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

Optimal. Leaf size=77 \[ \frac{x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 x}{5 d^4 \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0559055, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 x}{5 d^4 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.70146, size = 66, normalized size = 0.86 \[ \frac{2 \left (d + e x\right )}{5 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{x}{5 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{2 x}{5 d^{4} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0475341, size = 70, normalized size = 0.91 \[ \frac{\sqrt{d^2-e^2 x^2} \left (2 d^3+d^2 e x-4 d e^2 x^2+2 e^3 x^3\right )}{5 d^4 e (d-e x)^3 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 65, normalized size = 0.8 \[{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{3} \left ( 2\,{e}^{3}{x}^{3}-4\,{e}^{2}{x}^{2}d+x{d}^{2}e+2\,{d}^{3} \right ) }{5\,{d}^{4}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.716504, size = 105, normalized size = 1.36 \[ \frac{2 \, x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2 \, d}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2}} + \frac{2 \, x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.270689, size = 333, normalized size = 4.32 \[ \frac{2 \, e^{5} x^{6} + 2 \, d e^{4} x^{5} - 20 \, d^{2} e^{3} x^{4} + 15 \, d^{3} e^{2} x^{3} + 20 \, d^{4} e x^{2} - 20 \, d^{5} x -{\left (2 \, e^{4} x^{5} - 10 \, d e^{3} x^{4} + 5 \, d^{2} e^{2} x^{3} + 20 \, d^{3} e x^{2} - 20 \, d^{4} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d^{4} e^{6} x^{6} - 2 \, d^{5} e^{5} x^{5} - 4 \, d^{6} e^{4} x^{4} + 10 \, d^{7} e^{3} x^{3} - d^{8} e^{2} x^{2} - 8 \, d^{9} e x + 4 \, d^{10} +{\left (3 \, d^{5} e^{4} x^{4} - 6 \, d^{6} e^{3} x^{3} - d^{7} e^{2} x^{2} + 8 \, d^{8} e x - 4 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(2*e^5*x^6 + 2*d*e^4*x^5 - 20*d^2*e^3*x^4 + 15*d^3*e^2*x^3 + 20*d^4*e*x^2 -
20*d^5*x - (2*e^4*x^5 - 10*d*e^3*x^4 + 5*d^2*e^2*x^3 + 20*d^3*e*x^2 - 20*d^4*x)*
sqrt(-e^2*x^2 + d^2))/(d^4*e^6*x^6 - 2*d^5*e^5*x^5 - 4*d^6*e^4*x^4 + 10*d^7*e^3*
x^3 - d^8*e^2*x^2 - 8*d^9*e*x + 4*d^10 + (3*d^5*e^4*x^4 - 6*d^6*e^3*x^3 - d^7*e^
2*x^2 + 8*d^8*e*x - 4*d^9)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**2/(-(-d + e*x)*(d + e*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.288109, size = 82, normalized size = 1.06 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (x^{2}{\left (\frac{2 \, x^{2} e^{4}}{d^{4}} - \frac{5 \, e^{2}}{d^{2}}\right )} + 5\right )} x + 2 \, d e^{\left (-1\right )}\right )}}{5 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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