3.832 \(\int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 x}{21 d^6 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(4*x)/(21*d^4*(d^2 - e^2*x^2)^(3/2)) - 1/(7*d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2
)) - 1/(7*d^2*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (8*x)/(21*d^6*Sqrt[d^2 - e^2*
x^2])

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Rubi [A]  time = 0.11521, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 x}{21 d^6 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^2*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(4*x)/(21*d^4*(d^2 - e^2*x^2)^(3/2)) - 1/(7*d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2
)) - 1/(7*d^2*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (8*x)/(21*d^6*Sqrt[d^2 - e^2*
x^2])

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Rubi in Sympy [A]  time = 12.6264, size = 97, normalized size = 0.84 \[ - \frac{1}{7 d e \left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{1}{7 d^{2} e \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{4 x}{21 d^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{8 x}{21 d^{6} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**2/(-e**2*x**2+d**2)**(5/2),x)

[Out]

-1/(7*d*e*(d + e*x)**2*(d**2 - e**2*x**2)**(3/2)) - 1/(7*d**2*e*(d + e*x)*(d**2
- e**2*x**2)**(3/2)) + 4*x/(21*d**4*(d**2 - e**2*x**2)**(3/2)) + 8*x/(21*d**6*sq
rt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.0743298, size = 93, normalized size = 0.81 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-6 d^5+9 d^4 e x+24 d^3 e^2 x^2+4 d^2 e^3 x^3-16 d e^4 x^4-8 e^5 x^5\right )}{21 d^6 e (d-e x)^2 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^2*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-6*d^5 + 9*d^4*e*x + 24*d^3*e^2*x^2 + 4*d^2*e^3*x^3 - 16*d
*e^4*x^4 - 8*e^5*x^5))/(21*d^6*e*(d - e*x)^2*(d + e*x)^4)

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Maple [A]  time = 0.013, size = 88, normalized size = 0.8 \[ -{\frac{ \left ( -ex+d \right ) \left ( 8\,{e}^{5}{x}^{5}+16\,{e}^{4}{x}^{4}d-4\,{e}^{3}{x}^{3}{d}^{2}-24\,{e}^{2}{x}^{2}{d}^{3}-9\,x{d}^{4}e+6\,{d}^{5} \right ) }{ \left ( 21\,ex+21\,d \right ){d}^{6}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/21*(-e*x+d)*(8*e^5*x^5+16*d*e^4*x^4-4*d^2*e^3*x^3-24*d^3*e^2*x^2-9*d^4*e*x+6*
d^5)/(e*x+d)/d^6/e/(-e^2*x^2+d^2)^(5/2)

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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(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229929, size = 570, normalized size = 4.96 \[ -\frac{6 \, e^{9} x^{10} - 28 \, d e^{8} x^{9} - 158 \, d^{2} e^{7} x^{8} + 12 \, d^{3} e^{6} x^{7} + 602 \, d^{4} e^{5} x^{6} + 329 \, d^{5} e^{4} x^{5} - 784 \, d^{6} e^{3} x^{4} - 644 \, d^{7} e^{2} x^{3} + 336 \, d^{8} e x^{2} + 336 \, d^{9} x +{\left (8 \, e^{8} x^{9} + 46 \, d e^{7} x^{8} - 40 \, d^{2} e^{6} x^{7} - 336 \, d^{3} e^{5} x^{6} - 133 \, d^{4} e^{4} x^{5} + 616 \, d^{5} e^{3} x^{4} + 476 \, d^{6} e^{2} x^{3} - 336 \, d^{7} e x^{2} - 336 \, d^{8} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{21 \,{\left (d^{6} e^{10} x^{10} + 2 \, d^{7} e^{9} x^{9} - 13 \, d^{8} e^{8} x^{8} - 28 \, d^{9} e^{7} x^{7} + 27 \, d^{10} e^{6} x^{6} + 82 \, d^{11} e^{5} x^{5} - 3 \, d^{12} e^{4} x^{4} - 88 \, d^{13} e^{3} x^{3} - 28 \, d^{14} e^{2} x^{2} + 32 \, d^{15} e x + 16 \, d^{16} +{\left (5 \, d^{7} e^{8} x^{8} + 10 \, d^{8} e^{7} x^{7} - 20 \, d^{9} e^{6} x^{6} - 50 \, d^{10} e^{5} x^{5} + 11 \, d^{11} e^{4} x^{4} + 72 \, d^{12} e^{3} x^{3} + 20 \, d^{13} e^{2} x^{2} - 32 \, d^{14} e x - 16 \, d^{15}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

-1/21*(6*e^9*x^10 - 28*d*e^8*x^9 - 158*d^2*e^7*x^8 + 12*d^3*e^6*x^7 + 602*d^4*e^
5*x^6 + 329*d^5*e^4*x^5 - 784*d^6*e^3*x^4 - 644*d^7*e^2*x^3 + 336*d^8*e*x^2 + 33
6*d^9*x + (8*e^8*x^9 + 46*d*e^7*x^8 - 40*d^2*e^6*x^7 - 336*d^3*e^5*x^6 - 133*d^4
*e^4*x^5 + 616*d^5*e^3*x^4 + 476*d^6*e^2*x^3 - 336*d^7*e*x^2 - 336*d^8*x)*sqrt(-
e^2*x^2 + d^2))/(d^6*e^10*x^10 + 2*d^7*e^9*x^9 - 13*d^8*e^8*x^8 - 28*d^9*e^7*x^7
+ 27*d^10*e^6*x^6 + 82*d^11*e^5*x^5 - 3*d^12*e^4*x^4 - 88*d^13*e^3*x^3 - 28*d^1
4*e^2*x^2 + 32*d^15*e*x + 16*d^16 + (5*d^7*e^8*x^8 + 10*d^8*e^7*x^7 - 20*d^9*e^6
*x^6 - 50*d^10*e^5*x^5 + 11*d^11*e^4*x^4 + 72*d^12*e^3*x^3 + 20*d^13*e^2*x^2 - 3
2*d^14*e*x - 16*d^15)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**2/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)**2), x)

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GIAC/XCAS [A]  time = 0.608149, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^2),x, algorithm="giac")

[Out]

sage0*x