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

Optimal. Leaf size=188 \[ -\frac{11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{55 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e}+\frac{55}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2} \]

[Out]

(55*d^7*x*Sqrt[d^2 - e^2*x^2])/128 + (55*d^5*x*(d^2 - e^2*x^2)^(3/2))/192 + (11*
d^3*x*(d^2 - e^2*x^2)^(5/2))/48 - (11*d^2*(d^2 - e^2*x^2)^(7/2))/(56*e) - (11*d*
(d + e*x)*(d^2 - e^2*x^2)^(7/2))/(72*e) - ((d + e*x)^2*(d^2 - e^2*x^2)^(7/2))/(9
*e) + (55*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e)

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Rubi [A]  time = 0.177185, antiderivative size = 188, 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 \[ -\frac{11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{55 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e}+\frac{55}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(55*d^7*x*Sqrt[d^2 - e^2*x^2])/128 + (55*d^5*x*(d^2 - e^2*x^2)^(3/2))/192 + (11*
d^3*x*(d^2 - e^2*x^2)^(5/2))/48 - (11*d^2*(d^2 - e^2*x^2)^(7/2))/(56*e) - (11*d*
(d + e*x)*(d^2 - e^2*x^2)^(7/2))/(72*e) - ((d + e*x)^2*(d^2 - e^2*x^2)^(7/2))/(9
*e) + (55*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e)

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Rubi in Sympy [A]  time = 31.6048, size = 163, normalized size = 0.87 \[ \frac{55 d^{9} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{128 e} + \frac{55 d^{7} x \sqrt{d^{2} - e^{2} x^{2}}}{128} + \frac{55 d^{5} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{192} + \frac{11 d^{3} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{48} - \frac{11 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{56 e} - \frac{11 d \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{72 e} - \frac{\left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{9 e} \]

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)

[Out]

55*d**9*atan(e*x/sqrt(d**2 - e**2*x**2))/(128*e) + 55*d**7*x*sqrt(d**2 - e**2*x*
*2)/128 + 55*d**5*x*(d**2 - e**2*x**2)**(3/2)/192 + 11*d**3*x*(d**2 - e**2*x**2)
**(5/2)/48 - 11*d**2*(d**2 - e**2*x**2)**(7/2)/(56*e) - 11*d*(d + e*x)*(d**2 - e
**2*x**2)**(7/2)/(72*e) - (d + e*x)**2*(d**2 - e**2*x**2)**(7/2)/(9*e)

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Mathematica [A]  time = 0.136007, size = 135, normalized size = 0.72 \[ \frac{3465 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (-3712 d^8+4599 d^7 e x+10240 d^6 e^2 x^2+3066 d^5 e^3 x^3-8448 d^4 e^4 x^4-7224 d^3 e^5 x^5+1024 d^2 e^6 x^6+3024 d e^7 x^7+896 e^8 x^8\right )}{8064 e} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-3712*d^8 + 4599*d^7*e*x + 10240*d^6*e^2*x^2 + 3066*d^5*e^
3*x^3 - 8448*d^4*e^4*x^4 - 7224*d^3*e^5*x^5 + 1024*d^2*e^6*x^6 + 3024*d*e^7*x^7
+ 896*e^8*x^8) + 3465*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8064*e)

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Maple [A]  time = 0.012, size = 154, normalized size = 0.8 \[{\frac{11\,{d}^{3}x}{48} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{55\,{d}^{5}x}{192} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{55\,{d}^{7}x}{128}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{55\,{d}^{9}}{128}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{e{x}^{2}}{9} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{29\,{d}^{2}}{63\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{3\,dx}{8} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

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)

[Out]

11/48*d^3*x*(-e^2*x^2+d^2)^(5/2)+55/192*d^5*x*(-e^2*x^2+d^2)^(3/2)+55/128*d^7*x*
(-e^2*x^2+d^2)^(1/2)+55/128*d^9/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^
(1/2))-1/9*e*x^2*(-e^2*x^2+d^2)^(7/2)-29/63*d^2*(-e^2*x^2+d^2)^(7/2)/e-3/8*d*x*(
-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 0.794114, size = 197, normalized size = 1.05 \[ \frac{55 \, d^{9} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{128 \, \sqrt{e^{2}}} + \frac{55}{128} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{7} x + \frac{55}{192} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{5} x + \frac{11}{48} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{3} x - \frac{1}{9} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e x^{2} - \frac{3}{8} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x - \frac{29 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2}}{63 \, e} \]

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, algorithm="maxima")

[Out]

55/128*d^9*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) + 55/128*sqrt(-e^2*x^2 + d^2)*d
^7*x + 55/192*(-e^2*x^2 + d^2)^(3/2)*d^5*x + 11/48*(-e^2*x^2 + d^2)^(5/2)*d^3*x
- 1/9*(-e^2*x^2 + d^2)^(7/2)*e*x^2 - 3/8*(-e^2*x^2 + d^2)^(7/2)*d*x - 29/63*(-e^
2*x^2 + d^2)^(7/2)*d^2/e

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Fricas [A]  time = 0.287759, size = 860, normalized size = 4.57 \[ \frac{896 \, e^{18} x^{18} + 3024 \, d e^{17} x^{17} - 35712 \, d^{2} e^{16} x^{16} - 131208 \, d^{3} e^{15} x^{15} + 200448 \, d^{4} e^{14} x^{14} + 1145970 \, d^{5} e^{13} x^{13} + 26880 \, d^{6} e^{12} x^{12} - 4224339 \, d^{7} e^{11} x^{11} - 2862720 \, d^{8} e^{10} x^{10} + 7768929 \, d^{9} e^{9} x^{9} + 9289728 \, d^{10} e^{8} x^{8} - 6681528 \, d^{11} e^{7} x^{7} - 13848576 \, d^{12} e^{6} x^{6} + 843696 \, d^{13} e^{5} x^{5} + 10321920 \, d^{14} e^{4} x^{4} + 2452800 \, d^{15} e^{3} x^{3} - 3096576 \, d^{16} e^{2} x^{2} - 1177344 \, d^{17} e x - 6930 \,{\left (9 \, d^{10} e^{8} x^{8} - 120 \, d^{12} e^{6} x^{6} + 432 \, d^{14} e^{4} x^{4} - 576 \, d^{16} e^{2} x^{2} + 256 \, d^{18} -{\left (d^{9} e^{8} x^{8} - 40 \, d^{11} e^{6} x^{6} + 240 \, d^{13} e^{4} x^{4} - 448 \, d^{15} e^{2} x^{2} + 256 \, d^{17}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \,{\left (2688 \, d e^{16} x^{16} + 9072 \, d^{2} e^{15} x^{15} - 32768 \, d^{3} e^{14} x^{14} - 142632 \, d^{4} e^{13} x^{13} + 62720 \, d^{5} e^{12} x^{12} + 733614 \, d^{6} e^{11} x^{11} + 344064 \, d^{7} e^{10} x^{10} - 1729707 \, d^{8} e^{9} x^{9} - 1756160 \, d^{9} e^{8} x^{8} + 1902600 \, d^{10} e^{7} x^{7} + 3282944 \, d^{11} e^{6} x^{6} - 542864 \, d^{12} e^{5} x^{5} - 2924544 \, d^{13} e^{4} x^{4} - 621376 \, d^{14} e^{3} x^{3} + 1032192 \, d^{15} e^{2} x^{2} + 392448 \, d^{16} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{8064 \,{\left (9 \, d e^{9} x^{8} - 120 \, d^{3} e^{7} x^{6} + 432 \, d^{5} e^{5} x^{4} - 576 \, d^{7} e^{3} x^{2} + 256 \, d^{9} e -{\left (e^{9} x^{8} - 40 \, d^{2} e^{7} x^{6} + 240 \, d^{4} e^{5} x^{4} - 448 \, d^{6} e^{3} x^{2} + 256 \, d^{8} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8064*(896*e^18*x^18 + 3024*d*e^17*x^17 - 35712*d^2*e^16*x^16 - 131208*d^3*e^15
*x^15 + 200448*d^4*e^14*x^14 + 1145970*d^5*e^13*x^13 + 26880*d^6*e^12*x^12 - 422
4339*d^7*e^11*x^11 - 2862720*d^8*e^10*x^10 + 7768929*d^9*e^9*x^9 + 9289728*d^10*
e^8*x^8 - 6681528*d^11*e^7*x^7 - 13848576*d^12*e^6*x^6 + 843696*d^13*e^5*x^5 + 1
0321920*d^14*e^4*x^4 + 2452800*d^15*e^3*x^3 - 3096576*d^16*e^2*x^2 - 1177344*d^1
7*e*x - 6930*(9*d^10*e^8*x^8 - 120*d^12*e^6*x^6 + 432*d^14*e^4*x^4 - 576*d^16*e^
2*x^2 + 256*d^18 - (d^9*e^8*x^8 - 40*d^11*e^6*x^6 + 240*d^13*e^4*x^4 - 448*d^15*
e^2*x^2 + 256*d^17)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*
x)) + 3*(2688*d*e^16*x^16 + 9072*d^2*e^15*x^15 - 32768*d^3*e^14*x^14 - 142632*d^
4*e^13*x^13 + 62720*d^5*e^12*x^12 + 733614*d^6*e^11*x^11 + 344064*d^7*e^10*x^10
- 1729707*d^8*e^9*x^9 - 1756160*d^9*e^8*x^8 + 1902600*d^10*e^7*x^7 + 3282944*d^1
1*e^6*x^6 - 542864*d^12*e^5*x^5 - 2924544*d^13*e^4*x^4 - 621376*d^14*e^3*x^3 + 1
032192*d^15*e^2*x^2 + 392448*d^16*e*x)*sqrt(-e^2*x^2 + d^2))/(9*d*e^9*x^8 - 120*
d^3*e^7*x^6 + 432*d^5*e^5*x^4 - 576*d^7*e^3*x^2 + 256*d^9*e - (e^9*x^8 - 40*d^2*
e^7*x^6 + 240*d^4*e^5*x^4 - 448*d^6*e^3*x^2 + 256*d^8*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 69.7639, size = 1284, normalized size = 6.83 \[ \text{result too large to display} \]

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)

[Out]

d**7*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**6*e*Piecewise(
(x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) +
d**5*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)) - 5*d**4*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)/(15*
e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4, True)) - 5
*d**3*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*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**2*e**5*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sqr
t(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)) + 3*d*e**6*Pi
ecewise((-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)) + 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))

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GIAC/XCAS [A]  time = 0.283636, size = 158, normalized size = 0.84 \[ \frac{55}{128} \, d^{9} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{1}{8064} \,{\left (3712 \, d^{8} e^{\left (-1\right )} -{\left (4599 \, d^{7} + 2 \,{\left (5120 \, d^{6} e +{\left (1533 \, d^{5} e^{2} - 4 \,{\left (1056 \, d^{4} e^{3} +{\left (903 \, d^{3} e^{4} - 2 \,{\left (64 \, d^{2} e^{5} + 7 \,{\left (8 \, x e^{7} + 27 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

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, algorithm="giac")

[Out]

55/128*d^9*arcsin(x*e/d)*e^(-1)*sign(d) - 1/8064*(3712*d^8*e^(-1) - (4599*d^7 +
2*(5120*d^6*e + (1533*d^5*e^2 - 4*(1056*d^4*e^3 + (903*d^3*e^4 - 2*(64*d^2*e^5 +
7*(8*x*e^7 + 27*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)