Integrand size = 20, antiderivative size = 241 \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt [4]{3} e}-\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt [4]{3} e}-\frac {\log \left (\frac {\sqrt {6-3 e x}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\log \left (\frac {\sqrt {6-3 e x}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} \sqrt [4]{3} e} \]
-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*a rctan(-1+(-e*x+2)^(1/4)*2^(1/2)/(e*x+2)^(1/4))*2^(1/2)*3^(3/4)/e-1/3*arcta n(1+(-e*x+2)^(1/4)*2^(1/2)/(e*x+2)^(1/4))*2^(1/2)*3^(3/4)/e
Time = 0.99 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{4-e^2 x^2}}{\sqrt {2-e x}-\sqrt {2+e x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{4-e^2 x^2}}{\sqrt {2-e x}+\sqrt {2+e x}}\right )\right )}{\sqrt [4]{3} e} \]
(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)
Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.79, 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 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {6-3 e x}-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {6-3 e x}+\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{2 \sqrt {2}}\right )\right )}{\sqrt [4]{3} e}\) |
(-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)
3.12.70.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
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]]))
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]
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]
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]
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]
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]
\[\int \frac {1}{\left (-3 e x +6\right )^{\frac {1}{4}} \left (e x +2\right )^{\frac {3}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=-\left (\frac {1}{3}\right )^{\frac {1}{4}} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {3 \, \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e^{2} x - 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}}}{e x - 2}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e^{2} x - 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}}}{e x - 2}\right ) + i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (i \, e^{2} x - 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}}}{e x - 2}\right ) - i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-i \, e^{2} x + 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}}}{e x - 2}\right ) \]
-(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))
\[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\frac {3^{\frac {3}{4}} \int \frac {1}{\sqrt [4]{- e x + 2} \left (e x + 2\right )^{\frac {3}{4}}}\, dx}{3} \]
\[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\int { \frac {1}{{\left (e x + 2\right )}^{\frac {3}{4}} {\left (-3 \, e x + 6\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\int { \frac {1}{{\left (e x + 2\right )}^{\frac {3}{4}} {\left (-3 \, e x + 6\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx=\int \frac {1}{{\left (e\,x+2\right )}^{3/4}\,{\left (6-3\,e\,x\right )}^{1/4}} \,d x \]