3.22 \(\int \frac{x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{5 e^5 \sqrt{d^2-e^2 x^2}}-\frac{4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(x^4*(d + e*x))/(5*d*e*(d^2 - e^2*x^2)^(5/2)) - (4*d^2)/(15*e^5*(d^2 - e^2*x^2)^
(3/2)) + 4/(5*e^5*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.164312, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{5 e^5 \sqrt{d^2-e^2 x^2}}-\frac{4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*(d + e*x))/(5*d*e*(d^2 - e^2*x^2)^(5/2)) - (4*d^2)/(15*e^5*(d^2 - e^2*x^2)^
(3/2)) + 4/(5*e^5*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 17.9974, size = 70, normalized size = 0.83 \[ - \frac{4 d^{2}}{15 e^{5} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{4}{5 e^{5} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{x^{4} \left (d + e x\right )}{5 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.061653, size = 82, normalized size = 0.98 \[ \frac{\sqrt{d^2-e^2 x^2} \left (8 d^4-8 d^3 e x-12 d^2 e^2 x^2+12 d e^3 x^3+3 e^4 x^4\right )}{15 d e^5 (d-e x)^3 (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(8*d^4 - 8*d^3*e*x - 12*d^2*e^2*x^2 + 12*d*e^3*x^3 + 3*e^4*
x^4))/(15*d*e^5*(d - e*x)^3*(d + e*x)^2)

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Maple [A]  time = 0.012, size = 77, normalized size = 0.9 \[{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( 3\,{x}^{4}{e}^{4}+12\,{x}^{3}d{e}^{3}-12\,{d}^{2}{x}^{2}{e}^{2}-8\,{d}^{3}xe+8\,{d}^{4} \right ) }{15\,d{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(-e*x+d)*(e*x+d)^2*(3*e^4*x^4+12*d*e^3*x^3-12*d^2*e^2*x^2-8*d^3*e*x+8*d^4)/
d/e^5/(-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 0.720575, size = 215, normalized size = 2.56 \[ \frac{x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{4 \, d^{2} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} - \frac{3 \, d^{3} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{5}} + \frac{d x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}} + \frac{x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^4/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.282955, size = 336, normalized size = 4. \[ -\frac{3 \, e^{3} x^{8} - 20 \, d e^{2} x^{7} - 4 \, d^{2} e x^{6} + 24 \, d^{3} x^{5} + 4 \,{\left (2 \, e^{2} x^{7} + d e x^{6} - 6 \, d^{2} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d^{2} e^{7} x^{7} - 4 \, d^{3} e^{6} x^{6} - 16 \, d^{4} e^{5} x^{5} + 16 \, d^{5} e^{4} x^{4} + 20 \, d^{6} e^{3} x^{3} - 20 \, d^{7} e^{2} x^{2} - 8 \, d^{8} e x + 8 \, d^{9} -{\left (d e^{7} x^{7} - d^{2} e^{6} x^{6} - 9 \, d^{3} e^{5} x^{5} + 9 \, d^{4} e^{4} x^{4} + 16 \, d^{5} e^{3} x^{3} - 16 \, d^{6} e^{2} x^{2} - 8 \, d^{7} e x + 8 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^4/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/15*(3*e^3*x^8 - 20*d*e^2*x^7 - 4*d^2*e*x^6 + 24*d^3*x^5 + 4*(2*e^2*x^7 + d*e*
x^6 - 6*d^2*x^5)*sqrt(-e^2*x^2 + d^2))/(4*d^2*e^7*x^7 - 4*d^3*e^6*x^6 - 16*d^4*e
^5*x^5 + 16*d^5*e^4*x^4 + 20*d^6*e^3*x^3 - 20*d^7*e^2*x^2 - 8*d^8*e*x + 8*d^9 -
(d*e^7*x^7 - d^2*e^6*x^6 - 9*d^3*e^5*x^5 + 9*d^4*e^4*x^4 + 16*d^5*e^3*x^3 - 16*d
^6*e^2*x^2 - 8*d^7*e*x + 8*d^8)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 24.287, size = 418, normalized size = 4.98 \[ d \left (\begin{cases} - \frac{i x^{5}}{5 d^{7} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{x^{5}}{5 d^{7} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} \frac{8 d^{4}}{15 d^{4} e^{6} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{20 d^{2} e^{2} x^{2}}{15 d^{4} e^{6} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{15 e^{4} x^{4}}{15 d^{4} e^{6} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{6}}{6 \left (d^{2}\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((-I*x**5/(5*d**7*sqrt(-1 + e**2*x**2/d**2) - 10*d**5*e**2*x**2*sqrt(
-1 + e**2*x**2/d**2) + 5*d**3*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**
2/d**2) > 1), (x**5/(5*d**7*sqrt(1 - e**2*x**2/d**2) - 10*d**5*e**2*x**2*sqrt(1
- e**2*x**2/d**2) + 5*d**3*e**4*x**4*sqrt(1 - e**2*x**2/d**2)), True)) + e*Piece
wise((8*d**4/(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*x**2*sqrt(d**2
- e**2*x**2) + 15*e**10*x**4*sqrt(d**2 - e**2*x**2)) - 20*d**2*e**2*x**2/(15*d**
4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*x**2*sqrt(d**2 - e**2*x**2) + 15*e*
*10*x**4*sqrt(d**2 - e**2*x**2)) + 15*e**4*x**4/(15*d**4*e**6*sqrt(d**2 - e**2*x
**2) - 30*d**2*e**8*x**2*sqrt(d**2 - e**2*x**2) + 15*e**10*x**4*sqrt(d**2 - e**2
*x**2)), Ne(e, 0)), (x**6/(6*(d**2)**(7/2)), True))

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GIAC/XCAS [A]  time = 0.293953, size = 86, normalized size = 1.02 \[ -\frac{{\left (8 \, d^{4} e^{\left (-5\right )} +{\left (3 \, x^{2}{\left (\frac{x}{d} + 5 \, e^{\left (-1\right )}\right )} - 20 \, d^{2} e^{\left (-3\right )}\right )} x^{2}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^4/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

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