3.799 \(\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=143 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac{2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{e (d+e x)}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

[Out]

(2*Sqrt[d^2 - e^2*x^2])/(e*(d + e*x)) - (2*(d^2 - e^2*x^2)^(3/2))/(3*e*(d + e*x)
^3) + (2*(d^2 - e^2*x^2)^(5/2))/(5*e*(d + e*x)^5) - (2*(d^2 - e^2*x^2)^(7/2))/(7
*e*(d + e*x)^7) + ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]]/e

_______________________________________________________________________________________

Rubi [A]  time = 0.137663, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac{2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{e (d+e x)}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x]

[Out]

(2*Sqrt[d^2 - e^2*x^2])/(e*(d + e*x)) - (2*(d^2 - e^2*x^2)^(3/2))/(3*e*(d + e*x)
^3) + (2*(d^2 - e^2*x^2)^(5/2))/(5*e*(d + e*x)^5) - (2*(d^2 - e^2*x^2)^(7/2))/(7
*e*(d + e*x)^7) + ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]]/e

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.929, size = 117, normalized size = 0.82 \[ \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e} + \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{e \left (d + e x\right )} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{3}} + \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{5 e \left (d + e x\right )^{5}} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{7 e \left (d + e x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**8,x)

[Out]

atan(e*x/sqrt(d**2 - e**2*x**2))/e + 2*sqrt(d**2 - e**2*x**2)/(e*(d + e*x)) - 2*
(d**2 - e**2*x**2)**(3/2)/(3*e*(d + e*x)**3) + 2*(d**2 - e**2*x**2)**(5/2)/(5*e*
(d + e*x)**5) - 2*(d**2 - e**2*x**2)**(7/2)/(7*e*(d + e*x)**7)

_______________________________________________________________________________________

Mathematica [A]  time = 0.131577, size = 85, normalized size = 0.59 \[ \frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}+\frac{8 \sqrt{d^2-e^2 x^2} \left (19 d^3+76 d^2 e x+71 d e^2 x^2+44 e^3 x^3\right )}{105 e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x]

[Out]

(8*Sqrt[d^2 - e^2*x^2]*(19*d^3 + 76*d^2*e*x + 71*d*e^2*x^2 + 44*e^3*x^3))/(105*e
*(d + e*x)^4) + ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]]/e

_______________________________________________________________________________________

Maple [B]  time = 0.018, size = 496, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x)

[Out]

-1/7/e^9/d/(d/e+x)^8*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+1/35/e^8/d^2/(d/e+x)^7
*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)-2/105/e^7/d^3/(d/e+x)^6*(-(d/e+x)^2*e^2+2*
d*e*(d/e+x))^(9/2)+2/35/e^6/d^4/(d/e+x)^5*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+8
/35/e^5/d^5/(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+8/21/e^4/d^6/(d/e+x)^
3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+16/35/e^3/d^7/(d/e+x)^2*(-(d/e+x)^2*e^2+2
*d*e*(d/e+x))^(9/2)+16/35/e/d^7*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)+8/15/d^6*(-
(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)*x+2/3/d^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2
)*x+1/d^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+1/(e^2)^(1/2)*arctan((e^2)^(1/2
)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.236138, size = 608, normalized size = 4.25 \[ -\frac{2 \,{\left (100 \, e^{7} x^{7} + 812 \, d e^{6} x^{6} + 812 \, d^{2} e^{5} x^{5} + 140 \, d^{3} e^{4} x^{4} - 1400 \, d^{4} e^{3} x^{3} - 1680 \, d^{5} e^{2} x^{2} + 105 \,{\left (e^{7} x^{7} - 14 \, d^{2} e^{5} x^{5} - 28 \, d^{3} e^{4} x^{4} - 7 \, d^{4} e^{3} x^{3} + 28 \, d^{5} e^{2} x^{2} + 28 \, d^{6} e x + 8 \, d^{7} +{\left (e^{6} x^{6} + 7 \, d e^{5} x^{5} + 11 \, d^{2} e^{4} x^{4} - 7 \, d^{3} e^{3} x^{3} - 32 \, d^{4} e^{2} x^{2} - 28 \, d^{5} e x - 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 28 \,{\left (9 \, e^{6} x^{6} + 4 \, d e^{5} x^{5} - 25 \, d^{2} e^{4} x^{4} - 50 \, d^{3} e^{3} x^{3} - 60 \, d^{4} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}}{105 \,{\left (e^{8} x^{7} - 14 \, d^{2} e^{6} x^{5} - 28 \, d^{3} e^{5} x^{4} - 7 \, d^{4} e^{4} x^{3} + 28 \, d^{5} e^{3} x^{2} + 28 \, d^{6} e^{2} x + 8 \, d^{7} e +{\left (e^{7} x^{6} + 7 \, d e^{6} x^{5} + 11 \, d^{2} e^{5} x^{4} - 7 \, d^{3} e^{4} x^{3} - 32 \, d^{4} e^{3} x^{2} - 28 \, d^{5} e^{2} x - 8 \, d^{6} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^8,x, algorithm="fricas")

[Out]

-2/105*(100*e^7*x^7 + 812*d*e^6*x^6 + 812*d^2*e^5*x^5 + 140*d^3*e^4*x^4 - 1400*d
^4*e^3*x^3 - 1680*d^5*e^2*x^2 + 105*(e^7*x^7 - 14*d^2*e^5*x^5 - 28*d^3*e^4*x^4 -
7*d^4*e^3*x^3 + 28*d^5*e^2*x^2 + 28*d^6*e*x + 8*d^7 + (e^6*x^6 + 7*d*e^5*x^5 +
11*d^2*e^4*x^4 - 7*d^3*e^3*x^3 - 32*d^4*e^2*x^2 - 28*d^5*e*x - 8*d^6)*sqrt(-e^2*
x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 28*(9*e^6*x^6 + 4*d*e^5*
x^5 - 25*d^2*e^4*x^4 - 50*d^3*e^3*x^3 - 60*d^4*e^2*x^2)*sqrt(-e^2*x^2 + d^2))/(e
^8*x^7 - 14*d^2*e^6*x^5 - 28*d^3*e^5*x^4 - 7*d^4*e^4*x^3 + 28*d^5*e^3*x^2 + 28*d
^6*e^2*x + 8*d^7*e + (e^7*x^6 + 7*d*e^6*x^5 + 11*d^2*e^5*x^4 - 7*d^3*e^4*x^3 - 3
2*d^4*e^3*x^2 - 28*d^5*e^2*x - 8*d^6*e)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.497839, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^8,x, algorithm="giac")

[Out]

Done