3.789 \(\int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx\)

Optimal. Leaf size=212 \[ -\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{91 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}+\frac{91}{256} d^9 x \sqrt{d^2-e^2 x^2}+\frac{91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2} \]

[Out]

(91*d^9*x*Sqrt[d^2 - e^2*x^2])/256 + (91*d^7*x*(d^2 - e^2*x^2)^(3/2))/384 + (91*
d^5*x*(d^2 - e^2*x^2)^(5/2))/480 + (13*d^3*x*(d^2 - e^2*x^2)^(7/2))/80 - (13*d^2
*(d^2 - e^2*x^2)^(9/2))/(90*e) - (13*d*(d + e*x)*(d^2 - e^2*x^2)^(9/2))/(110*e)
- ((d + e*x)^2*(d^2 - e^2*x^2)^(9/2))/(11*e) + (91*d^11*ArcTan[(e*x)/Sqrt[d^2 -
e^2*x^2]])/(256*e)

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Rubi [A]  time = 0.252574, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{91 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}+\frac{91}{256} d^9 x \sqrt{d^2-e^2 x^2}+\frac{91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(d^2 - e^2*x^2)^(7/2),x]

[Out]

(91*d^9*x*Sqrt[d^2 - e^2*x^2])/256 + (91*d^7*x*(d^2 - e^2*x^2)^(3/2))/384 + (91*
d^5*x*(d^2 - e^2*x^2)^(5/2))/480 + (13*d^3*x*(d^2 - e^2*x^2)^(7/2))/80 - (13*d^2
*(d^2 - e^2*x^2)^(9/2))/(90*e) - (13*d*(d + e*x)*(d^2 - e^2*x^2)^(9/2))/(110*e)
- ((d + e*x)^2*(d^2 - e^2*x^2)^(9/2))/(11*e) + (91*d^11*ArcTan[(e*x)/Sqrt[d^2 -
e^2*x^2]])/(256*e)

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Rubi in Sympy [A]  time = 34.8527, size = 185, normalized size = 0.87 \[ \frac{91 d^{11} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{256 e} + \frac{91 d^{9} x \sqrt{d^{2} - e^{2} x^{2}}}{256} + \frac{91 d^{7} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{384} + \frac{91 d^{5} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{480} + \frac{13 d^{3} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{80} - \frac{13 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{90 e} - \frac{13 d \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{110 e} - \frac{\left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{11 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(-e**2*x**2+d**2)**(7/2),x)

[Out]

91*d**11*atan(e*x/sqrt(d**2 - e**2*x**2))/(256*e) + 91*d**9*x*sqrt(d**2 - e**2*x
**2)/256 + 91*d**7*x*(d**2 - e**2*x**2)**(3/2)/384 + 91*d**5*x*(d**2 - e**2*x**2
)**(5/2)/480 + 13*d**3*x*(d**2 - e**2*x**2)**(7/2)/80 - 13*d**2*(d**2 - e**2*x**
2)**(9/2)/(90*e) - 13*d*(d + e*x)*(d**2 - e**2*x**2)**(9/2)/(110*e) - (d + e*x)*
*2*(d**2 - e**2*x**2)**(9/2)/(11*e)

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Mathematica [A]  time = 0.219693, size = 177, normalized size = 0.83 \[ \frac{91 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}+\sqrt{d^2-e^2 x^2} \left (-\frac{35 d^{10}}{99 e}+\frac{165 d^9 x}{256}+\frac{131}{99} d^8 e x^2+\frac{37}{384} d^7 e^2 x^3-\frac{58}{33} d^6 e^3 x^4-\frac{539}{480} d^5 e^4 x^5+\frac{86}{99} d^4 e^5 x^6+\frac{83}{80} d^3 e^6 x^7+\frac{1}{99} d^2 e^7 x^8-\frac{3}{10} d e^8 x^9-\frac{e^9 x^{10}}{11}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(d^2 - e^2*x^2)^(7/2),x]

[Out]

Sqrt[d^2 - e^2*x^2]*((-35*d^10)/(99*e) + (165*d^9*x)/256 + (131*d^8*e*x^2)/99 +
(37*d^7*e^2*x^3)/384 - (58*d^6*e^3*x^4)/33 - (539*d^5*e^4*x^5)/480 + (86*d^4*e^5
*x^6)/99 + (83*d^3*e^6*x^7)/80 + (d^2*e^7*x^8)/99 - (3*d*e^8*x^9)/10 - (e^9*x^10
)/11) + (91*d^11*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e)

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Maple [A]  time = 0.033, size = 174, normalized size = 0.8 \[{\frac{13\,{d}^{3}x}{80} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{91\,{d}^{5}x}{480} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{91\,{d}^{7}x}{384} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{91\,{d}^{9}x}{256}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{91\,{d}^{11}}{256}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{e{x}^{2}}{11} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}}-{\frac{35\,{d}^{2}}{99\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}}-{\frac{3\,dx}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(-e^2*x^2+d^2)^(7/2),x)

[Out]

13/80*d^3*x*(-e^2*x^2+d^2)^(7/2)+91/480*d^5*x*(-e^2*x^2+d^2)^(5/2)+91/384*d^7*x*
(-e^2*x^2+d^2)^(3/2)+91/256*d^9*x*(-e^2*x^2+d^2)^(1/2)+91/256*d^11/(e^2)^(1/2)*a
rctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/11*e*x^2*(-e^2*x^2+d^2)^(9/2)-35/99*
d^2*(-e^2*x^2+d^2)^(9/2)/e-3/10*d*x*(-e^2*x^2+d^2)^(9/2)

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Maxima [A]  time = 0.804236, size = 224, normalized size = 1.06 \[ \frac{91 \, d^{11} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{256 \, \sqrt{e^{2}}} + \frac{91}{256} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{9} x + \frac{91}{384} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{7} x + \frac{91}{480} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{5} x + \frac{13}{80} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{3} x - \frac{1}{11} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} e x^{2} - \frac{3}{10} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} d x - \frac{35 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} d^{2}}{99 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^3,x, algorithm="maxima")

[Out]

91/256*d^11*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) + 91/256*sqrt(-e^2*x^2 + d^2)*
d^9*x + 91/384*(-e^2*x^2 + d^2)^(3/2)*d^7*x + 91/480*(-e^2*x^2 + d^2)^(5/2)*d^5*
x + 13/80*(-e^2*x^2 + d^2)^(7/2)*d^3*x - 1/11*(-e^2*x^2 + d^2)^(9/2)*e*x^2 - 3/1
0*(-e^2*x^2 + d^2)^(9/2)*d*x - 35/99*(-e^2*x^2 + d^2)^(9/2)*d^2/e

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Fricas [A]  time = 0.227832, size = 1038, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^3,x, algorithm="fricas")

[Out]

-1/126720*(11520*e^22*x^22 + 38016*d*e^21*x^21 - 704000*d^2*e^20*x^20 - 2450448*
d^3*e^19*x^19 + 7110400*d^4*e^18*x^18 + 31732008*d^5*e^17*x^17 - 20951040*d^6*e^
16*x^16 - 179618538*d^7*e^15*x^15 - 31806720*d^8*e^14*x^14 + 553822335*d^9*e^13*
x^13 + 363686400*d^10*e^12*x^12 - 1002294777*d^11*e^11*x^11 - 1064870400*d^12*e^
10*x^10 + 1037391036*d^13*e^9*x^9 + 1689600000*d^14*e^8*x^8 - 466100976*d^15*e^7
*x^7 - 1590927360*d^16*e^6*x^6 - 148194816*d^17*e^5*x^5 + 843448320*d^18*e^4*x^4
+ 259311360*d^19*e^3*x^3 - 194641920*d^20*e^2*x^2 - 83635200*d^21*e*x + 90090*(
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 + 2
816*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))*a
rctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 11*(11520*d*e^20*x^20 + 38016*d^2*e^1
9*x^19 - 231680*d^3*e^18*x^18 - 891792*d^4*e^17*x^17 + 1205760*d^5*e^16*x^16 + 7
029528*d^6*e^15*x^15 - 668160*d^7*e^14*x^14 - 27315090*d^8*e^13*x^13 - 13674240*
d^9*e^12*x^12 + 59488605*d^10*e^11*x^11 + 55119360*d^11*e^10*x^10 - 73357572*d^1
2*e^9*x^9 - 104509440*d^13*e^8*x^8 + 42644784*d^14*e^7*x^7 + 112926720*d^15*e^6*
x^6 + 4536576*d^16*e^5*x^5 - 67829760*d^17*e^4*x^4 - 19772160*d^18*e^3*x^3 + 176
94720*d^19*e^2*x^2 + 7603200*d^20*e*x)*sqrt(-e^2*x^2 + d^2))/(11*d*e^11*x^10 - 2
20*d^3*e^9*x^8 + 1232*d^5*e^7*x^6 - 2816*d^7*e^5*x^4 + 2816*d^9*e^3*x^2 - 1024*d
^11*e - (e^11*x^10 - 60*d^2*e^9*x^8 + 560*d^4*e^7*x^6 - 1792*d^6*e^5*x^4 + 2304*
d^8*e^3*x^2 - 1024*d^10*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 108.931, size = 1496, normalized size = 7.06 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(-e**2*x**2+d**2)**(7/2),x)

[Out]

d**9*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)) + 3*d**8*e*Piecewise(
(x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) -
8*d**6*e**3*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**2*x**2*sqr
t(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)) - 6*d**5*e**4*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*sqr
t(-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)) + 6*d**4*e**5*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(1
05*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)) + 8*d**3*e**6*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)) - 3*d*e**8*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*sq
rt(-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/(2
56*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*sqr
t(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**9*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 = 0.238705, size = 186, normalized size = 0.88 \[ \frac{91}{256} \, d^{11} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{1}{126720} \,{\left (44800 \, d^{10} e^{\left (-1\right )} -{\left (81675 \, d^{9} + 2 \,{\left (83840 \, d^{8} e +{\left (6105 \, d^{7} e^{2} - 4 \,{\left (27840 \, d^{6} e^{3} +{\left (17787 \, d^{5} e^{4} - 2 \,{\left (6880 \, d^{4} e^{5} +{\left (8217 \, d^{3} e^{6} + 8 \,{\left (10 \, d^{2} e^{7} - 9 \,{\left (10 \, x e^{9} + 33 \, d e^{8}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^3,x, algorithm="giac")

[Out]

91/256*d^11*arcsin(x*e/d)*e^(-1)*sign(d) - 1/126720*(44800*d^10*e^(-1) - (81675*
d^9 + 2*(83840*d^8*e + (6105*d^7*e^2 - 4*(27840*d^6*e^3 + (17787*d^5*e^4 - 2*(68
80*d^4*e^5 + (8217*d^3*e^6 + 8*(10*d^2*e^7 - 9*(10*x*e^9 + 33*d*e^8)*x)*x)*x)*x)
*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)