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

Optimal. Leaf size=148 \[ \frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{35 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e}+\frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2} \]

[Out]

(35*d^7*x*Sqrt[d^2 - e^2*x^2])/128 + (35*d^5*x*(d^2 - e^2*x^2)^(3/2))/192 + (7*d
^3*x*(d^2 - e^2*x^2)^(5/2))/48 + (d*x*(d^2 - e^2*x^2)^(7/2))/8 - (d^2 - e^2*x^2)
^(9/2)/(9*e) + (35*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e)

_______________________________________________________________________________________

Rubi [A]  time = 0.129248, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{35 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e}+\frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(35*d^7*x*Sqrt[d^2 - e^2*x^2])/128 + (35*d^5*x*(d^2 - e^2*x^2)^(3/2))/192 + (7*d
^3*x*(d^2 - e^2*x^2)^(5/2))/48 + (d*x*(d^2 - e^2*x^2)^(7/2))/8 - (d^2 - e^2*x^2)
^(9/2)/(9*e) + (35*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.2234, size = 128, normalized size = 0.86 \[ \frac{35 d^{9} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{128 e} + \frac{35 d^{7} x \sqrt{d^{2} - e^{2} x^{2}}}{128} + \frac{35 d^{5} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{192} + \frac{7 d^{3} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{48} + \frac{d x \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{8} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{9 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35*d**9*atan(e*x/sqrt(d**2 - e**2*x**2))/(128*e) + 35*d**7*x*sqrt(d**2 - e**2*x*
*2)/128 + 35*d**5*x*(d**2 - e**2*x**2)**(3/2)/192 + 7*d**3*x*(d**2 - e**2*x**2)*
*(5/2)/48 + d*x*(d**2 - e**2*x**2)**(7/2)/8 - (d**2 - e**2*x**2)**(9/2)/(9*e)

_______________________________________________________________________________________

Mathematica [A]  time = 0.153161, size = 135, normalized size = 0.91 \[ \frac{315 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (-128 d^8+837 d^7 e x+512 d^6 e^2 x^2-978 d^5 e^3 x^3-768 d^4 e^4 x^4+600 d^3 e^5 x^5+512 d^2 e^6 x^6-144 d e^7 x^7-128 e^8 x^8\right )}{1152 e} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-128*d^8 + 837*d^7*e*x + 512*d^6*e^2*x^2 - 978*d^5*e^3*x^3
- 768*d^4*e^4*x^4 + 600*d^3*e^5*x^5 + 512*d^2*e^6*x^6 - 144*d*e^7*x^7 - 128*e^8
*x^8) + 315*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(1152*e)

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 131, normalized size = 0.9 \[{\frac{dx}{8} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{7\,{d}^{3}x}{48} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{d}^{5}x}{192} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{35\,{d}^{7}x}{128}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{35\,{d}^{9}}{128}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{9\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*d*x*(-e^2*x^2+d^2)^(7/2)+7/48*d^3*x*(-e^2*x^2+d^2)^(5/2)+35/192*d^5*x*(-e^2*
x^2+d^2)^(3/2)+35/128*d^7*x*(-e^2*x^2+d^2)^(1/2)+35/128*d^9/(e^2)^(1/2)*arctan((
e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/9*(-e^2*x^2+d^2)^(9/2)/e

_______________________________________________________________________________________

Maxima [A]  time = 0.801128, size = 166, normalized size = 1.12 \[ \frac{35 \, d^{9} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{128 \, \sqrt{e^{2}}} + \frac{35}{128} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{7} x + \frac{35}{192} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{5} x + \frac{7}{48} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{3} x + \frac{1}{8} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}}}{9 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35/128*d^9*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) + 35/128*sqrt(-e^2*x^2 + d^2)*d
^7*x + 35/192*(-e^2*x^2 + d^2)^(3/2)*d^5*x + 7/48*(-e^2*x^2 + d^2)^(5/2)*d^3*x +
1/8*(-e^2*x^2 + d^2)^(7/2)*d*x - 1/9*(-e^2*x^2 + d^2)^(9/2)/e

_______________________________________________________________________________________

Fricas [A]  time = 0.231889, size = 860, normalized size = 5.81 \[ -\frac{128 \, e^{18} x^{18} + 144 \, d e^{17} x^{17} - 5760 \, d^{2} e^{16} x^{16} - 6504 \, d^{3} e^{15} x^{15} + 57600 \, d^{4} e^{14} x^{14} + 65898 \, d^{5} e^{13} x^{13} - 263424 \, d^{6} e^{12} x^{12} - 308007 \, d^{7} e^{11} x^{11} + 678528 \, d^{8} e^{10} x^{10} + 822333 \, d^{9} e^{9} x^{9} - 1069056 \, d^{10} e^{8} x^{8} - 1366488 \, d^{11} e^{7} x^{7} + 1044480 \, d^{12} e^{6} x^{6} + 1417968 \, d^{13} e^{5} x^{5} - 589824 \, d^{14} e^{4} x^{4} - 839616 \, d^{15} e^{3} x^{3} + 147456 \, d^{16} e^{2} x^{2} + 214272 \, d^{17} e x + 630 \,{\left (9 \, d^{10} e^{8} x^{8} - 120 \, d^{12} e^{6} x^{6} + 432 \, d^{14} e^{4} x^{4} - 576 \, d^{16} e^{2} x^{2} + 256 \, d^{18} -{\left (d^{9} e^{8} x^{8} - 40 \, d^{11} e^{6} x^{6} + 240 \, d^{13} e^{4} x^{4} - 448 \, d^{15} e^{2} x^{2} + 256 \, d^{17}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \,{\left (384 \, d e^{16} x^{16} + 432 \, d^{2} e^{15} x^{15} - 6656 \, d^{3} e^{14} x^{14} - 7560 \, d^{4} e^{13} x^{13} + 41216 \, d^{5} e^{12} x^{12} + 47670 \, d^{6} e^{11} x^{11} - 130560 \, d^{7} e^{10} x^{10} - 155679 \, d^{8} e^{9} x^{9} + 240640 \, d^{9} e^{8} x^{8} + 301800 \, d^{10} e^{7} x^{7} - 268288 \, d^{11} e^{6} x^{6} - 359504 \, d^{12} e^{5} x^{5} + 172032 \, d^{13} e^{4} x^{4} + 244160 \, d^{14} e^{3} x^{3} - 49152 \, d^{15} e^{2} x^{2} - 71424 \, d^{16} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1152 \,{\left (9 \, d e^{9} x^{8} - 120 \, d^{3} e^{7} x^{6} + 432 \, d^{5} e^{5} x^{4} - 576 \, d^{7} e^{3} x^{2} + 256 \, d^{9} e -{\left (e^{9} x^{8} - 40 \, d^{2} e^{7} x^{6} + 240 \, d^{4} e^{5} x^{4} - 448 \, d^{6} e^{3} x^{2} + 256 \, d^{8} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1152*(128*e^18*x^18 + 144*d*e^17*x^17 - 5760*d^2*e^16*x^16 - 6504*d^3*e^15*x^
15 + 57600*d^4*e^14*x^14 + 65898*d^5*e^13*x^13 - 263424*d^6*e^12*x^12 - 308007*d
^7*e^11*x^11 + 678528*d^8*e^10*x^10 + 822333*d^9*e^9*x^9 - 1069056*d^10*e^8*x^8
- 1366488*d^11*e^7*x^7 + 1044480*d^12*e^6*x^6 + 1417968*d^13*e^5*x^5 - 589824*d^
14*e^4*x^4 - 839616*d^15*e^3*x^3 + 147456*d^16*e^2*x^2 + 214272*d^17*e*x + 630*(
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*(384*d
*e^16*x^16 + 432*d^2*e^15*x^15 - 6656*d^3*e^14*x^14 - 7560*d^4*e^13*x^13 + 41216
*d^5*e^12*x^12 + 47670*d^6*e^11*x^11 - 130560*d^7*e^10*x^10 - 155679*d^8*e^9*x^9
+ 240640*d^9*e^8*x^8 + 301800*d^10*e^7*x^7 - 268288*d^11*e^6*x^6 - 359504*d^12*
e^5*x^5 + 172032*d^13*e^4*x^4 + 244160*d^14*e^3*x^3 - 49152*d^15*e^2*x^2 - 71424
*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))

_______________________________________________________________________________________

Sympy [A]  time = 73.1962, size = 1284, normalized size = 8.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(-e**2*x**2+d**2)**(7/2),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)) + 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)) - 3
*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)) - 3*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)) + 3
*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)) +
3*d**2*e**5*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*s
qrt(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)) - 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))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.237669, size = 161, normalized size = 1.09 \[ \frac{35}{128} \, d^{9} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{1}{1152} \,{\left (128 \, d^{8} e^{\left (-1\right )} -{\left (837 \, d^{7} + 2 \,{\left (256 \, d^{6} e -{\left (489 \, d^{5} e^{2} + 4 \,{\left (96 \, d^{4} e^{3} -{\left (75 \, d^{3} e^{4} + 2 \,{\left (32 \, d^{2} e^{5} -{\left (8 \, x e^{7} + 9 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35/128*d^9*arcsin(x*e/d)*e^(-1)*sign(d) - 1/1152*(128*d^8*e^(-1) - (837*d^7 + 2*
(256*d^6*e - (489*d^5*e^2 + 4*(96*d^4*e^3 - (75*d^3*e^4 + 2*(32*d^2*e^5 - (8*x*e
^7 + 9*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)