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

Optimal. Leaf size=223 \[ -\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}+\frac{19 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e^3}+\frac{19 d^9 x \sqrt{d^2-e^2 x^2}}{256 e^2}+\frac{19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac{19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac{d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3} \]

[Out]

(19*d^9*x*Sqrt[d^2 - e^2*x^2])/(256*e^2) + (19*d^7*x*(d^2 - e^2*x^2)^(3/2))/(384
*e^2) + (19*d^5*x*(d^2 - e^2*x^2)^(5/2))/(480*e^2) - (37*d^2*x^2*(d^2 - e^2*x^2)
^(7/2))/(99*e) - (3*d*x^3*(d^2 - e^2*x^2)^(7/2))/10 - (e*x^4*(d^2 - e^2*x^2)^(7/
2))/11 - (d^3*(5920*d + 13167*e*x)*(d^2 - e^2*x^2)^(7/2))/(55440*e^3) + (19*d^11
*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e^3)

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Rubi [A]  time = 0.53622, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}+\frac{19 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e^3}+\frac{19 d^9 x \sqrt{d^2-e^2 x^2}}{256 e^2}+\frac{19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac{19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac{d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(19*d^9*x*Sqrt[d^2 - e^2*x^2])/(256*e^2) + (19*d^7*x*(d^2 - e^2*x^2)^(3/2))/(384
*e^2) + (19*d^5*x*(d^2 - e^2*x^2)^(5/2))/(480*e^2) - (37*d^2*x^2*(d^2 - e^2*x^2)
^(7/2))/(99*e) - (3*d*x^3*(d^2 - e^2*x^2)^(7/2))/10 - (e*x^4*(d^2 - e^2*x^2)^(7/
2))/11 - (d^3*(5920*d + 13167*e*x)*(d^2 - e^2*x^2)^(7/2))/(55440*e^3) + (19*d^11
*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e^3)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.197481, size = 177, normalized size = 0.79 \[ \frac{19 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e^3}+\sqrt{d^2-e^2 x^2} \left (-\frac{74 d^{10}}{693 e^3}-\frac{19 d^9 x}{256 e^2}-\frac{37 d^8 x^2}{693 e}+\frac{109 d^7 x^3}{384}+\frac{164}{231} d^6 e x^4+\frac{109}{480} d^5 e^2 x^5-\frac{514}{693} d^4 e^3 x^6-\frac{53}{80} d^3 e^4 x^7+\frac{10}{99} d^2 e^5 x^8+\frac{3}{10} d e^6 x^9+\frac{e^7 x^{10}}{11}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d^2 - e^2*x^2]*((-74*d^10)/(693*e^3) - (19*d^9*x)/(256*e^2) - (37*d^8*x^2)/
(693*e) + (109*d^7*x^3)/384 + (164*d^6*e*x^4)/231 + (109*d^5*e^2*x^5)/480 - (514
*d^4*e^3*x^6)/693 - (53*d^3*e^4*x^7)/80 + (10*d^2*e^5*x^8)/99 + (3*d*e^6*x^9)/10
+ (e^7*x^10)/11) + (19*d^11*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e^3)

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Maple [A]  time = 0.017, size = 216, normalized size = 1. \[ -{\frac{19\,{d}^{3}x}{80\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{19\,{d}^{5}x}{480\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{19\,{d}^{7}x}{384\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{19\,{d}^{9}x}{256\,{e}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{19\,{d}^{11}}{256\,{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{e{x}^{4}}{11} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{37\,{d}^{2}{x}^{2}}{99\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{74\,{d}^{4}}{693\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{3\,d{x}^{3}}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-19/80*d^3*x*(-e^2*x^2+d^2)^(7/2)/e^2+19/480*d^5*x*(-e^2*x^2+d^2)^(5/2)/e^2+19/3
84*d^7*x*(-e^2*x^2+d^2)^(3/2)/e^2+19/256*d^9*x*(-e^2*x^2+d^2)^(1/2)/e^2+19/256*d
^11/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/11*e*x^4*(-e^2*
x^2+d^2)^(7/2)-37/99*d^2*x^2*(-e^2*x^2+d^2)^(7/2)/e-74/693/e^3*d^4*(-e^2*x^2+d^2
)^(7/2)-3/10*d*x^3*(-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 0.798492, size = 281, normalized size = 1.26 \[ \frac{19 \, d^{11} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{256 \, \sqrt{e^{2}} e^{2}} + \frac{19 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{9} x}{256 \, e^{2}} - \frac{1}{11} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e x^{4} + \frac{19 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{7} x}{384 \, e^{2}} - \frac{3}{10} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x^{3} + \frac{19 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{5} x}{480 \, e^{2}} - \frac{37 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2} x^{2}}{99 \, e} - \frac{19 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{3} x}{80 \, e^{2}} - \frac{74 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{4}}{693 \, e^{3}} \]

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^2,x, algorithm="maxima")

[Out]

19/256*d^11*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^2) + 19/256*sqrt(-e^2*x^2 +
d^2)*d^9*x/e^2 - 1/11*(-e^2*x^2 + d^2)^(7/2)*e*x^4 + 19/384*(-e^2*x^2 + d^2)^(3
/2)*d^7*x/e^2 - 3/10*(-e^2*x^2 + d^2)^(7/2)*d*x^3 + 19/480*(-e^2*x^2 + d^2)^(5/2
)*d^5*x/e^2 - 37/99*(-e^2*x^2 + d^2)^(7/2)*d^2*x^2/e - 19/80*(-e^2*x^2 + d^2)^(7
/2)*d^3*x/e^2 - 74/693*(-e^2*x^2 + d^2)^(7/2)*d^4/e^3

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Fricas [A]  time = 0.288256, size = 1014, normalized size = 4.55 \[ \text{result too large to display} \]

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^2,x, algorithm="fricas")

[Out]

1/887040*(80640*e^22*x^22 + 266112*d*e^21*x^21 - 4829440*d^2*e^20*x^20 - 1682049
6*d^3*e^19*x^19 + 43873280*d^4*e^18*x^18 + 201038376*d^5*e^17*x^17 - 93350400*d^
6*e^16*x^16 - 1002282666*d^7*e^15*x^15 - 326810880*d^8*e^14*x^14 + 2581643295*d^
9*e^13*x^13 + 2039304960*d^10*e^12*x^12 - 3606334809*d^11*e^11*x^11 - 4416276480
*d^12*e^10*x^10 + 2420282172*d^13*e^9*x^9 + 4934307840*d^14*e^8*x^8 - 85957872*d
^15*e^7*x^7 - 2857451520*d^16*e^6*x^6 - 901350912*d^17*e^5*x^5 + 681246720*d^18*
e^4*x^4 + 476931840*d^19*e^3*x^3 - 67415040*d^21*e*x - 131670*(11*d^12*e^10*x^10
- 220*d^14*e^8*x^8 + 1232*d^16*e^6*x^6 - 2816*d^18*e^4*x^4 + 2816*d^20*e^2*x^2
- 1024*d^22 - (d^11*e^10*x^10 - 60*d^13*e^8*x^8 + 560*d^15*e^6*x^6 - 1792*d^17*e
^4*x^4 + 2304*d^19*e^2*x^2 - 1024*d^21)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(
-e^2*x^2 + d^2))/(e*x)) + 11*(80640*d*e^20*x^20 + 266112*d^2*e^19*x^19 - 1523200
*d^3*e^18*x^18 - 5909904*d^4*e^17*x^17 + 6581760*d^5*e^16*x^16 + 41759256*d^6*e^
15*x^15 + 3179520*d^7*e^14*x^14 - 137719890*d^8*e^13*x^13 - 88623360*d^9*e^12*x^
12 + 236025405*d^10*e^11*x^11 + 255252480*d^11*e^10*x^10 - 197264004*d^12*e^9*x^
9 - 341913600*d^13*e^8*x^8 + 34441008*d^14*e^7*x^7 + 228802560*d^15*e^6*x^6 + 62
560512*d^16*e^5*x^5 - 61931520*d^17*e^4*x^4 - 40293120*d^18*e^3*x^3 + 6128640*d^
20*e*x)*sqrt(-e^2*x^2 + d^2))/(11*d*e^13*x^10 - 220*d^3*e^11*x^8 + 1232*d^5*e^9*
x^6 - 2816*d^7*e^7*x^4 + 2816*d^9*e^5*x^2 - 1024*d^11*e^3 - (e^13*x^10 - 60*d^2*
e^11*x^8 + 560*d^4*e^9*x^6 - 1792*d^6*e^7*x^4 + 2304*d^8*e^5*x^2 - 1024*d^10*e^3
)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 115.11, size = 1681, normalized size = 7.54 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**7*Piecewise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 + e**2*
x**2/d**2)) - 3*I*d*x**3/(8*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**5/(4*d*sqrt(-
1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**4*asin(e*x/d)/(8*e**3) - d**
3*x/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**2)) -
e**2*x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True)) + 3*d**6*e*Piecewise((-2*d**4*s
qrt(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**5*e**2*
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)) - 5*d**4*e**3
*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)) - 5*d**3*e**4*Piecewis
e((-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)) + d**2*e**5*Piecewise((-16*d**8*sqrt(d**2 - e**2*x**2)/(315*e**8
) - 8*d**6*x**2*sqrt(d**2 - e**2*x**2)/(315*e**6) - 2*d**4*x**4*sqrt(d**2 - e**2
*x**2)/(105*e**4) - d**2*x**6*sqrt(d**2 - e**2*x**2)/(63*e**2) + x**8*sqrt(d**2
- e**2*x**2)/9, Ne(e, 0)), (x**8*sqrt(d**2)/8, True)) + 3*d*e**6*Piecewise((-7*I
*d**10*acosh(e*x/d)/(256*e**9) + 7*I*d**9*x/(256*e**8*sqrt(-1 + e**2*x**2/d**2))
- 7*I*d**7*x**3/(768*e**6*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d**5*x**5/(1920*e**4
*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**7/(480*e**2*sqrt(-1 + e**2*x**2/d**2)) -
9*I*d*x**9/(80*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**11/(10*d*sqrt(-1 + e**2*x
**2/d**2)), Abs(e**2*x**2/d**2) > 1), (7*d**10*asin(e*x/d)/(256*e**9) - 7*d**9*x
/(256*e**8*sqrt(1 - e**2*x**2/d**2)) + 7*d**7*x**3/(768*e**6*sqrt(1 - e**2*x**2/
d**2)) + 7*d**5*x**5/(1920*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**7/(480*e**2*
sqrt(1 - e**2*x**2/d**2)) + 9*d*x**9/(80*sqrt(1 - e**2*x**2/d**2)) - e**2*x**11/
(10*d*sqrt(1 - e**2*x**2/d**2)), True)) + e**7*Piecewise((-128*d**10*sqrt(d**2 -
e**2*x**2)/(3465*e**10) - 64*d**8*x**2*sqrt(d**2 - e**2*x**2)/(3465*e**8) - 16*
d**6*x**4*sqrt(d**2 - e**2*x**2)/(1155*e**6) - 8*d**4*x**6*sqrt(d**2 - e**2*x**2
)/(693*e**4) - d**2*x**8*sqrt(d**2 - e**2*x**2)/(99*e**2) + x**10*sqrt(d**2 - e*
*2*x**2)/11, Ne(e, 0)), (x**10*sqrt(d**2)/10, True))

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GIAC/XCAS [A]  time = 0.289068, size = 188, normalized size = 0.84 \[ \frac{19}{256} \, d^{11} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )}{\rm sign}\left (d\right ) - \frac{1}{887040} \,{\left (94720 \, d^{10} e^{\left (-3\right )} +{\left (65835 \, d^{9} e^{\left (-2\right )} + 2 \,{\left (23680 \, d^{8} e^{\left (-1\right )} -{\left (125895 \, d^{7} + 4 \,{\left (78720 \, d^{6} e +{\left (25179 \, d^{5} e^{2} - 2 \,{\left (41120 \, d^{4} e^{3} + 7 \,{\left (5247 \, d^{3} e^{4} - 8 \,{\left (100 \, d^{2} e^{5} + 9 \,{\left (10 \, x e^{7} + 33 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

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^2,x, algorithm="giac")

[Out]

19/256*d^11*arcsin(x*e/d)*e^(-3)*sign(d) - 1/887040*(94720*d^10*e^(-3) + (65835*
d^9*e^(-2) + 2*(23680*d^8*e^(-1) - (125895*d^7 + 4*(78720*d^6*e + (25179*d^5*e^2
- 2*(41120*d^4*e^3 + 7*(5247*d^3*e^4 - 8*(100*d^2*e^5 + 9*(10*x*e^7 + 33*d*e^6)
*x)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)