\(\int \frac {x}{(d+e x) (d^2-e^2 x^2)^{3/2}} \, dx\) [131]

Optimal result

Integrand size = 25, antiderivative size = 58 \[ \int \frac {x}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {x}{3 d^2 e \sqrt {d^2-e^2 x^2}}+\frac {1}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {807, 197} \[ \int \frac {x}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {x}{3 d^2 e \sqrt {d^2-e^2 x^2}}+\frac {1}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}} \]

[In]

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

[Out]

x/(3*d^2*e*Sqrt[d^2 - e^2*x^2]) + 1/(3*e^2*(d + e*x)*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 807

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d
+ e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m + p + 1))), x] + Dist[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d
)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2
+ a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) &&
NeQ[m + p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 e} \\ & = \frac {x}{3 d^2 e \sqrt {d^2-e^2 x^2}}+\frac {1}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int \frac {x}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (d^2+d e x+e^2 x^2\right )}{3 d^2 e^2 (d-e x) (d+e x)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.76

method	result	size
gosper	\(\frac {\left (-e x +d \right ) \left (e^{2} x^{2}+d e x +d^{2}\right )}{3 d^{2} e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\)	\(44\)
trager	\(\frac {\left (e^{2} x^{2}+d e x +d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{2} e^{2} \left (e x +d \right )^{2} \left (-e x +d \right )}\)	\(53\)
default	\(\frac {x}{d^{2} e \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {d \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{e^{2}}\)	\(129\)

[In]

int(x/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (50) = 100\).

Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.74 \[ \int \frac {x}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3} - {\left (e^{2} x^{2} + d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{2} e^{5} x^{3} + d^{3} e^{4} x^{2} - d^{4} e^{3} x - d^{5} e^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate(x/(e*x+d)/(-e**2*x**2+d**2)**(3/2),x)

[Out]

Integral(x/((-(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.16 \[ \int \frac {x}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {1}{3 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{3} x + \sqrt {-e^{2} x^{2} + d^{2}} d e^{2}\right )}} + \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.90 \[ \int \frac {x}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2+d\,e\,x+e^2\,x^2\right )}{3\,d^2\,e^2\,{\left (d+e\,x\right )}^2\,\left (d-e\,x\right )} \]

[In]

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

[Out]

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