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

Optimal. Leaf size=87 \[ \frac{x}{15 d^2 e^2 \sqrt{d^2-e^2 x^2}}+\frac{d (d+e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7 (d+e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(d*(d + e*x)^2)/(5*e^3*(d^2 - e^2*x^2)^(5/2)) - (7*(d + e*x))/(15*e^3*(d^2 - e^2
*x^2)^(3/2)) + x/(15*d^2*e^2*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.221954, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x}{15 d^2 e^2 \sqrt{d^2-e^2 x^2}}+\frac{d (d+e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7 (d+e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(d*(d + e*x)^2)/(5*e^3*(d^2 - e^2*x^2)^(5/2)) - (7*(d + e*x))/(15*e^3*(d^2 - e^2
*x^2)^(3/2)) + x/(15*d^2*e^2*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 39.8406, size = 75, normalized size = 0.86 \[ \frac{d}{5 e^{3} \left (d - e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{7}{15 e^{3} \left (d - e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} + \frac{x}{15 d^{2} e^{2} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d/(5*e**3*(d - e*x)**2*sqrt(d**2 - e**2*x**2)) - 7/(15*e**3*(d - e*x)*sqrt(d**2
- e**2*x**2)) + x/(15*d**2*e**2*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.0574321, size = 70, normalized size = 0.8 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-4 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )}{15 d^2 e^3 (d-e x)^3 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 66, normalized size = 0.8 \[ -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{3} \left ( -{e}^{3}{x}^{3}+2\,d{e}^{2}{x}^{2}-8\,{d}^{2}ex+4\,{d}^{3} \right ) }{15\,{d}^{2}{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.722242, size = 177, normalized size = 2.03 \[ \frac{x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2 \, d x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} - \frac{d^{2} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{4 \, d^{3}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} + \frac{x}{30 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} + \frac{x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^3/(-e^2*x^2 + d^2)^(5/2) + 2/3*d*x^2/((-e^2*x^2 + d^2)^(5/2)*e) - 1/10*d^2
*x/((-e^2*x^2 + d^2)^(5/2)*e^2) - 4/15*d^3/((-e^2*x^2 + d^2)^(5/2)*e^3) + 1/30*x
/((-e^2*x^2 + d^2)^(3/2)*e^2) + 1/15*x/(sqrt(-e^2*x^2 + d^2)*d^2*e^2)

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Fricas [A]  time = 0.274306, size = 277, normalized size = 3.18 \[ -\frac{4 \, e^{3} x^{6} - 11 \, d e^{2} x^{5} - 10 \, d^{2} e x^{4} + 20 \, d^{3} x^{3} +{\left (e^{2} x^{5} + 10 \, d e x^{4} - 20 \, d^{2} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{2} e^{6} x^{6} - 2 \, d^{3} e^{5} x^{5} - 4 \, d^{4} e^{4} x^{4} + 10 \, d^{5} e^{3} x^{3} - d^{6} e^{2} x^{2} - 8 \, d^{7} e x + 4 \, d^{8} +{\left (3 \, d^{3} e^{4} x^{4} - 6 \, d^{4} e^{3} x^{3} - d^{5} e^{2} x^{2} + 8 \, d^{6} e x - 4 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(4*e^3*x^6 - 11*d*e^2*x^5 - 10*d^2*e*x^4 + 20*d^3*x^3 + (e^2*x^5 + 10*d*e*
x^4 - 20*d^2*x^3)*sqrt(-e^2*x^2 + d^2))/(d^2*e^6*x^6 - 2*d^3*e^5*x^5 - 4*d^4*e^4
*x^4 + 10*d^5*e^3*x^3 - d^6*e^2*x^2 - 8*d^7*e*x + 4*d^8 + (3*d^3*e^4*x^4 - 6*d^4
*e^3*x^3 - d^5*e^2*x^2 + 8*d^6*e*x - 4*d^7)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.30857, size = 82, normalized size = 0.94 \[ \frac{{\left (4 \, d^{3} e^{\left (-3\right )} -{\left (x{\left (\frac{x^{2} e^{2}}{d^{2}} + 5\right )} + 10 \, d e^{\left (-1\right )}\right )} x^{2}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(4*d^3*e^(-3) - (x*(x^2*e^2/d^2 + 5) + 10*d*e^(-1))*x^2)*sqrt(-x^2*e^2 + d^
2)/(x^2*e^2 - d^2)^3