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

Optimal. Leaf size=83 \[ -\frac{2 x}{15 d^4 e \sqrt{d^2-e^2 x^2}}-\frac{x}{15 d^2 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{d+e x}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.022972, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {778, 192, 191} \[ -\frac{2 x}{15 d^4 e \sqrt{d^2-e^2 x^2}}-\frac{x}{15 d^2 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{d+e x}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{x (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{d+e x}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 e}\\ &=\frac{d+e x}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x}{15 d^2 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2 e}\\ &=\frac{d+e x}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x}{15 d^2 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{15 d^4 e \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0389166, size = 82, normalized size = 0.99 \[ \frac{3 d^2 e^2 x^2-3 d^3 e x+3 d^4+2 d e^3 x^3-2 e^4 x^4}{15 d^4 e^2 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.049, size = 77, normalized size = 0.9 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( -2\,{e}^{4}{x}^{4}+2\,{e}^{3}{x}^{3}d+3\,{x}^{2}{d}^{2}{e}^{2}-3\,x{d}^{3}e+3\,{d}^{4} \right ) }{15\,{d}^{4}{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.00599, size = 117, normalized size = 1.41 \begin{align*} \frac{x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2} e} - \frac{2 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4} e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.01013, size = 339, normalized size = 4.08 \begin{align*} \frac{3 \, e^{5} x^{5} - 3 \, d e^{4} x^{4} - 6 \, d^{2} e^{3} x^{3} + 6 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x - 3 \, d^{5} +{\left (2 \, e^{4} x^{4} - 2 \, d e^{3} x^{3} - 3 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x - 3 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{4} e^{7} x^{5} - d^{5} e^{6} x^{4} - 2 \, d^{6} e^{5} x^{3} + 2 \, d^{7} e^{4} x^{2} + d^{8} e^{3} x - d^{9} e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 11.4744, size = 434, normalized size = 5.23 \begin{align*} d \left (\begin{cases} \frac{1}{5 d^{4} e^{2} \sqrt{d^{2} - e^{2} x^{2}} - 10 d^{2} e^{4} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 5 e^{6} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 \left (d^{2}\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) + e \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}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*Piecewise((1/(5*d**4*e**2*sqrt(d**2 - e**2*x**2) - 10*d**2*e**4*x**2*sqrt(d**2 - e**2*x**2) + 5*e**6*x**4*sq
rt(d**2 - e**2*x**2)), Ne(e, 0)), (x**2/(2*(d**2)**(7/2)), True)) + e*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)/Abs(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))

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Giac [A]  time = 1.18807, size = 77, normalized size = 0.93 \begin{align*} \frac{{\left (x^{3}{\left (\frac{2 \, x^{2} e^{3}}{d^{4}} - \frac{5 \, e}{d^{2}}\right )} - 3 \, d e^{\left (-2\right )}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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