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

Optimal. Leaf size=90 \[ \frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 d+3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.165658, antiderivative size = 90, 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 \[ \frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 d+3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.3052, size = 80, normalized size = 0.89 \[ \frac{x^{3} \left (d + e x\right )}{5 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{2}{15 d e^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{x^{2} \left (d - 3 e x\right )}{15 d^{2} e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0618098, size = 82, normalized size = 0.91 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-2 d^4+2 d^3 e x+3 d^2 e^2 x^2-3 d e^3 x^3+3 e^4 x^4\right )}{15 d^2 e^4 (d-e x)^3 (d+e x)^2} \]

Antiderivative was successfully verified.

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

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Maple [A]  time = 0.013, size = 77, normalized size = 0.9 \[ -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( -3\,{x}^{4}{e}^{4}+3\,{x}^{3}d{e}^{3}-3\,{x}^{2}{d}^{2}{e}^{2}-2\,{d}^{3}xe+2\,{d}^{4} \right ) }{15\,{d}^{2}{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.719988, size = 181, normalized size = 2.01 \[ \frac{x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{3 \, d^{2} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} - \frac{2 \, d^{3}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{3}} + \frac{x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.280792, size = 369, normalized size = 4.1 \[ -\frac{3 \, e^{4} x^{8} + 5 \, d e^{3} x^{7} - 29 \, d^{2} e^{2} x^{6} - 6 \, d^{3} e x^{5} + 30 \, d^{4} x^{4} - 2 \,{\left (e^{3} x^{7} - 7 \, d e^{2} x^{6} - 3 \, d^{2} e x^{5} + 15 \, d^{3} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d^{3} e^{7} x^{7} - 4 \, d^{4} e^{6} x^{6} - 16 \, d^{5} e^{5} x^{5} + 16 \, d^{6} e^{4} x^{4} + 20 \, d^{7} e^{3} x^{3} - 20 \, d^{8} e^{2} x^{2} - 8 \, d^{9} e x + 8 \, d^{10} -{\left (d^{2} e^{7} x^{7} - d^{3} e^{6} x^{6} - 9 \, d^{4} e^{5} x^{5} + 9 \, d^{5} e^{4} x^{4} + 16 \, d^{6} e^{3} x^{3} - 16 \, d^{7} e^{2} x^{2} - 8 \, d^{8} e x + 8 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(3*e^4*x^8 + 5*d*e^3*x^7 - 29*d^2*e^2*x^6 - 6*d^3*e*x^5 + 30*d^4*x^4 - 2*(
e^3*x^7 - 7*d*e^2*x^6 - 3*d^2*e*x^5 + 15*d^3*x^4)*sqrt(-e^2*x^2 + d^2))/(4*d^3*e
^7*x^7 - 4*d^4*e^6*x^6 - 16*d^5*e^5*x^5 + 16*d^6*e^4*x^4 + 20*d^7*e^3*x^3 - 20*d
^8*e^2*x^2 - 8*d^9*e*x + 8*d^10 - (d^2*e^7*x^7 - d^3*e^6*x^6 - 9*d^4*e^5*x^5 + 9
*d^5*e^4*x^4 + 16*d^6*e^3*x^3 - 16*d^7*e^2*x^2 - 8*d^8*e*x + 8*d^9)*sqrt(-e^2*x^
2 + d^2))

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Sympy [A]  time = 23.6504, size = 337, normalized size = 3.74 \[ d \left (\begin{cases} - \frac{2 d^{2}}{15 d^{4} e^{4} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{8} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{5 e^{2} x^{2}}{15 d^{4} e^{4} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{8} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \left (d^{2}\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} - \frac{i x^{5}}{5 d^{7} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{x^{5}}{5 d^{7} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*Piecewise((-2*d**2/(15*d**4*e**4*sqrt(d**2 - e**2*x**2) - 30*d**2*e**6*x**2*sq
rt(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)) + e
*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))

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GIAC/XCAS [A]  time = 0.301329, size = 78, normalized size = 0.87 \[ \frac{{\left (2 \, d^{3} e^{\left (-4\right )} -{\left (\frac{3 \, x^{3} e}{d^{2}} + 5 \, d e^{\left (-2\right )}\right )} x^{2}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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