Integrand size = 17, antiderivative size = 85 \[ \int \frac {1}{(2+3 x)^{3/4} (4+3 x)} \, dx=-\frac {\arctan \left (\frac {\sqrt {2}-\sqrt {2+3 x}}{2^{3/4} \sqrt [4]{2+3 x}}\right )}{3 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{2+3 x}}{\sqrt {2}+\sqrt {2+3 x}}\right )}{3 \sqrt [4]{2}} \] Output:
-1/6*arctan(1/2*(2^(1/2)-(2+3*x)^(1/2))*2^(1/4)/(2+3*x)^(1/4))*2^(3/4)+1/6 *arctanh(2^(3/4)*(2+3*x)^(1/4)/(2^(1/2)+(2+3*x)^(1/2)))*2^(3/4)
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(2+3 x)^{3/4} (4+3 x)} \, dx=\frac {\arctan \left (\frac {-\sqrt {2}+\sqrt {2+3 x}}{2^{3/4} \sqrt [4]{2+3 x}}\right )+\text {arctanh}\left (\frac {2 \sqrt [4]{4+6 x}}{2+\sqrt {4+6 x}}\right )}{3 \sqrt [4]{2}} \] Input:
Integrate[1/((2 + 3*x)^(3/4)*(4 + 3*x)),x]
Output:
(ArcTan[(-Sqrt[2] + Sqrt[2 + 3*x])/(2^(3/4)*(2 + 3*x)^(1/4))] + ArcTanh[(2 *(4 + 6*x)^(1/4))/(2 + Sqrt[4 + 6*x])])/(3*2^(1/4))
Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.84, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {73, 755, 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.
| \(\displaystyle \int \frac {1}{(3 x+2)^{3/4} (3 x+4)} \, dx\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {4}{3} \int \frac {1}{3 x+4}d\sqrt [4]{3 x+2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {4}{3} \left (\frac {\int \frac {\sqrt {2}-\sqrt {3 x+2}}{3 x+4}d\sqrt [4]{3 x+2}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {3 x+2}+\sqrt {2}}{3 x+4}d\sqrt [4]{3 x+2}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {4}{3} \left (\frac {\int \frac {\sqrt {2}-\sqrt {3 x+2}}{3 x+4}d\sqrt [4]{3 x+2}}{2 \sqrt {2}}+\frac {\frac {1}{2} \int \frac {1}{\sqrt {3 x+2}-2^{3/4} \sqrt [4]{3 x+2}+\sqrt {2}}d\sqrt [4]{3 x+2}+\frac {1}{2} \int \frac {1}{\sqrt {3 x+2}+2^{3/4} \sqrt [4]{3 x+2}+\sqrt {2}}d\sqrt [4]{3 x+2}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {4}{3} \left (\frac {\frac {\int \frac {1}{-\sqrt {3 x+2}-1}d\left (1-\sqrt [4]{2} \sqrt [4]{3 x+2}\right )}{2^{3/4}}-\frac {\int \frac {1}{-\sqrt {3 x+2}-1}d\left (\sqrt [4]{2} \sqrt [4]{3 x+2}+1\right )}{2^{3/4}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}-\sqrt {3 x+2}}{3 x+4}d\sqrt [4]{3 x+2}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4}{3} \left (\frac {\int \frac {\sqrt {2}-\sqrt {3 x+2}}{3 x+4}d\sqrt [4]{3 x+2}}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\sqrt [4]{2} \sqrt [4]{3 x+2}+1\right )}{2^{3/4}}-\frac {\arctan \left (1-\sqrt [4]{2} \sqrt [4]{3 x+2}\right )}{2^{3/4}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {4}{3} \left (\frac {-\frac {\int -\frac {2^{3/4}-2 \sqrt [4]{3 x+2}}{\sqrt {3 x+2}-2^{3/4} \sqrt [4]{3 x+2}+\sqrt {2}}d\sqrt [4]{3 x+2}}{2\ 2^{3/4}}-\frac {\int -\frac {2^{3/4} \left (\sqrt [4]{2} \sqrt [4]{3 x+2}+1\right )}{\sqrt {3 x+2}+2^{3/4} \sqrt [4]{3 x+2}+\sqrt {2}}d\sqrt [4]{3 x+2}}{2\ 2^{3/4}}}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\sqrt [4]{2} \sqrt [4]{3 x+2}+1\right )}{2^{3/4}}-\frac {\arctan \left (1-\sqrt [4]{2} \sqrt [4]{3 x+2}\right )}{2^{3/4}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{3} \left (\frac {\frac {\int \frac {2^{3/4}-2 \sqrt [4]{3 x+2}}{\sqrt {3 x+2}-2^{3/4} \sqrt [4]{3 x+2}+\sqrt {2}}d\sqrt [4]{3 x+2}}{2\ 2^{3/4}}+\frac {\int \frac {2^{3/4} \left (\sqrt [4]{2} \sqrt [4]{3 x+2}+1\right )}{\sqrt {3 x+2}+2^{3/4} \sqrt [4]{3 x+2}+\sqrt {2}}d\sqrt [4]{3 x+2}}{2\ 2^{3/4}}}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\sqrt [4]{2} \sqrt [4]{3 x+2}+1\right )}{2^{3/4}}-\frac {\arctan \left (1-\sqrt [4]{2} \sqrt [4]{3 x+2}\right )}{2^{3/4}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{3} \left (\frac {\frac {\int \frac {2^{3/4}-2 \sqrt [4]{3 x+2}}{\sqrt {3 x+2}-2^{3/4} \sqrt [4]{3 x+2}+\sqrt {2}}d\sqrt [4]{3 x+2}}{2\ 2^{3/4}}+\frac {1}{2} \int \frac {\sqrt [4]{2} \sqrt [4]{3 x+2}+1}{\sqrt {3 x+2}+2^{3/4} \sqrt [4]{3 x+2}+\sqrt {2}}d\sqrt [4]{3 x+2}}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\sqrt [4]{2} \sqrt [4]{3 x+2}+1\right )}{2^{3/4}}-\frac {\arctan \left (1-\sqrt [4]{2} \sqrt [4]{3 x+2}\right )}{2^{3/4}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4}{3} \left (\frac {\frac {\arctan \left (\sqrt [4]{2} \sqrt [4]{3 x+2}+1\right )}{2^{3/4}}-\frac {\arctan \left (1-\sqrt [4]{2} \sqrt [4]{3 x+2}\right )}{2^{3/4}}}{2 \sqrt {2}}+\frac {\frac {\log \left (\sqrt {3 x+2}+2^{3/4} \sqrt [4]{3 x+2}+\sqrt {2}\right )}{2\ 2^{3/4}}-\frac {\log \left (\sqrt {3 x+2}-2^{3/4} \sqrt [4]{3 x+2}+\sqrt {2}\right )}{2\ 2^{3/4}}}{2 \sqrt {2}}\right )\) |
Input:
Int[1/((2 + 3*x)^(3/4)*(4 + 3*x)),x]
Output:
(4*((-(ArcTan[1 - 2^(1/4)*(2 + 3*x)^(1/4)]/2^(3/4)) + ArcTan[1 + 2^(1/4)*( 2 + 3*x)^(1/4)]/2^(3/4))/(2*Sqrt[2]) + (-1/2*Log[Sqrt[2] - 2^(3/4)*(2 + 3* x)^(1/4) + Sqrt[2 + 3*x]]/2^(3/4) + Log[Sqrt[2] + 2^(3/4)*(2 + 3*x)^(1/4) + Sqrt[2 + 3*x]]/(2*2^(3/4)))/(2*Sqrt[2])))/3
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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[((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]
Time = 0.63 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(\frac {2^{\frac {3}{4}} \left (\ln \left (\frac {\sqrt {2+3 x}+2^{\frac {3}{4}} \left (2+3 x \right )^{\frac {1}{4}}+\sqrt {2}}{\sqrt {2+3 x}-2^{\frac {3}{4}} \left (2+3 x \right )^{\frac {1}{4}}+\sqrt {2}}\right )+2 \arctan \left (2^{\frac {1}{4}} \left (2+3 x \right )^{\frac {1}{4}}+1\right )+2 \arctan \left (2^{\frac {1}{4}} \left (2+3 x \right )^{\frac {1}{4}}-1\right )\right )}{12}\) | \(88\) |
| default | \(\frac {2^{\frac {3}{4}} \left (\ln \left (\frac {\sqrt {2+3 x}+2^{\frac {3}{4}} \left (2+3 x \right )^{\frac {1}{4}}+\sqrt {2}}{\sqrt {2+3 x}-2^{\frac {3}{4}} \left (2+3 x \right )^{\frac {1}{4}}+\sqrt {2}}\right )+2 \arctan \left (2^{\frac {1}{4}} \left (2+3 x \right )^{\frac {1}{4}}+1\right )+2 \arctan \left (2^{\frac {1}{4}} \left (2+3 x \right )^{\frac {1}{4}}-1\right )\right )}{12}\) | \(88\) |
| pseudoelliptic | \(\frac {2^{\frac {3}{4}} \left (\ln \left (\frac {\sqrt {2+3 x}+2^{\frac {3}{4}} \left (2+3 x \right )^{\frac {1}{4}}+\sqrt {2}}{\sqrt {2+3 x}-2^{\frac {3}{4}} \left (2+3 x \right )^{\frac {1}{4}}+\sqrt {2}}\right )+2 \arctan \left (2^{\frac {1}{4}} \left (2+3 x \right )^{\frac {1}{4}}+1\right )+2 \arctan \left (2^{\frac {1}{4}} \left (2+3 x \right )^{\frac {1}{4}}-1\right )\right )}{12}\) | \(88\) |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (2+3 x \right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {2+3 x}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (2+3 x \right )^{\frac {1}{4}}+3 x}{3 x +4}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (2+3 x \right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {2+3 x}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} \left (2+3 x \right )^{\frac {1}{4}}+3 x}{3 x +4}\right )}{6}\) | \(169\) |
Input:
int(1/(2+3*x)^(3/4)/(3*x+4),x,method=_RETURNVERBOSE)
Output:
1/12*2^(3/4)*(ln(((2+3*x)^(1/2)+2^(3/4)*(2+3*x)^(1/4)+2^(1/2))/((2+3*x)^(1 /2)-2^(3/4)*(2+3*x)^(1/4)+2^(1/2)))+2*arctan(2^(1/4)*(2+3*x)^(1/4)+1)+2*ar ctan(2^(1/4)*(2+3*x)^(1/4)-1))
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(2+3 x)^{3/4} (4+3 x)} \, dx=\frac {1}{6} \cdot 2^{\frac {3}{4}} \arctan \left (2^{\frac {1}{4}} {\left (3 \, x + 2\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{6} \cdot 2^{\frac {3}{4}} \arctan \left (2^{\frac {1}{4}} {\left (3 \, x + 2\right )}^{\frac {1}{4}} - 1\right ) + \frac {1}{12} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (3 \, x + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {3 \, x + 2}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (3 \, x + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {3 \, x + 2}\right ) \] Input:
integrate(1/(2+3*x)^(3/4)/(4+3*x),x, algorithm="fricas")
Output:
1/6*2^(3/4)*arctan(2^(1/4)*(3*x + 2)^(1/4) + 1) + 1/6*2^(3/4)*arctan(2^(1/ 4)*(3*x + 2)^(1/4) - 1) + 1/12*2^(3/4)*log(2^(3/4)*(3*x + 2)^(1/4) + sqrt( 2) + sqrt(3*x + 2)) - 1/12*2^(3/4)*log(-2^(3/4)*(3*x + 2)^(1/4) + sqrt(2) + sqrt(3*x + 2))
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.64 \[ \int \frac {1}{(2+3 x)^{3/4} (4+3 x)} \, dx=- \frac {\sqrt [4]{2} e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \cdot \sqrt [4]{3} \sqrt [4]{x + \frac {2}{3}} e^{\frac {i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {1}{4}\right )}{24 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt [4]{2} i e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \cdot \sqrt [4]{3} \sqrt [4]{x + \frac {2}{3}} e^{\frac {3 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {1}{4}\right )}{24 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt [4]{2} e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \cdot \sqrt [4]{3} \sqrt [4]{x + \frac {2}{3}} e^{\frac {5 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {1}{4}\right )}{24 \Gamma \left (\frac {5}{4}\right )} - \frac {\sqrt [4]{2} i e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \cdot \sqrt [4]{3} \sqrt [4]{x + \frac {2}{3}} e^{\frac {7 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {1}{4}\right )}{24 \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate(1/(2+3*x)**(3/4)/(4+3*x),x)
Output:
-2**(1/4)*exp(-I*pi/4)*log(-2**(3/4)*3**(1/4)*(x + 2/3)**(1/4)*exp_polar(I *pi/4)/2 + 1)*gamma(1/4)/(24*gamma(5/4)) + 2**(1/4)*I*exp(-I*pi/4)*log(-2* *(3/4)*3**(1/4)*(x + 2/3)**(1/4)*exp_polar(3*I*pi/4)/2 + 1)*gamma(1/4)/(24 *gamma(5/4)) + 2**(1/4)*exp(-I*pi/4)*log(-2**(3/4)*3**(1/4)*(x + 2/3)**(1/ 4)*exp_polar(5*I*pi/4)/2 + 1)*gamma(1/4)/(24*gamma(5/4)) - 2**(1/4)*I*exp( -I*pi/4)*log(-2**(3/4)*3**(1/4)*(x + 2/3)**(1/4)*exp_polar(7*I*pi/4)/2 + 1 )*gamma(1/4)/(24*gamma(5/4))
Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(2+3 x)^{3/4} (4+3 x)} \, dx=\frac {1}{6} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (3 \, x + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (3 \, x + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (3 \, x + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {3 \, x + 2}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (3 \, x + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {3 \, x + 2}\right ) \] Input:
integrate(1/(2+3*x)^(3/4)/(4+3*x),x, algorithm="maxima")
Output:
1/6*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(3*x + 2)^(1/4))) + 1/6*2^(3/4 )*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(3*x + 2)^(1/4))) + 1/12*2^(3/4)*log(2^ (3/4)*(3*x + 2)^(1/4) + sqrt(2) + sqrt(3*x + 2)) - 1/12*2^(3/4)*log(-2^(3/ 4)*(3*x + 2)^(1/4) + sqrt(2) + sqrt(3*x + 2))
Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(2+3 x)^{3/4} (4+3 x)} \, dx=\frac {1}{6} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (3 \, x + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (3 \, x + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (3 \, x + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {3 \, x + 2}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (3 \, x + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {3 \, x + 2}\right ) \] Input:
integrate(1/(2+3*x)^(3/4)/(4+3*x),x, algorithm="giac")
Output:
1/6*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(3*x + 2)^(1/4))) + 1/6*2^(3/4 )*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(3*x + 2)^(1/4))) + 1/12*2^(3/4)*log(2^ (3/4)*(3*x + 2)^(1/4) + sqrt(2) + sqrt(3*x + 2)) - 1/12*2^(3/4)*log(-2^(3/ 4)*(3*x + 2)^(1/4) + sqrt(2) + sqrt(3*x + 2))
Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(2+3 x)^{3/4} (4+3 x)} \, dx=2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (3\,x+2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (3\,x+2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right ) \] Input:
int(1/((3*x + 2)^(3/4)*(3*x + 4)),x)
Output:
2^(3/4)*atan(2^(1/4)*(3*x + 2)^(1/4)*(1/2 - 1i/2))*(1/6 + 1i/6) + 2^(3/4)* atan(2^(1/4)*(3*x + 2)^(1/4)*(1/2 + 1i/2))*(1/6 - 1i/6)
Time = 0.17 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(2+3 x)^{3/4} (4+3 x)} \, dx=\frac {\sqrt {2}\, 2^{\frac {1}{4}} \left (2 \mathit {atan} \left (\frac {\left (2 \left (3 x +2\right )^{\frac {1}{4}}-\sqrt {2}\, 2^{\frac {1}{4}}\right ) 2^{\frac {3}{4}}}{2 \sqrt {2}}\right )+2 \mathit {atan} \left (\frac {\left (2 \left (3 x +2\right )^{\frac {1}{4}}+\sqrt {2}\, 2^{\frac {1}{4}}\right ) 2^{\frac {3}{4}}}{2 \sqrt {2}}\right )-\mathrm {log}\left (-\left (3 x +2\right )^{\frac {1}{4}} \sqrt {2}\, 2^{\frac {1}{4}}+\sqrt {3 x +2}+\sqrt {2}\right )+\mathrm {log}\left (\left (3 x +2\right )^{\frac {1}{4}} \sqrt {2}\, 2^{\frac {1}{4}}+\sqrt {3 x +2}+\sqrt {2}\right )\right )}{12} \] Input:
int(1/(2+3*x)^(3/4)/(4+3*x),x)
Output:
(sqrt(2)*2**(1/4)*(2*atan((2*(3*x + 2)**(1/4) - sqrt(2)*2**(1/4))/(sqrt(2) *2**(1/4))) + 2*atan((2*(3*x + 2)**(1/4) + sqrt(2)*2**(1/4))/(sqrt(2)*2**( 1/4))) - log( - (3*x + 2)**(1/4)*sqrt(2)*2**(1/4) + sqrt(3*x + 2) + sqrt(2 )) + log((3*x + 2)**(1/4)*sqrt(2)*2**(1/4) + sqrt(3*x + 2) + sqrt(2))))/12