### 3.841 $$\int \frac{d+e x}{(d^2-e^2 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=56 $\frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}+\frac{d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0123644, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {639, 191} $\frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}+\frac{d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/(d^2 - e^2*x^2)^(5/2),x]

[Out]

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

Rule 639

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

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

Mathematica [A]  time = 0.0237811, size = 51, normalized size = 0.91 $\frac{d^2+2 d e x-2 e^2 x^2}{3 d^3 e (d-e x) \sqrt{d^2-e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)/(d^2 - e^2*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.044, size = 53, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) ^{2} \left ( -ex+d \right ) \left ( -2\,{e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) }{3\,{d}^{3}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(-e^2*x^2+d^2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.20168, size = 81, normalized size = 1.45 \begin{align*} \frac{x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d} + \frac{1}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} + \frac{2 \, x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.08311, size = 192, normalized size = 3.43 \begin{align*} \frac{e^{3} x^{3} - d e^{2} x^{2} - d^{2} e x + d^{3} -{\left (2 \, e^{2} x^{2} - 2 \, d e x - d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (d^{3} e^{4} x^{3} - d^{4} e^{3} x^{2} - d^{5} e^{2} x + d^{6} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 4.95444, size = 298, normalized size = 5.32 \begin{align*} d \left (\begin{cases} \frac{3 i d^{2} x}{- 3 d^{7} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{2 i e^{2} x^{3}}{- 3 d^{7} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\- \frac{3 d^{2} x}{- 3 d^{7} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{2 e^{2} x^{3}}{- 3 d^{7} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} - \frac{1}{- 3 d^{2} e^{2} \sqrt{d^{2} - e^{2} x^{2}} + 3 e^{4} x^{2} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 \left (d^{2}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

d*Piecewise((3*I*d**2*x/(-3*d**7*sqrt(-1 + e**2*x**2/d**2) + 3*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)) - 2*I
*e**2*x**3/(-3*d**7*sqrt(-1 + e**2*x**2/d**2) + 3*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Ab
s(d**2) > 1), (-3*d**2*x/(-3*d**7*sqrt(1 - e**2*x**2/d**2) + 3*d**5*e**2*x**2*sqrt(1 - e**2*x**2/d**2)) + 2*e*
*2*x**3/(-3*d**7*sqrt(1 - e**2*x**2/d**2) + 3*d**5*e**2*x**2*sqrt(1 - e**2*x**2/d**2)), True)) + e*Piecewise((
-1/(-3*d**2*e**2*sqrt(d**2 - e**2*x**2) + 3*e**4*x**2*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**2/(2*(d**2)**(5/
2)), True))

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Giac [A]  time = 1.42761, size = 70, normalized size = 1.25 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (x{\left (\frac{2 \, x^{2} e^{2}}{d^{3}} - \frac{3}{d}\right )} - e^{\left (-1\right )}\right )}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

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