3.46 \(\int \frac{x^3 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=97 \[ \frac{d^2 (d+e x)^2}{5 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 d (d+e x)}{5 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{5 d+2 e x}{5 d e^4 \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.278389, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{d^2 (d+e x)^2}{5 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 d (d+e x)}{5 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{5 d+2 e x}{5 d e^4 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 44.0344, size = 97, normalized size = 1. \[ \frac{d^{2}}{5 e^{4} \left (d - e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{4 d}{5 e^{4} \left (d - e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} + \frac{1}{e^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{2 x}{5 d e^{3} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(e*x+d)**2/(-e**2*x**2+d**2)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.0549273, size = 70, normalized size = 0.72 \[ \frac{\sqrt{d^2-e^2 x^2} \left (2 d^3-4 d^2 e x+d e^2 x^2+2 e^3 x^3\right )}{5 d e^4 (d-e x)^3 (d+e x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

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

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Maple [A]  time = 0.011, size = 65, normalized size = 0.7 \[{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{3} \left ( 2\,{e}^{3}{x}^{3}+d{e}^{2}{x}^{2}-4\,{d}^{2}ex+2\,{d}^{3} \right ) }{5\,d{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(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+d*e^2*x^2-4*d^2*e*x+2*d^3)/d/e^4/(-e^2*x^2+d^2
)^(7/2)

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Maxima [A]  time = 0.714582, size = 209, normalized size = 2.15 \[ \frac{x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{d x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} - \frac{d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{3 \, d^{3} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} + \frac{2 \, d^{4}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{d x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{3}} + \frac{2 \, x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*x^3/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.269581, size = 247, normalized size = 2.55 \[ \frac{2 \, e^{2} x^{6} + 2 \, d e x^{5} - 5 \, d^{2} x^{4} -{\left (2 \, e x^{5} - 5 \, d x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d e^{6} x^{6} - 2 \, d^{2} e^{5} x^{5} - 4 \, d^{3} e^{4} x^{4} + 10 \, d^{4} e^{3} x^{3} - d^{5} e^{2} x^{2} - 8 \, d^{6} e x + 4 \, d^{7} +{\left (3 \, d^{2} e^{4} x^{4} - 6 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2} + 8 \, d^{5} e x - 4 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*x^3/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(e*x+d)**2/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(x**3*(d + e*x)**2/(-(-d + e*x)*(d + e*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.296317, size = 85, normalized size = 0.88 \[ -\frac{{\left (2 \, d^{4} e^{\left (-4\right )} +{\left (x^{2}{\left (\frac{2 \, x e}{d} + 5\right )} - 5 \, d^{2} e^{\left (-2\right )}\right )} x^{2}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{5 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*x^3/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

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