\(\int \frac {d+e x}{(d^2-e^2 x^2)^{5/2}} \, dx\) [841]

Optimal result

Integrand size = 22, antiderivative size = 56 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {653, 197} \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}} \]

[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 197

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]

Rule 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.04 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\left (d^2+2 d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{3 d^3 e (d-e x)^2 (d+e x)} \]

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

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95

method	result	size
gosper	\(\frac {\left (e x +d \right )^{2} \left (-e x +d \right ) \left (-2 x^{2} e^{2}+2 d e x +d^{2}\right )}{3 d^{3} e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\)	\(53\)
trager	\(\frac {\left (-2 x^{2} e^{2}+2 d e x +d^{2}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{3 d^{3} \left (-e x +d \right )^{2} e \left (e x +d \right )}\)	\(55\)
default	\(d \left (\frac {x}{3 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}\right )+\frac {1}{3 e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}\)	\(64\)

[In]

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

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (48) = 96\).

Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.82 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {e^{3} x^{3} - d e^{2} x^{2} - d^{2} e x + d^{3} - {\left (2 \, e^{2} x^{2} - 2 \, d e x - d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{3} e^{4} x^{3} - d^{4} e^{3} x^{2} - d^{5} e^{2} x + d^{6} e\right )}} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.66 (sec) , antiderivative size = 296, normalized size of antiderivative = 5.29 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=d \left (\begin {cases} \frac {3 i d^{2} x}{- 3 d^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {2 i e^{2} x^{3}}{- 3 d^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {3 d^{2} x}{- 3 d^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {2 e^{2} x^{3}}{- 3 d^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} - \frac {1}{- 3 d^{2} e^{2} \sqrt {d^{2} - e^{2} x^{2}} + 3 e^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{2}}{2 \left (d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \]

[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/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))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {1}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} + \frac {2 \, x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} \]

[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)

Giac [F]

\[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {e x + d}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.81 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2+2\,d\,e\,x-2\,e^2\,x^2\right )}{3\,d^3\,e\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^2} \]

[In]

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

[Out]

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