Integrand size = 27, antiderivative size = 67 \[ \int \frac {\sqrt {4+3 x}}{(5-2 x) \left (2+3 x+x^2\right )} \, dx=\frac {2}{9} \sqrt {2} \arctan \left (\frac {\sqrt {4+3 x}}{\sqrt {2}}\right )-\frac {2}{7} \text {arctanh}\left (\sqrt {4+3 x}\right )+\frac {2}{63} \sqrt {46} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right ) \] Output:
2/9*2^(1/2)*arctan(1/2*2^(1/2)*(4+3*x)^(1/2))-2/7*arctanh((4+3*x)^(1/2))+2 /63*46^(1/2)*arctanh(1/23*46^(1/2)*(4+3*x)^(1/2))
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {4+3 x}}{(5-2 x) \left (2+3 x+x^2\right )} \, dx=\frac {2}{63} \left (7 \sqrt {2} \arctan \left (\sqrt {2+\frac {3 x}{2}}\right )-9 \text {arctanh}\left (\sqrt {4+3 x}\right )+\sqrt {46} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right )\right ) \] Input:
Integrate[Sqrt[4 + 3*x]/((5 - 2*x)*(2 + 3*x + x^2)),x]
Output:
(2*(7*Sqrt[2]*ArcTan[Sqrt[2 + (3*x)/2]] - 9*ArcTanh[Sqrt[4 + 3*x]] + Sqrt[ 46]*ArcTanh[Sqrt[2/23]*Sqrt[4 + 3*x]]))/63
Time = 0.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1199, 2009}
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.
\(\displaystyle \int \frac {\sqrt {3 x+4}}{(5-2 x) \left (x^2+3 x+2\right )} \, dx\) |
\(\Big \downarrow \) 1199 |
\(\displaystyle \frac {2}{3} \int \left (\frac {2}{3 (3 x+6)}+\frac {46}{21 (23-2 (3 x+4))}-\frac {3}{7 (-3 x-3)}\right )d\sqrt {3 x+4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} \left (\frac {1}{3} \sqrt {2} \arctan \left (\frac {\sqrt {3 x+4}}{\sqrt {2}}\right )-\frac {3}{7} \text {arctanh}\left (\sqrt {3 x+4}\right )+\frac {1}{21} \sqrt {46} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )\right )\) |
Input:
Int[Sqrt[4 + 3*x]/((5 - 2*x)*(2 + 3*x + x^2)),x]
Output:
(2*((Sqrt[2]*ArcTan[Sqrt[4 + 3*x]/Sqrt[2]])/3 - (3*ArcTanh[Sqrt[4 + 3*x]]) /7 + (Sqrt[46]*ArcTanh[Sqrt[2/23]*Sqrt[4 + 3*x]])/21))/3
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e Subs t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer Q[n] && FractionQ[m]
Time = 2.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(-\frac {\ln \left (\sqrt {3 x +4}+1\right )}{7}+\frac {\ln \left (\sqrt {3 x +4}-1\right )}{7}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )}{9}+\frac {2 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{63}\) | \(62\) |
default | \(-\frac {\ln \left (\sqrt {3 x +4}+1\right )}{7}+\frac {\ln \left (\sqrt {3 x +4}-1\right )}{7}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )}{9}+\frac {2 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{63}\) | \(62\) |
pseudoelliptic | \(-\frac {\ln \left (\sqrt {3 x +4}+1\right )}{7}+\frac {\ln \left (\sqrt {3 x +4}-1\right )}{7}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )}{9}+\frac {2 \sqrt {2}\, \sqrt {23}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{63}\) | \(65\) |
trager | \(-\frac {\ln \left (\frac {2 \sqrt {3 x +4}+5+3 x}{x +1}\right )}{7}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {3 x \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )-4 \sqrt {3 x +4}}{2+x}\right )}{9}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right ) \ln \left (\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right ) x +31 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right )-92 \sqrt {3 x +4}}{-5+2 x}\right )}{63}\) | \(112\) |
Input:
int((3*x+4)^(1/2)/(5-2*x)/(x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
-1/7*ln((3*x+4)^(1/2)+1)+1/7*ln((3*x+4)^(1/2)-1)+2/9*2^(1/2)*arctan(1/2*2^ (1/2)*(3*x+4)^(1/2))+2/63*46^(1/2)*arctanh(1/23*46^(1/2)*(3*x+4)^(1/2))
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {4+3 x}}{(5-2 x) \left (2+3 x+x^2\right )} \, dx=\frac {2}{9} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) + \frac {1}{63} \, \sqrt {46} \log \left (\frac {2 \, \sqrt {46} \sqrt {3 \, x + 4} + 6 \, x + 31}{2 \, x - 5}\right ) - \frac {1}{7} \, \log \left (\sqrt {3 \, x + 4} + 1\right ) + \frac {1}{7} \, \log \left (\sqrt {3 \, x + 4} - 1\right ) \] Input:
integrate((4+3*x)^(1/2)/(5-2*x)/(x^2+3*x+2),x, algorithm="fricas")
Output:
2/9*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(3*x + 4)) + 1/63*sqrt(46)*log((2*sqrt( 46)*sqrt(3*x + 4) + 6*x + 31)/(2*x - 5)) - 1/7*log(sqrt(3*x + 4) + 1) + 1/ 7*log(sqrt(3*x + 4) - 1)
Time = 1.61 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {4+3 x}}{(5-2 x) \left (2+3 x+x^2\right )} \, dx=- \frac {\sqrt {46} \left (\log {\left (\sqrt {3 x + 4} - \frac {\sqrt {46}}{2} \right )} - \log {\left (\sqrt {3 x + 4} + \frac {\sqrt {46}}{2} \right )}\right )}{63} + \frac {\log {\left (\sqrt {3 x + 4} - 1 \right )}}{7} - \frac {\log {\left (\sqrt {3 x + 4} + 1 \right )}}{7} + \frac {2 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {3 x + 4}}{2} \right )}}{9} \] Input:
integrate((4+3*x)**(1/2)/(5-2*x)/(x**2+3*x+2),x)
Output:
-sqrt(46)*(log(sqrt(3*x + 4) - sqrt(46)/2) - log(sqrt(3*x + 4) + sqrt(46)/ 2))/63 + log(sqrt(3*x + 4) - 1)/7 - log(sqrt(3*x + 4) + 1)/7 + 2*sqrt(2)*a tan(sqrt(2)*sqrt(3*x + 4)/2)/9
Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {4+3 x}}{(5-2 x) \left (2+3 x+x^2\right )} \, dx=\frac {2}{9} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) - \frac {1}{63} \, \sqrt {46} \log \left (-\frac {\sqrt {46} - 2 \, \sqrt {3 \, x + 4}}{\sqrt {46} + 2 \, \sqrt {3 \, x + 4}}\right ) - \frac {1}{7} \, \log \left (\sqrt {3 \, x + 4} + 1\right ) + \frac {1}{7} \, \log \left (\sqrt {3 \, x + 4} - 1\right ) \] Input:
integrate((4+3*x)^(1/2)/(5-2*x)/(x^2+3*x+2),x, algorithm="maxima")
Output:
2/9*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(3*x + 4)) - 1/63*sqrt(46)*log(-(sqrt(4 6) - 2*sqrt(3*x + 4))/(sqrt(46) + 2*sqrt(3*x + 4))) - 1/7*log(sqrt(3*x + 4 ) + 1) + 1/7*log(sqrt(3*x + 4) - 1)
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {4+3 x}}{(5-2 x) \left (2+3 x+x^2\right )} \, dx=\frac {2}{9} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) - \frac {1}{63} \, \sqrt {46} \log \left (\frac {{\left | -2 \, \sqrt {46} + 4 \, \sqrt {3 \, x + 4} \right |}}{2 \, {\left (\sqrt {46} + 2 \, \sqrt {3 \, x + 4}\right )}}\right ) - \frac {1}{7} \, \log \left (\sqrt {3 \, x + 4} + 1\right ) + \frac {1}{7} \, \log \left ({\left | \sqrt {3 \, x + 4} - 1 \right |}\right ) \] Input:
integrate((4+3*x)^(1/2)/(5-2*x)/(x^2+3*x+2),x, algorithm="giac")
Output:
2/9*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(3*x + 4)) - 1/63*sqrt(46)*log(1/2*abs( -2*sqrt(46) + 4*sqrt(3*x + 4))/(sqrt(46) + 2*sqrt(3*x + 4))) - 1/7*log(sqr t(3*x + 4) + 1) + 1/7*log(abs(sqrt(3*x + 4) - 1))
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {4+3 x}}{(5-2 x) \left (2+3 x+x^2\right )} \, dx=\frac {2\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {3\,x+4}}{2}\right )}{9}-\frac {2\,\mathrm {atanh}\left (\sqrt {3\,x+4}\right )}{7}+\frac {2\,\sqrt {46}\,\mathrm {atanh}\left (\frac {\sqrt {46}\,\sqrt {3\,x+4}}{23}\right )}{63} \] Input:
int(-(3*x + 4)^(1/2)/((2*x - 5)*(3*x + x^2 + 2)),x)
Output:
(2*2^(1/2)*atan((2^(1/2)*(3*x + 4)^(1/2))/2))/9 - (2*atanh((3*x + 4)^(1/2) ))/7 + (2*46^(1/2)*atanh((46^(1/2)*(3*x + 4)^(1/2))/23))/63
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {4+3 x}}{(5-2 x) \left (2+3 x+x^2\right )} \, dx=\frac {2 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {3 x +4}}{\sqrt {2}}\right )}{9}-\frac {\sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}-\sqrt {46}\right )}{63}+\frac {\sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}+\sqrt {46}\right )}{63}+\frac {\mathrm {log}\left (\sqrt {3 x +4}-1\right )}{7}-\frac {\mathrm {log}\left (\sqrt {3 x +4}+1\right )}{7} \] Input:
int((4+3*x)^(1/2)/(5-2*x)/(x^2+3*x+2),x)
Output:
(14*sqrt(2)*atan(sqrt(3*x + 4)/sqrt(2)) - sqrt(46)*log(2*sqrt(3*x + 4) - s qrt(46)) + sqrt(46)*log(2*sqrt(3*x + 4) + sqrt(46)) + 9*log(sqrt(3*x + 4) - 1) - 9*log(sqrt(3*x + 4) + 1))/63