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

Optimal. Leaf size=172 \[ -\frac{d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac{d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac{3 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^4}-\frac{3 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^3}-\frac{d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3} \]

[Out]

(-3*d^6*x*Sqrt[d^2 - e^2*x^2])/(128*e^3) - (d^4*x*(d^2 - e^2*x^2)^(3/2))/(64*e^3
) - (d*x^2*(d^2 - e^2*x^2)^(5/2))/(7*e^2) + (x^3*(d^2 - e^2*x^2)^(5/2))/(8*e) -
(d^2*(32*d - 35*e*x)*(d^2 - e^2*x^2)^(5/2))/(560*e^4) - (3*d^8*ArcTan[(e*x)/Sqrt
[d^2 - e^2*x^2]])/(128*e^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.388477, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac{d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac{3 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^4}-\frac{3 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^3}-\frac{d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-3*d^6*x*Sqrt[d^2 - e^2*x^2])/(128*e^3) - (d^4*x*(d^2 - e^2*x^2)^(3/2))/(64*e^3
) - (d*x^2*(d^2 - e^2*x^2)^(5/2))/(7*e^2) + (x^3*(d^2 - e^2*x^2)^(5/2))/(8*e) -
(d^2*(32*d - 35*e*x)*(d^2 - e^2*x^2)^(5/2))/(560*e^4) - (3*d^8*ArcTan[(e*x)/Sqrt
[d^2 - e^2*x^2]])/(128*e^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.2577, size = 151, normalized size = 0.88 \[ - \frac{3 d^{8} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{128 e^{4}} - \frac{3 d^{6} x \sqrt{d^{2} - e^{2} x^{2}}}{128 e^{3}} - \frac{d^{4} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{64 e^{3}} - \frac{d^{2} \left (96 d - 105 e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{1680 e^{4}} - \frac{d x^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{7 e^{2}} + \frac{x^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{8 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*d**8*atan(e*x/sqrt(d**2 - e**2*x**2))/(128*e**4) - 3*d**6*x*sqrt(d**2 - e**2*
x**2)/(128*e**3) - d**4*x*(d**2 - e**2*x**2)**(3/2)/(64*e**3) - d**2*(96*d - 105
*e*x)*(d**2 - e**2*x**2)**(5/2)/(1680*e**4) - d*x**2*(d**2 - e**2*x**2)**(5/2)/(
7*e**2) + x**3*(d**2 - e**2*x**2)**(5/2)/(8*e)

_______________________________________________________________________________________

Mathematica [A]  time = 0.121789, size = 124, normalized size = 0.72 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-256 d^7+105 d^6 e x-128 d^5 e^2 x^2+70 d^4 e^3 x^3+1024 d^3 e^4 x^4-840 d^2 e^5 x^5-640 d e^6 x^6+560 e^7 x^7\right )-105 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4480 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 305, normalized size = 1.8 \[ -{\frac{x}{8\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{d}^{2}x}{16\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{15\,{d}^{4}x}{64\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{45\,{d}^{6}x}{128\,{e}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{45\,{d}^{8}}{128\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{d}{7\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{d}^{3}}{5\,{e}^{4}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{{d}^{4}x}{4\,{e}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{6}x}{8\,{e}^{3}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{3\,{d}^{8}}{8\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/e^3*x*(-e^2*x^2+d^2)^(7/2)+3/16*d^2/e^3*x*(-e^2*x^2+d^2)^(5/2)+15/64*d^4*x*
(-e^2*x^2+d^2)^(3/2)/e^3+45/128*d^6*x*(-e^2*x^2+d^2)^(1/2)/e^3+45/128*d^8/e^3/(e
^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+1/7*d/e^4*(-e^2*x^2+d^2)^(7
/2)-1/5*d^3/e^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)-1/4*d^4/e^3*(-(x+d/e)^2*e^2
+2*d*e*(x+d/e))^(3/2)*x-3/8*d^6/e^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x-3/8*d
^8/e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.295509, size = 744, normalized size = 4.33 \[ -\frac{4480 \, d e^{15} x^{15} - 5120 \, d^{2} e^{14} x^{14} - 56000 \, d^{3} e^{13} x^{13} + 64512 \, d^{4} e^{12} x^{12} + 226800 \, d^{5} e^{11} x^{11} - 265216 \, d^{6} e^{10} x^{10} - 413000 \, d^{7} e^{9} x^{9} + 492800 \, d^{8} e^{8} x^{8} + 350280 \, d^{9} e^{7} x^{7} - 430080 \, d^{10} e^{6} x^{6} - 101360 \, d^{11} e^{5} x^{5} + 143360 \, d^{12} e^{4} x^{4} - 24640 \, d^{13} e^{3} x^{3} + 13440 \, d^{15} e x - 210 \,{\left (d^{8} e^{8} x^{8} - 32 \, d^{10} e^{6} x^{6} + 160 \, d^{12} e^{4} x^{4} - 256 \, d^{14} e^{2} x^{2} + 128 \, d^{16} + 8 \,{\left (d^{9} e^{6} x^{6} - 10 \, d^{11} e^{4} x^{4} + 24 \, d^{13} e^{2} x^{2} - 16 \, d^{15}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (560 \, e^{15} x^{15} - 640 \, d e^{14} x^{14} - 18760 \, d^{2} e^{13} x^{13} + 21504 \, d^{3} e^{12} x^{12} + 116550 \, d^{4} e^{11} x^{11} - 135296 \, d^{5} e^{10} x^{10} - 279895 \, d^{6} e^{9} x^{9} + 331520 \, d^{7} e^{8} x^{8} + 294560 \, d^{8} e^{7} x^{7} - 358400 \, d^{9} e^{6} x^{6} - 108640 \, d^{10} e^{5} x^{5} + 143360 \, d^{11} e^{4} x^{4} - 17920 \, d^{12} e^{3} x^{3} + 13440 \, d^{14} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{4480 \,{\left (e^{12} x^{8} - 32 \, d^{2} e^{10} x^{6} + 160 \, d^{4} e^{8} x^{4} - 256 \, d^{6} e^{6} x^{2} + 128 \, d^{8} e^{4} + 8 \,{\left (d e^{10} x^{6} - 10 \, d^{3} e^{8} x^{4} + 24 \, d^{5} e^{6} x^{2} - 16 \, d^{7} e^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*x^3/(e*x + d),x, algorithm="fricas")

[Out]

-1/4480*(4480*d*e^15*x^15 - 5120*d^2*e^14*x^14 - 56000*d^3*e^13*x^13 + 64512*d^4
*e^12*x^12 + 226800*d^5*e^11*x^11 - 265216*d^6*e^10*x^10 - 413000*d^7*e^9*x^9 +
492800*d^8*e^8*x^8 + 350280*d^9*e^7*x^7 - 430080*d^10*e^6*x^6 - 101360*d^11*e^5*
x^5 + 143360*d^12*e^4*x^4 - 24640*d^13*e^3*x^3 + 13440*d^15*e*x - 210*(d^8*e^8*x
^8 - 32*d^10*e^6*x^6 + 160*d^12*e^4*x^4 - 256*d^14*e^2*x^2 + 128*d^16 + 8*(d^9*e
^6*x^6 - 10*d^11*e^4*x^4 + 24*d^13*e^2*x^2 - 16*d^15)*sqrt(-e^2*x^2 + d^2))*arct
an(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (560*e^15*x^15 - 640*d*e^14*x^14 - 18760
*d^2*e^13*x^13 + 21504*d^3*e^12*x^12 + 116550*d^4*e^11*x^11 - 135296*d^5*e^10*x^
10 - 279895*d^6*e^9*x^9 + 331520*d^7*e^8*x^8 + 294560*d^8*e^7*x^7 - 358400*d^9*e
^6*x^6 - 108640*d^10*e^5*x^5 + 143360*d^11*e^4*x^4 - 17920*d^12*e^3*x^3 + 13440*
d^14*e*x)*sqrt(-e^2*x^2 + d^2))/(e^12*x^8 - 32*d^2*e^10*x^6 + 160*d^4*e^8*x^4 -
256*d^6*e^6*x^2 + 128*d^8*e^4 + 8*(d*e^10*x^6 - 10*d^3*e^8*x^4 + 24*d^5*e^6*x^2
- 16*d^7*e^4)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

Sympy [A]  time = 82.4995, size = 775, normalized size = 4.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**2*x**2*sqrt(d**2 -
e**2*x**2)/(15*e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**
2)/4, True)) - d**2*e*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e
**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**3/(48*e**2*sqrt(-1 + e**2*x**2/d**2))
- 5*I*d*x**5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1 + e**2*x
**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e
**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5
*d*x**5/(24*sqrt(1 - e**2*x**2/d**2)) - e**2*x**7/(6*d*sqrt(1 - e**2*x**2/d**2))
, True)) - d*e**2*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*
x**2*sqrt(d**2 - e**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e*
*2) + x**6*sqrt(d**2 - e**2*x**2)/7, Ne(e, 0)), (x**6*sqrt(d**2)/6, True)) + e**
3*Piecewise((-5*I*d**8*acosh(e*x/d)/(128*e**7) + 5*I*d**7*x/(128*e**6*sqrt(-1 +
e**2*x**2/d**2)) - 5*I*d**5*x**3/(384*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x
**5/(192*e**2*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d*x**7/(48*sqrt(-1 + e**2*x**2/d*
*2)) + I*e**2*x**9/(8*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (5
*d**8*asin(e*x/d)/(128*e**7) - 5*d**7*x/(128*e**6*sqrt(1 - e**2*x**2/d**2)) + 5*
d**5*x**3/(384*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**5/(192*e**2*sqrt(1 - e**
2*x**2/d**2)) + 7*d*x**7/(48*sqrt(1 - e**2*x**2/d**2)) - e**2*x**9/(8*d*sqrt(1 -
e**2*x**2/d**2)), True))

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*x^3/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError