\(\int \frac {x^2}{(d+e x) (d^2-e^2 x^2)^{3/2}} \, dx\) [130]

Optimal result

Integrand size = 27, antiderivative size = 60 \[ \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}}-\frac {x^2}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {869, 12, 267} \[ \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}}-\frac {x^2}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 267

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

Rule 869

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

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (2 d^2+2 d e x-e^2 x^2\right )}{3 d e^3 (d-e x) (d+e x)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.80

method	result	size
gosper	\(\frac {\left (-e x +d \right ) \left (-e^{2} x^{2}+2 d e x +2 d^{2}\right )}{3 d \,e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\)	\(48\)
trager	\(\frac {\left (-e^{2} x^{2}+2 d e x +2 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3 d \,e^{3} \left (e x +d \right )^{2} \left (-e x +d \right )}\)	\(57\)
default	\(\frac {1}{e^{3} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {x}{d \,e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {d^{2} \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{e^{3}}\)	\(149\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.72 \[ \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, e^{3} x^{3} + 2 \, d e^{2} x^{2} - 2 \, d^{2} e x - 2 \, d^{3} + {\left (e^{2} x^{2} - 2 \, d e x - 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d e^{6} x^{3} + d^{2} e^{5} x^{2} - d^{3} e^{4} x - d^{4} e^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.43 \[ \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {d}{3 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{4} x + \sqrt {-e^{2} x^{2} + d^{2}} d e^{3}\right )}} - \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2}} + \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} e^{3}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^2+2\,d\,e\,x-e^2\,x^2\right )}{3\,d\,e^3\,{\left (d+e\,x\right )}^2\,\left (d-e\,x\right )} \]

[In]

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

[Out]

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