\(\int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx\) [280]

Optimal result

Integrand size = 20, antiderivative size = 151 \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=-\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x}}\right )}{\sqrt [4]{3} e}+\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x}}\right )}{\sqrt [4]{3} e}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x} \left (1+\frac {\sqrt {2+e x}}{\sqrt {2-e x}}\right )}\right )}{\sqrt [4]{3} e} \] Output:

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

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{4-e^2 x^2}}{\sqrt {2-e x}-\sqrt {2+e x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{4-e^2 x^2}}{\sqrt {2-e x}+\sqrt {2+e x}}\right )\right )}{\sqrt [4]{3} e} \] Input:

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

Output:

(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)
 

Rubi [A] (warning: unable to verify)

Time = 0.30 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.26, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {73, 27, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 73

\(\displaystyle -\frac {4 \int \frac {3^{3/4} \sqrt {6-3 e x}}{(3 e x+6)^{3/4}}d\sqrt [4]{6-3 e x}}{3 e}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {4 \int \frac {\sqrt {6-3 e x}}{(3 e x+6)^{3/4}}d\sqrt [4]{6-3 e x}}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 854

\(\displaystyle -\frac {4 \int \frac {\sqrt {6-3 e x}}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 826

\(\displaystyle -\frac {4 \left (\frac {1}{2} \int \frac {\sqrt {6-3 e x}+1}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}-\frac {1}{2} \int \frac {1-\sqrt {6-3 e x}}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {6-3 e x}-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+\frac {1}{2} \int \frac {1}{\sqrt {6-3 e x}+\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )-\frac {1}{2} \int \frac {1-\sqrt {6-3 e x}}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {6-3 e x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {6-3 e x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {6-3 e x}}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {6-3 e x}}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {6-3 e x}-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {6-3 e x}+\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{2 \sqrt {2}}\right )\right )}{\sqrt [4]{3} e}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 217

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 826

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

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[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])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
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 1479

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=-\frac {2 \cdot 12^{\frac {3}{4}} \arctan \left (\frac {3 \, e x + 12^{\frac {1}{4}} {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}} - 6}{3 \, {\left (e x - 2\right )}}\right ) + 2 \cdot 12^{\frac {3}{4}} \arctan \left (-\frac {3 \, e x - 12^{\frac {1}{4}} {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}} - 6}{3 \, {\left (e x - 2\right )}}\right ) + 12^{\frac {3}{4}} \log \left (\frac {12^{\frac {3}{4}} {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}} + 6 \, \sqrt {3} {\left (e x - 2\right )} - 6 \, \sqrt {e x + 2} \sqrt {-3 \, e x + 6}}{e x - 2}\right ) - 12^{\frac {3}{4}} \log \left (-\frac {12^{\frac {3}{4}} {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}} - 6 \, \sqrt {3} {\left (e x - 2\right )} + 6 \, \sqrt {e x + 2} \sqrt {-3 \, e x + 6}}{e x - 2}\right )}{12 \, e} \] Input:

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

Output:

-1/12*(2*12^(3/4)*arctan(1/3*(3*e*x + 12^(1/4)*(e*x + 2)^(1/4)*(-3*e*x + 6 
)^(3/4) - 6)/(e*x - 2)) + 2*12^(3/4)*arctan(-1/3*(3*e*x - 12^(1/4)*(e*x + 
2)^(1/4)*(-3*e*x + 6)^(3/4) - 6)/(e*x - 2)) + 12^(3/4)*log((12^(3/4)*(e*x 
+ 2)^(1/4)*(-3*e*x + 6)^(3/4) + 6*sqrt(3)*(e*x - 2) - 6*sqrt(e*x + 2)*sqrt 
(-3*e*x + 6))/(e*x - 2)) - 12^(3/4)*log(-(12^(3/4)*(e*x + 2)^(1/4)*(-3*e*x 
 + 6)^(3/4) - 6*sqrt(3)*(e*x - 2) + 6*sqrt(e*x + 2)*sqrt(-3*e*x + 6))/(e*x 
 - 2)))/e
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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