\(\int \frac {(12-3 e^2 x^2)^{3/2}}{(2+e x)^{7/2}} \, dx\) [907]

Optimal result

Integrand size = 24, antiderivative size = 73 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{7/2}} \, dx=-\frac {9 \sqrt {3} \sqrt {2-e x}}{e}-\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (2+e x)}+\frac {18 \sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \]

[Out]

-3*(-e*x+2)^(3/2)*3^(1/2)/e/(e*x+2)+18*arctanh(1/2*(-e*x+2)^(1/2))*3^(1/2)/e-9*3^(1/2)*(-e*x+2)^(1/2)/e

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {641, 43, 52, 65, 212} \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{7/2}} \, dx=\frac {18 \sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e}-\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (e x+2)}-\frac {9 \sqrt {3} \sqrt {2-e x}}{e} \]

[In]

Int[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(7/2),x]

[Out]

(-9*Sqrt[3]*Sqrt[2 - e*x])/e - (3*Sqrt[3]*(2 - e*x)^(3/2))/(e*(2 + e*x)) + (18*Sqrt[3]*ArcTanh[Sqrt[2 - e*x]/2
])/e

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rubi steps \begin{align*} \text {integral}& = \int \frac {(6-3 e x)^{3/2}}{(2+e x)^2} \, dx \\ & = -\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (2+e x)}-\frac {9}{2} \int \frac {\sqrt {6-3 e x}}{2+e x} \, dx \\ & = -\frac {9 \sqrt {3} \sqrt {2-e x}}{e}-\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (2+e x)}-54 \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = -\frac {9 \sqrt {3} \sqrt {2-e x}}{e}-\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (2+e x)}+\frac {36 \text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{e} \\ & = -\frac {9 \sqrt {3} \sqrt {2-e x}}{e}-\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (2+e x)}+\frac {18 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{7/2}} \, dx=\frac {6 \sqrt {3} \left (-\frac {(4+e x) \sqrt {4-e^2 x^2}}{(2+e x)^{3/2}}+3 \text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )\right )}{e} \]

[In]

Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(7/2),x]

[Out]

(6*Sqrt[3]*(-(((4 + e*x)*Sqrt[4 - e^2*x^2])/(2 + e*x)^(3/2)) + 3*ArcTanh[(2*Sqrt[2 + e*x])/Sqrt[4 - e^2*x^2]])
)/e

Maple [A] (verified)

Time = 2.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.36

method	result	size
default	\(\frac {6 \sqrt {-x^{2} e^{2}+4}\, \left (3 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x -e x \sqrt {-3 e x +6}+6 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )-4 \sqrt {-3 e x +6}\right ) \sqrt {3}}{\left (e x +2\right )^{\frac {3}{2}} \sqrt {-3 e x +6}\, e}\)	\(99\)
risch	\(\frac {18 \left (e x -2\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}-\frac {72 \left (-\frac {\sqrt {-3 e x +6}}{2 \left (-3 e x -6\right )}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )}{4}\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{e \sqrt {-3 x^{2} e^{2}+12}}\)	\(141\)

[In]

int((-3*e^2*x^2+12)^(3/2)/(e*x+2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

6*(-e^2*x^2+4)^(1/2)*(3*3^(1/2)*arctanh(1/6*(-3*e*x+6)^(1/2)*3^(1/2))*e*x-e*x*(-3*e*x+6)^(1/2)+6*3^(1/2)*arcta
nh(1/6*(-3*e*x+6)^(1/2)*3^(1/2))-4*(-3*e*x+6)^(1/2))/(e*x+2)^(3/2)/(-3*e*x+6)^(1/2)*3^(1/2)/e

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (59) = 118\).

Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.67 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{7/2}} \, dx=\frac {3 \, {\left (3 \, \sqrt {3} {\left (e^{2} x^{2} + 4 \, e x + 4\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x - 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 2 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 4\right )} \sqrt {e x + 2}\right )}}{e^{3} x^{2} + 4 \, e^{2} x + 4 \, e} \]

[In]

integrate((-3*e^2*x^2+12)^(3/2)/(e*x+2)^(7/2),x, algorithm="fricas")

[Out]

3*(3*sqrt(3)*(e^2*x^2 + 4*e*x + 4)*log(-(3*e^2*x^2 - 12*e*x - 4*sqrt(3)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2) -
36)/(e^2*x^2 + 4*e*x + 4)) - 2*sqrt(-3*e^2*x^2 + 12)*(e*x + 4)*sqrt(e*x + 2))/(e^3*x^2 + 4*e^2*x + 4*e)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((-3*e**2*x**2+12)**(3/2)/(e*x+2)**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{7/2}} \, dx=\int { \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}}}{{\left (e x + 2\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((-3*e^2*x^2+12)^(3/2)/(e*x+2)^(7/2),x, algorithm="maxima")

[Out]

integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(7/2), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.88 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{7/2}} \, dx=-\frac {3 \, \sqrt {3} {\left (2 \, \sqrt {-e x + 2} + \frac {4 \, \sqrt {-e x + 2}}{e x + 2} - 3 \, \log \left (\sqrt {-e x + 2} + 2\right ) + 3 \, \log \left (-\sqrt {-e x + 2} + 2\right )\right )}}{e} \]

[In]

integrate((-3*e^2*x^2+12)^(3/2)/(e*x+2)^(7/2),x, algorithm="giac")

[Out]

-3*sqrt(3)*(2*sqrt(-e*x + 2) + 4*sqrt(-e*x + 2)/(e*x + 2) - 3*log(sqrt(-e*x + 2) + 2) + 3*log(-sqrt(-e*x + 2)
+ 2))/e

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{7/2}} \, dx=\int \frac {{\left (12-3\,e^2\,x^2\right )}^{3/2}}{{\left (e\,x+2\right )}^{7/2}} \,d x \]

[In]

int((12 - 3*e^2*x^2)^(3/2)/(e*x + 2)^(7/2),x)

[Out]

int((12 - 3*e^2*x^2)^(3/2)/(e*x + 2)^(7/2), x)