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

Optimal. Leaf size=94 \[ \frac{x^2 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d-e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{15 d^3 e^2 \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.170861, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{x^2 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d-e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{15 d^3 e^2 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.9699, size = 80, normalized size = 0.85 \[ \frac{x^{2} \left (d + e x\right )}{5 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{2 d - 2 e x}{15 d e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2 x}{15 d^{3} e^{2} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.058219, size = 82, normalized size = 0.87 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-2 d^4+2 d^3 e x+3 d^2 e^2 x^2+2 d e^3 x^3-2 e^4 x^4\right )}{15 d^3 e^3 (d-e x)^3 (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 77, normalized size = 0.8 \[ -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( 2\,{e}^{4}{x}^{4}-2\,{x}^{3}d{e}^{3}-3\,{x}^{2}{d}^{2}{e}^{2}-2\,x{d}^{3}e+2\,{d}^{4} \right ) }{15\,{d}^{3}{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.725597, size = 151, normalized size = 1.61 \[ \frac{x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{2 \, d^{2}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} - \frac{x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d e^{2}} - \frac{2 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.283243, size = 398, normalized size = 4.23 \[ \frac{2 \, e^{5} x^{8} - 10 \, d e^{4} x^{7} - 11 \, d^{2} e^{3} x^{6} + 46 \, d^{3} e^{2} x^{5} + 10 \, d^{4} e x^{4} - 40 \, d^{5} x^{3} + 2 \,{\left (e^{4} x^{7} + 3 \, d e^{3} x^{6} - 13 \, d^{2} e^{2} x^{5} - 5 \, d^{3} e x^{4} + 20 \, d^{4} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d^{4} e^{7} x^{7} - 4 \, d^{5} e^{6} x^{6} - 16 \, d^{6} e^{5} x^{5} + 16 \, d^{7} e^{4} x^{4} + 20 \, d^{8} e^{3} x^{3} - 20 \, d^{9} e^{2} x^{2} - 8 \, d^{10} e x + 8 \, d^{11} -{\left (d^{3} e^{7} x^{7} - d^{4} e^{6} x^{6} - 9 \, d^{5} e^{5} x^{5} + 9 \, d^{6} e^{4} x^{4} + 16 \, d^{7} e^{3} x^{3} - 16 \, d^{8} e^{2} x^{2} - 8 \, d^{9} e x + 8 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(2*e^5*x^8 - 10*d*e^4*x^7 - 11*d^2*e^3*x^6 + 46*d^3*e^2*x^5 + 10*d^4*e*x^4
- 40*d^5*x^3 + 2*(e^4*x^7 + 3*d*e^3*x^6 - 13*d^2*e^2*x^5 - 5*d^3*e*x^4 + 20*d^4*
x^3)*sqrt(-e^2*x^2 + d^2))/(4*d^4*e^7*x^7 - 4*d^5*e^6*x^6 - 16*d^6*e^5*x^5 + 16*
d^7*e^4*x^4 + 20*d^8*e^3*x^3 - 20*d^9*e^2*x^2 - 8*d^10*e*x + 8*d^11 - (d^3*e^7*x
^7 - d^4*e^6*x^6 - 9*d^5*e^5*x^5 + 9*d^6*e^4*x^4 + 16*d^7*e^3*x^3 - 16*d^8*e^2*x
^2 - 8*d^9*e*x + 8*d^10)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 24.2424, size = 513, normalized size = 5.46 \[ d \left (\begin{cases} - \frac{5 i d^{2} x^{3}}{15 d^{9} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{2 i e^{2} x^{5}}{15 d^{9} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{5 d^{2} x^{3}}{15 d^{9} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{2 e^{2} x^{5}}{15 d^{9} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} - \frac{2 d^{2}}{15 d^{4} e^{4} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{8} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{5 e^{2} x^{2}}{15 d^{4} e^{4} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{8} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \left (d^{2}\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.295831, size = 86, normalized size = 0.91 \[ \frac{{\left ({\left (x{\left (\frac{2 \, x^{2} e^{2}}{d^{3}} - \frac{5}{d}\right )} - 5 \, e^{\left (-1\right )}\right )} x^{2} + 2 \, d^{2} e^{\left (-3\right )}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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