\(\int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx\) [1170]

Optimal result

Integrand size = 20, antiderivative size = 241 \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt [4]{3} e}-\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt [4]{3} e}-\frac {\log \left (\frac {\sqrt {6-3 e x}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\log \left (\frac {\sqrt {6-3 e x}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} \sqrt [4]{3} e} \]

[Out]

-1/6*ln(3^(1/2)-(-e*x+2)^(1/4)*6^(1/2)/(e*x+2)^(1/4)+3^(1/2)*(-e*x+2)^(1/2)/(e*x+2)^(1/2))*3^(3/4)/e*2^(1/2)+1
/6*ln(3^(1/2)+(-e*x+2)^(1/4)*6^(1/2)/(e*x+2)^(1/4)+3^(1/2)*(-e*x+2)^(1/2)/(e*x+2)^(1/2))*3^(3/4)/e*2^(1/2)-1/3
*arctan(-1+(-e*x+2)^(1/4)*2^(1/2)/(e*x+2)^(1/4))*2^(1/2)*3^(3/4)/e-1/3*arctan(1+(-e*x+2)^(1/4)*2^(1/2)/(e*x+2)
^(1/4))*2^(1/2)*3^(3/4)/e

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt [4]{3} e}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt [4]{3} e}-\frac {\log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} \sqrt [4]{3} e} \]

[In]

Int[1/((6 - 3*e*x)^(1/4)*(2 + e*x)^(3/4)),x]

[Out]

(Sqrt[2]*ArcTan[1 - (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)])/(3^(1/4)*e) - (Sqrt[2]*ArcTan[1 + (Sqrt[2]*(2
- e*x)^(1/4))/(2 + e*x)^(1/4)])/(3^(1/4)*e) - Log[(Sqrt[6 - 3*e*x] - Sqrt[6]*(2 - e*x)^(1/4)*(2 + e*x)^(1/4) +
Sqrt[3]*Sqrt[2 + e*x])/Sqrt[2 + e*x]]/(Sqrt[2]*3^(1/4)*e) + Log[(Sqrt[6 - 3*e*x] + Sqrt[6]*(2 - e*x)^(1/4)*(2
+ e*x)^(1/4) + Sqrt[3]*Sqrt[2 + e*x])/Sqrt[2 + e*x]]/(Sqrt[2]*3^(1/4)*e)

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 210

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

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = -\frac {4 \text {Subst}\left (\int \frac {x^2}{\left (4-\frac {x^4}{3}\right )^{3/4}} \, dx,x,\sqrt [4]{6-3 e x}\right )}{3 e} \\ & = -\frac {4 \text {Subst}\left (\int \frac {x^2}{1+\frac {x^4}{3}} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{3 e} \\ & = \frac {2 \text {Subst}\left (\int \frac {\sqrt {3}-x^2}{1+\frac {x^4}{3}} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{3 e}-\frac {2 \text {Subst}\left (\int \frac {\sqrt {3}+x^2}{1+\frac {x^4}{3}} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{3 e} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{e}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{e}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{3}+2 x}{-\sqrt {3}-\sqrt {2} \sqrt [4]{3} x-x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} \sqrt [4]{3} e}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{3}-2 x}{-\sqrt {3}+\sqrt {2} \sqrt [4]{3} x-x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} \sqrt [4]{3} e} \\ & = -\frac {\log \left (\frac {\sqrt {2-e x}-\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\log \left (\frac {\sqrt {2-e x}+\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} \sqrt [4]{3} e}-\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt [4]{3} e}+\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt [4]{3} e} \\ & = \frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt [4]{3} e}-\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt [4]{3} e}-\frac {\log \left (\frac {\sqrt {2-e x}-\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\log \left (\frac {\sqrt {2-e x}+\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} \sqrt [4]{3} e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{4-e^2 x^2}}{\sqrt {2-e x}-\sqrt {2+e x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{4-e^2 x^2}}{\sqrt {2-e x}+\sqrt {2+e x}}\right )\right )}{\sqrt [4]{3} e} \]

[In]

Integrate[1/((6 - 3*e*x)^(1/4)*(2 + e*x)^(3/4)),x]

[Out]

(Sqrt[2]*(ArcTan[(Sqrt[2]*(4 - e^2*x^2)^(1/4))/(Sqrt[2 - e*x] - Sqrt[2 + e*x])] + ArcTanh[(Sqrt[2]*(4 - e^2*x^
2)^(1/4))/(Sqrt[2 - e*x] + Sqrt[2 + e*x])]))/(3^(1/4)*e)

Maple [F]

\[\int \frac {1}{\left (-3 e x +6\right )^{\frac {1}{4}} \left (e x +2\right )^{\frac {3}{4}}}d x\]

[In]

int(1/(-3*e*x+6)^(1/4)/(e*x+2)^(3/4),x)

[Out]

int(1/(-3*e*x+6)^(1/4)/(e*x+2)^(3/4),x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=-\left (\frac {1}{3}\right )^{\frac {1}{4}} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {3 \, \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e^{2} x - 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}}}{e x - 2}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e^{2} x - 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}}}{e x - 2}\right ) + i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (i \, e^{2} x - 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}}}{e x - 2}\right ) - i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-i \, e^{2} x + 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}}}{e x - 2}\right ) \]

[In]

integrate(1/(-3*e*x+6)^(1/4)/(e*x+2)^(3/4),x, algorithm="fricas")

[Out]

-(1/3)^(1/4)*(-1/e^4)^(1/4)*log((3*(1/3)^(1/4)*(e^2*x - 2*e)*(-1/e^4)^(1/4) + (e*x + 2)^(1/4)*(-3*e*x + 6)^(3/
4))/(e*x - 2)) + (1/3)^(1/4)*(-1/e^4)^(1/4)*log(-(3*(1/3)^(1/4)*(e^2*x - 2*e)*(-1/e^4)^(1/4) - (e*x + 2)^(1/4)
*(-3*e*x + 6)^(3/4))/(e*x - 2)) + I*(1/3)^(1/4)*(-1/e^4)^(1/4)*log(-(3*(1/3)^(1/4)*(I*e^2*x - 2*I*e)*(-1/e^4)^
(1/4) - (e*x + 2)^(1/4)*(-3*e*x + 6)^(3/4))/(e*x - 2)) - I*(1/3)^(1/4)*(-1/e^4)^(1/4)*log(-(3*(1/3)^(1/4)*(-I*
e^2*x + 2*I*e)*(-1/e^4)^(1/4) - (e*x + 2)^(1/4)*(-3*e*x + 6)^(3/4))/(e*x - 2))

Sympy [F]

\[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\frac {3^{\frac {3}{4}} \int \frac {1}{\sqrt [4]{- e x + 2} \left (e x + 2\right )^{\frac {3}{4}}}\, dx}{3} \]

[In]

integrate(1/(-3*e*x+6)**(1/4)/(e*x+2)**(3/4),x)

[Out]

3**(3/4)*Integral(1/((-e*x + 2)**(1/4)*(e*x + 2)**(3/4)), x)/3

Maxima [F]

\[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\int { \frac {1}{{\left (e x + 2\right )}^{\frac {3}{4}} {\left (-3 \, e x + 6\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(-3*e*x+6)^(1/4)/(e*x+2)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((e*x + 2)^(3/4)*(-3*e*x + 6)^(1/4)), x)

Giac [F]

\[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\int { \frac {1}{{\left (e x + 2\right )}^{\frac {3}{4}} {\left (-3 \, e x + 6\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(-3*e*x+6)^(1/4)/(e*x+2)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((e*x + 2)^(3/4)*(-3*e*x + 6)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\int \frac {1}{{\left (e\,x+2\right )}^{3/4}\,{\left (6-3\,e\,x\right )}^{1/4}} \,d x \]

[In]

int(1/((e*x + 2)^(3/4)*(6 - 3*e*x)^(1/4)),x)

[Out]

int(1/((e*x + 2)^(3/4)*(6 - 3*e*x)^(1/4)), x)