wise((-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**5*e**2*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)) - 5*d**4*e**3*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)) - 5*d**3*e**4*Piecewise((I*d/(x*
sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(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)) + d**2*e**5*Piecewise((d**2/(e*
x*sqrt(d**2/(e**2*x**2) - 1)) - d*acosh(d/(e*x)) - e*x/sqrt(d**2/(e**2*x**2) - 1
), Abs(d**2/(e**2*x**2)) > 1), (-I*d**2/(e*x*sqrt(-d**2/(e**2*x**2) + 1)) + I*d*
asin(d/(e*x)) + I*e*x/sqrt(-d**2/(e**2*x**2) + 1), True)) + 3*d*e**6*Piecewise((
-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**3/(
2*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**2*asin(e*x/d)/(2*e
) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True)) + e**7*Piecewise((x**2*sqrt(d**2)/2,
Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True))

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GIAC/XCAS [A]  time = 0.288121, 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^6,x, algorithm="giac")

[Out]

Done