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

Optimal. Leaf size=179 \[ \frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{77 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}+\frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2} \]

[Out]

(77*d^8*x*Sqrt[d^2 - e^2*x^2])/256 + (77*d^6*x*(d^2 - e^2*x^2)^(3/2))/384 + (77*
d^4*x*(d^2 - e^2*x^2)^(5/2))/480 + (11*d^2*x*(d^2 - e^2*x^2)^(7/2))/80 - (11*d*(
d^2 - e^2*x^2)^(9/2))/(90*e) - ((d + e*x)*(d^2 - e^2*x^2)^(9/2))/(10*e) + (77*d^
10*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e)

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Rubi [A]  time = 0.183303, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{77 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}+\frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(77*d^8*x*Sqrt[d^2 - e^2*x^2])/256 + (77*d^6*x*(d^2 - e^2*x^2)^(3/2))/384 + (77*
d^4*x*(d^2 - e^2*x^2)^(5/2))/480 + (11*d^2*x*(d^2 - e^2*x^2)^(7/2))/80 - (11*d*(
d^2 - e^2*x^2)^(9/2))/(90*e) - ((d + e*x)*(d^2 - e^2*x^2)^(9/2))/(10*e) + (77*d^
10*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e)

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Rubi in Sympy [A]  time = 28.087, size = 156, normalized size = 0.87 \[ \frac{77 d^{10} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{256 e} + \frac{77 d^{8} x \sqrt{d^{2} - e^{2} x^{2}}}{256} + \frac{77 d^{6} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{384} + \frac{77 d^{4} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{480} + \frac{11 d^{2} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{80} - \frac{11 d \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{90 e} - \frac{\left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{10 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

77*d**10*atan(e*x/sqrt(d**2 - e**2*x**2))/(256*e) + 77*d**8*x*sqrt(d**2 - e**2*x
**2)/256 + 77*d**6*x*(d**2 - e**2*x**2)**(3/2)/384 + 77*d**4*x*(d**2 - e**2*x**2
)**(5/2)/480 + 11*d**2*x*(d**2 - e**2*x**2)**(7/2)/80 - 11*d*(d**2 - e**2*x**2)*
*(9/2)/(90*e) - (d + e*x)*(d**2 - e**2*x**2)**(9/2)/(10*e)

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Mathematica [A]  time = 0.189161, size = 164, normalized size = 0.92 \[ \frac{77 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}+\sqrt{d^2-e^2 x^2} \left (-\frac{2 d^9}{9 e}+\frac{179 d^8 x}{256}+\frac{8}{9} d^7 e x^2-\frac{205}{384} d^6 e^2 x^3-\frac{4}{3} d^5 e^3 x^4-\frac{13}{480} d^4 e^4 x^5+\frac{8}{9} d^3 e^5 x^6+\frac{21}{80} d^2 e^6 x^7-\frac{2}{9} d e^7 x^8-\frac{e^8 x^9}{10}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d^2 - e^2*x^2]*((-2*d^9)/(9*e) + (179*d^8*x)/256 + (8*d^7*e*x^2)/9 - (205*d
^6*e^2*x^3)/384 - (4*d^5*e^3*x^4)/3 - (13*d^4*e^4*x^5)/480 + (8*d^3*e^5*x^6)/9 +
(21*d^2*e^6*x^7)/80 - (2*d*e^7*x^8)/9 - (e^8*x^9)/10) + (77*d^10*ArcTan[(e*x)/S
qrt[d^2 - e^2*x^2]])/(256*e)

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Maple [A]  time = 0.012, size = 151, normalized size = 0.8 \[{\frac{11\,{d}^{2}x}{80} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{77\,{d}^{4}x}{480} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{77\,{d}^{6}x}{384} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{77\,{d}^{8}x}{256}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{77\,{d}^{10}}{256}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{x}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}}-{\frac{2\,d}{9\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/80*d^2*x*(-e^2*x^2+d^2)^(7/2)+77/480*d^4*x*(-e^2*x^2+d^2)^(5/2)+77/384*d^6*x*
(-e^2*x^2+d^2)^(3/2)+77/256*d^8*x*(-e^2*x^2+d^2)^(1/2)+77/256*d^10/(e^2)^(1/2)*a
rctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/10*x*(-e^2*x^2+d^2)^(9/2)-2/9*d*(-e^
2*x^2+d^2)^(9/2)/e

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Maxima [A]  time = 0.800054, size = 193, normalized size = 1.08 \[ \frac{77 \, d^{10} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{256 \, \sqrt{e^{2}}} + \frac{77}{256} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{8} x + \frac{77}{384} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{6} x + \frac{77}{480} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{4} x + \frac{11}{80} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2} x - \frac{1}{10} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} x - \frac{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} d}{9 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

77/256*d^10*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) + 77/256*sqrt(-e^2*x^2 + d^2)*
d^8*x + 77/384*(-e^2*x^2 + d^2)^(3/2)*d^6*x + 77/480*(-e^2*x^2 + d^2)^(5/2)*d^4*
x + 11/80*(-e^2*x^2 + d^2)^(7/2)*d^2*x - 1/10*(-e^2*x^2 + d^2)^(9/2)*x - 2/9*(-e
^2*x^2 + d^2)^(9/2)*d/e

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Fricas [A]  time = 0.243538, size = 949, normalized size = 5.3 \[ \frac{11520 \, d e^{19} x^{19} + 25600 \, d^{2} e^{18} x^{18} - 226080 \, d^{3} e^{17} x^{17} - 537600 \, d^{4} e^{16} x^{16} + 1475664 \, d^{5} e^{15} x^{15} + 4024320 \, d^{6} e^{14} x^{14} - 4461300 \, d^{7} e^{13} x^{13} - 15575040 \, d^{8} e^{12} x^{12} + 6031710 \, d^{9} e^{11} x^{11} + 35842560 \, d^{10} e^{10} x^{10} + 722310 \, d^{11} e^{9} x^{9} - 51916800 \, d^{12} e^{8} x^{8} - 15104640 \, d^{13} e^{7} x^{7} + 47308800 \, d^{14} e^{6} x^{6} + 22947936 \, d^{15} e^{5} x^{5} - 25067520 \, d^{16} e^{4} x^{4} - 15521280 \, d^{17} e^{3} x^{3} + 5898240 \, d^{18} e^{2} x^{2} + 4124160 \, d^{19} e x - 6930 \,{\left (d^{10} e^{10} x^{10} - 50 \, d^{12} e^{8} x^{8} + 400 \, d^{14} e^{6} x^{6} - 1120 \, d^{16} e^{4} x^{4} + 1280 \, d^{18} e^{2} x^{2} - 512 \, d^{20} + 2 \,{\left (5 \, d^{11} e^{8} x^{8} - 80 \, d^{13} e^{6} x^{6} + 336 \, d^{15} e^{4} x^{4} - 512 \, d^{17} e^{2} x^{2} + 256 \, d^{19}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (1152 \, e^{19} x^{19} + 2560 \, d e^{18} x^{18} - 60624 \, d^{2} e^{17} x^{17} - 138240 \, d^{3} e^{16} x^{16} + 612312 \, d^{4} e^{15} x^{15} + 1551360 \, d^{5} e^{14} x^{14} - 2509290 \, d^{6} e^{13} x^{13} - 7741440 \, d^{7} e^{12} x^{12} + 4670685 \, d^{8} e^{11} x^{11} + 21404160 \, d^{9} e^{10} x^{10} - 1947234 \, d^{10} e^{9} x^{9} - 35819520 \, d^{11} e^{8} x^{8} - 8162352 \, d^{12} e^{7} x^{7} + 36986880 \, d^{13} e^{6} x^{6} + 16733856 \, d^{14} e^{5} x^{5} - 22118400 \, d^{15} e^{4} x^{4} - 13459200 \, d^{16} e^{3} x^{3} + 5898240 \, d^{17} e^{2} x^{2} + 4124160 \, d^{18} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{11520 \,{\left (e^{11} x^{10} - 50 \, d^{2} e^{9} x^{8} + 400 \, d^{4} e^{7} x^{6} - 1120 \, d^{6} e^{5} x^{4} + 1280 \, d^{8} e^{3} x^{2} - 512 \, d^{10} e + 2 \,{\left (5 \, d e^{9} x^{8} - 80 \, d^{3} e^{7} x^{6} + 336 \, d^{5} e^{5} x^{4} - 512 \, d^{7} e^{3} x^{2} + 256 \, d^{9} e\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)^(7/2)*(e*x + d)^2,x, algorithm="fricas")

[Out]

1/11520*(11520*d*e^19*x^19 + 25600*d^2*e^18*x^18 - 226080*d^3*e^17*x^17 - 537600
*d^4*e^16*x^16 + 1475664*d^5*e^15*x^15 + 4024320*d^6*e^14*x^14 - 4461300*d^7*e^1
3*x^13 - 15575040*d^8*e^12*x^12 + 6031710*d^9*e^11*x^11 + 35842560*d^10*e^10*x^1
0 + 722310*d^11*e^9*x^9 - 51916800*d^12*e^8*x^8 - 15104640*d^13*e^7*x^7 + 473088
00*d^14*e^6*x^6 + 22947936*d^15*e^5*x^5 - 25067520*d^16*e^4*x^4 - 15521280*d^17*
e^3*x^3 + 5898240*d^18*e^2*x^2 + 4124160*d^19*e*x - 6930*(d^10*e^10*x^10 - 50*d^
12*e^8*x^8 + 400*d^14*e^6*x^6 - 1120*d^16*e^4*x^4 + 1280*d^18*e^2*x^2 - 512*d^20
+ 2*(5*d^11*e^8*x^8 - 80*d^13*e^6*x^6 + 336*d^15*e^4*x^4 - 512*d^17*e^2*x^2 + 2
56*d^19)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (1152
*e^19*x^19 + 2560*d*e^18*x^18 - 60624*d^2*e^17*x^17 - 138240*d^3*e^16*x^16 + 612
312*d^4*e^15*x^15 + 1551360*d^5*e^14*x^14 - 2509290*d^6*e^13*x^13 - 7741440*d^7*
e^12*x^12 + 4670685*d^8*e^11*x^11 + 21404160*d^9*e^10*x^10 - 1947234*d^10*e^9*x^
9 - 35819520*d^11*e^8*x^8 - 8162352*d^12*e^7*x^7 + 36986880*d^13*e^6*x^6 + 16733
856*d^14*e^5*x^5 - 22118400*d^15*e^4*x^4 - 13459200*d^16*e^3*x^3 + 5898240*d^17*
e^2*x^2 + 4124160*d^18*e*x)*sqrt(-e^2*x^2 + d^2))/(e^11*x^10 - 50*d^2*e^9*x^8 +
400*d^4*e^7*x^6 - 1120*d^6*e^5*x^4 + 1280*d^8*e^3*x^2 - 512*d^10*e + 2*(5*d*e^9*
x^8 - 80*d^3*e^7*x^6 + 336*d^5*e^5*x^4 - 512*d^7*e^3*x^2 + 256*d^9*e)*sqrt(-e^2*
x^2 + d^2))

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Sympy [A]  time = 97.2097, size = 1413, normalized size = 7.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**8*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)) + 2*d**7*e*Piecewise(
(x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) -
2*d**6*e**2*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)) - 6*d**5*e**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)/(1
5*e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4, True)) +
6*d**3*e**5*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)) + 2*d**2*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)) - 2*d*e**7*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)) - e**8*Piecewis
e((-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/(19
20*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/(48
0*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))

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GIAC/XCAS [A]  time = 0.236549, size = 173, normalized size = 0.97 \[ \frac{77}{256} \, d^{10} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{1}{11520} \,{\left (2560 \, d^{9} e^{\left (-1\right )} -{\left (8055 \, d^{8} + 2 \,{\left (5120 \, d^{7} e -{\left (3075 \, d^{6} e^{2} + 4 \,{\left (1920 \, d^{5} e^{3} +{\left (39 \, d^{4} e^{4} - 2 \,{\left (640 \, d^{3} e^{5} +{\left (189 \, d^{2} e^{6} - 8 \,{\left (9 \, x e^{8} + 20 \, d e^{7}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

77/256*d^10*arcsin(x*e/d)*e^(-1)*sign(d) - 1/11520*(2560*d^9*e^(-1) - (8055*d^8
+ 2*(5120*d^7*e - (3075*d^6*e^2 + 4*(1920*d^5*e^3 + (39*d^4*e^4 - 2*(640*d^3*e^5
+ (189*d^2*e^6 - 8*(9*x*e^8 + 20*d*e^7)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 +
d^2)