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

Optimal. Leaf size=80 \[ \frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(d + e*x)/(5*d*e*(d^2 - e^2*x^2)^(5/2)) + (4*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) +
(8*x)/(15*d^5*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.0530993, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(d + e*x)/(5*d*e*(d^2 - e^2*x^2)^(5/2)) + (4*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) +
(8*x)/(15*d^5*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 7.79355, size = 66, normalized size = 0.82 \[ \frac{d + e x}{5 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{4 x}{15 d^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{8 x}{15 d^{5} \sqrt{d^{2} - e^{2} x^{2}}} \]

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]

(d + e*x)/(5*d*e*(d**2 - e**2*x**2)**(5/2)) + 4*x/(15*d**3*(d**2 - e**2*x**2)**(
3/2)) + 8*x/(15*d**5*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.0456302, size = 82, normalized size = 1.02 \[ \frac{\sqrt{d^2-e^2 x^2} \left (3 d^4+12 d^3 e x-12 d^2 e^2 x^2-8 d e^3 x^3+8 e^4 x^4\right )}{15 d^5 e (d-e x)^3 (d+e x)^2} \]

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]*(3*d^4 + 12*d^3*e*x - 12*d^2*e^2*x^2 - 8*d*e^3*x^3 + 8*e^4*
x^4))/(15*d^5*e*(d - e*x)^3*(d + e*x)^2)

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Maple [A]  time = 0.012, size = 77, normalized size = 1. \[{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( 8\,{e}^{4}{x}^{4}-8\,{e}^{3}{x}^{3}d-12\,{e}^{2}{x}^{2}{d}^{2}+12\,x{d}^{3}e+3\,{d}^{4} \right ) }{15\,{d}^{5}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(-e*x+d)*(e*x+d)^2*(8*e^4*x^4-8*d*e^3*x^3-12*d^2*e^2*x^2+12*d^3*e*x+3*d^4)/
d^5/e/(-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 0.713501, size = 108, normalized size = 1.35 \[ \frac{x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d} + \frac{1}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{4 \, x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{3}} + \frac{8 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*x/((-e^2*x^2 + d^2)^(5/2)*d) + 1/5/((-e^2*x^2 + d^2)^(5/2)*e) + 4/15*x/((-e^
2*x^2 + d^2)^(3/2)*d^3) + 8/15*x/(sqrt(-e^2*x^2 + d^2)*d^5)

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Fricas [A]  time = 0.286452, size = 452, normalized size = 5.65 \[ -\frac{8 \, e^{7} x^{8} - 20 \, d e^{6} x^{7} - 64 \, d^{2} e^{5} x^{6} + 124 \, d^{3} e^{4} x^{5} + 115 \, d^{4} e^{3} x^{4} - 220 \, d^{5} e^{2} x^{3} - 60 \, d^{6} e x^{2} + 120 \, d^{7} x +{\left (3 \, e^{6} x^{7} + 29 \, d e^{5} x^{6} - 59 \, d^{2} e^{4} x^{5} - 85 \, d^{3} e^{3} x^{4} + 160 \, d^{4} e^{2} x^{3} + 60 \, d^{5} e x^{2} - 120 \, d^{6} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d^{6} e^{7} x^{7} - 4 \, d^{7} e^{6} x^{6} - 16 \, d^{8} e^{5} x^{5} + 16 \, d^{9} e^{4} x^{4} + 20 \, d^{10} e^{3} x^{3} - 20 \, d^{11} e^{2} x^{2} - 8 \, d^{12} e x + 8 \, d^{13} -{\left (d^{5} e^{7} x^{7} - d^{6} e^{6} x^{6} - 9 \, d^{7} e^{5} x^{5} + 9 \, d^{8} e^{4} x^{4} + 16 \, d^{9} e^{3} x^{3} - 16 \, d^{10} e^{2} x^{2} - 8 \, d^{11} e x + 8 \, d^{12}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(8*e^7*x^8 - 20*d*e^6*x^7 - 64*d^2*e^5*x^6 + 124*d^3*e^4*x^5 + 115*d^4*e^3
*x^4 - 220*d^5*e^2*x^3 - 60*d^6*e*x^2 + 120*d^7*x + (3*e^6*x^7 + 29*d*e^5*x^6 -
59*d^2*e^4*x^5 - 85*d^3*e^3*x^4 + 160*d^4*e^2*x^3 + 60*d^5*e*x^2 - 120*d^6*x)*sq
rt(-e^2*x^2 + d^2))/(4*d^6*e^7*x^7 - 4*d^7*e^6*x^6 - 16*d^8*e^5*x^5 + 16*d^9*e^4
*x^4 + 20*d^10*e^3*x^3 - 20*d^11*e^2*x^2 - 8*d^12*e*x + 8*d^13 - (d^5*e^7*x^7 -
d^6*e^6*x^6 - 9*d^7*e^5*x^5 + 9*d^8*e^4*x^4 + 16*d^9*e^3*x^3 - 16*d^10*e^2*x^2 -
8*d^11*e*x + 8*d^12)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 24.0965, size = 604, normalized size = 7.55 \[ d \left (\begin{cases} - \frac{15 i d^{4} x}{15 d^{11} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{20 i d^{2} e^{2} x^{3}}{15 d^{11} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{8 i e^{4} x^{5}}{15 d^{11} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{15 d^{4} x}{15 d^{11} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{20 d^{2} e^{2} x^{3}}{15 d^{11} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{8 e^{4} x^{5}}{15 d^{11} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} \frac{1}{5 d^{4} e^{2} \sqrt{d^{2} - e^{2} x^{2}} - 10 d^{2} e^{4} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 5 e^{6} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 \left (d^{2}\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) \]

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*Piecewise((-15*I*d**4*x/(15*d**11*sqrt(-1 + e**2*x**2/d**2) - 30*d**9*e**2*x**
2*sqrt(-1 + e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)) + 20*
I*d**2*e**2*x**3/(15*d**11*sqrt(-1 + e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(-1
+ e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)) - 8*I*e**4*x**
5/(15*d**11*sqrt(-1 + e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(-1 + e**2*x**2/d*
*2) + 15*d**7*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (1
5*d**4*x/(15*d**11*sqrt(1 - e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(1 - e**2*x*
*2/d**2) + 15*d**7*e**4*x**4*sqrt(1 - e**2*x**2/d**2)) - 20*d**2*e**2*x**3/(15*d
**11*sqrt(1 - e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 15*
d**7*e**4*x**4*sqrt(1 - e**2*x**2/d**2)) + 8*e**4*x**5/(15*d**11*sqrt(1 - e**2*x
**2/d**2) - 30*d**9*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(
1 - e**2*x**2/d**2)), True)) + e*Piecewise((1/(5*d**4*e**2*sqrt(d**2 - e**2*x**2
) - 10*d**2*e**4*x**2*sqrt(d**2 - e**2*x**2) + 5*e**6*x**4*sqrt(d**2 - e**2*x**2
)), Ne(e, 0)), (x**2/(2*(d**2)**(7/2)), True))

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GIAC/XCAS [A]  time = 0.298916, size = 88, normalized size = 1.1 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (4 \, x^{2}{\left (\frac{2 \, x^{2} e^{4}}{d^{5}} - \frac{5 \, e^{2}}{d^{3}}\right )} + \frac{15}{d}\right )} x + 3 \, e^{\left (-1\right )}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*((4*x^2*(2*x^2*e^4/d^5 - 5*e^2/d^3) + 15/d)*x + 3*e^(
-1))/(x^2*e^2 - d^2)^3