Integrand size = 20, antiderivative size = 46 \[ \int \frac {1}{(4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {3}{\sqrt {4+3 x}}+\frac {\arctan \left (\frac {\sqrt {4+3 x}}{\sqrt {2}}\right )}{\sqrt {2}}-2 \text {arctanh}\left (\sqrt {4+3 x}\right ) \] Output:
3/(4+3*x)^(1/2)+1/2*2^(1/2)*arctan(1/2*2^(1/2)*(4+3*x)^(1/2))-2*arctanh((4 +3*x)^(1/2))
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {3}{\sqrt {4+3 x}}+\frac {\arctan \left (\sqrt {2+\frac {3 x}{2}}\right )}{\sqrt {2}}-2 \text {arctanh}\left (\sqrt {4+3 x}\right ) \] Input:
Integrate[1/((4 + 3*x)^(3/2)*(2 + 3*x + x^2)),x]
Output:
3/Sqrt[4 + 3*x] + ArcTan[Sqrt[2 + (3*x)/2]]/Sqrt[2] - 2*ArcTanh[Sqrt[4 + 3 *x]]
Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1147, 25, 1197, 27, 1480, 217, 219}
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.
\(\displaystyle \int \frac {1}{(3 x+4)^{3/2} \left (x^2+3 x+2\right )} \, dx\) |
\(\Big \downarrow \) 1147 |
\(\displaystyle \frac {3}{\sqrt {3 x+4}}-\frac {1}{2} \int -\frac {3 x+5}{\sqrt {3 x+4} \left (x^2+3 x+2\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \int \frac {3 x+5}{\sqrt {3 x+4} \left (x^2+3 x+2\right )}dx+\frac {3}{\sqrt {3 x+4}}\) |
\(\Big \downarrow \) 1197 |
\(\displaystyle \int -\frac {3 (3 x+5)}{-(3 x+4)^2-3 x-2}d\sqrt {3 x+4}+\frac {3}{\sqrt {3 x+4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{\sqrt {3 x+4}}-3 \int \frac {3 x+5}{-(3 x+4)^2-3 x-2}d\sqrt {3 x+4}\) |
\(\Big \downarrow \) 1480 |
\(\displaystyle \frac {3}{\sqrt {3 x+4}}-3 \left (\frac {1}{3} \int \frac {1}{-3 x-6}d\sqrt {3 x+4}+\frac {2}{3} \int \frac {1}{-3 x-3}d\sqrt {3 x+4}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {3}{\sqrt {3 x+4}}-3 \left (\frac {2}{3} \int \frac {1}{-3 x-3}d\sqrt {3 x+4}-\frac {\arctan \left (\frac {\sqrt {3 x+4}}{\sqrt {2}}\right )}{3 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{\sqrt {3 x+4}}-3 \left (\frac {2}{3} \text {arctanh}\left (\sqrt {3 x+4}\right )-\frac {\arctan \left (\frac {\sqrt {3 x+4}}{\sqrt {2}}\right )}{3 \sqrt {2}}\right )\) |
Input:
Int[1/((4 + 3*x)^(3/2)*(2 + 3*x + x^2)),x]
Output:
3/Sqrt[4 + 3*x] - 3*(-1/3*ArcTan[Sqrt[4 + 3*x]/Sqrt[2]]/Sqrt[2] + (2*ArcTa nh[Sqrt[4 + 3*x]])/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_)^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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol ] :> Simp[e*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Sim p[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c*e*x , x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[m, - 1]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 1.74 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {3}{\sqrt {3 x +4}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )}{2}+\ln \left (\sqrt {3 x +4}-1\right )-\ln \left (\sqrt {3 x +4}+1\right )\) | \(51\) |
default | \(\frac {3}{\sqrt {3 x +4}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )}{2}+\ln \left (\sqrt {3 x +4}-1\right )-\ln \left (\sqrt {3 x +4}+1\right )\) | \(51\) |
risch | \(\frac {3}{\sqrt {3 x +4}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )}{2}+\ln \left (\sqrt {3 x +4}-1\right )-\ln \left (\sqrt {3 x +4}+1\right )\) | \(51\) |
trager | \(\frac {3}{\sqrt {3 x +4}}+\ln \left (\frac {-5-3 x +2 \sqrt {3 x +4}}{x +1}\right )-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {3 x \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )+4 \sqrt {3 x +4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{2+x}\right )}{4}\) | \(74\) |
pseudoelliptic | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right ) \sqrt {3 x +4}+2 \ln \left (\sqrt {3 x +4}-1\right ) \sqrt {3 x +4}-2 \ln \left (\sqrt {3 x +4}+1\right ) \sqrt {3 x +4}+6}{2 \sqrt {3 x +4}}\) | \(74\) |
Input:
int(1/(3*x+4)^(3/2)/(x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
3/(3*x+4)^(1/2)+1/2*2^(1/2)*arctan(1/2*2^(1/2)*(3*x+4)^(1/2))+ln((3*x+4)^( 1/2)-1)-ln((3*x+4)^(1/2)+1)
Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {\sqrt {2} {\left (3 \, x + 4\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) - 2 \, {\left (3 \, x + 4\right )} \log \left (\sqrt {3 \, x + 4} + 1\right ) + 2 \, {\left (3 \, x + 4\right )} \log \left (\sqrt {3 \, x + 4} - 1\right ) + 6 \, \sqrt {3 \, x + 4}}{2 \, {\left (3 \, x + 4\right )}} \] Input:
integrate(1/(4+3*x)^(3/2)/(x^2+3*x+2),x, algorithm="fricas")
Output:
1/2*(sqrt(2)*(3*x + 4)*arctan(1/2*sqrt(2)*sqrt(3*x + 4)) - 2*(3*x + 4)*log (sqrt(3*x + 4) + 1) + 2*(3*x + 4)*log(sqrt(3*x + 4) - 1) + 6*sqrt(3*x + 4) )/(3*x + 4)
Time = 1.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\log {\left (\sqrt {3 x + 4} - 1 \right )} - \log {\left (\sqrt {3 x + 4} + 1 \right )} + \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {3 x + 4}}{2} \right )}}{2} + \frac {3}{\sqrt {3 x + 4}} \] Input:
integrate(1/(4+3*x)**(3/2)/(x**2+3*x+2),x)
Output:
log(sqrt(3*x + 4) - 1) - log(sqrt(3*x + 4) + 1) + sqrt(2)*atan(sqrt(2)*sqr t(3*x + 4)/2)/2 + 3/sqrt(3*x + 4)
Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) + \frac {3}{\sqrt {3 \, x + 4}} - \log \left (\sqrt {3 \, x + 4} + 1\right ) + \log \left (\sqrt {3 \, x + 4} - 1\right ) \] Input:
integrate(1/(4+3*x)^(3/2)/(x^2+3*x+2),x, algorithm="maxima")
Output:
1/2*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(3*x + 4)) + 3/sqrt(3*x + 4) - log(sqrt (3*x + 4) + 1) + log(sqrt(3*x + 4) - 1)
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) + \frac {3}{\sqrt {3 \, x + 4}} - \log \left (\sqrt {3 \, x + 4} + 1\right ) + \log \left ({\left | \sqrt {3 \, x + 4} - 1 \right |}\right ) \] Input:
integrate(1/(4+3*x)^(3/2)/(x^2+3*x+2),x, algorithm="giac")
Output:
1/2*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(3*x + 4)) + 3/sqrt(3*x + 4) - log(sqrt (3*x + 4) + 1) + log(abs(sqrt(3*x + 4) - 1))
Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {3\,x+4}}{2}\right )}{2}-2\,\mathrm {atanh}\left (\sqrt {3\,x+4}\right )+\frac {3}{\sqrt {3\,x+4}} \] Input:
int(1/((3*x + 4)^(3/2)*(3*x + x^2 + 2)),x)
Output:
(2^(1/2)*atan((2^(1/2)*(3*x + 4)^(1/2))/2))/2 - 2*atanh((3*x + 4)^(1/2)) + 3/(3*x + 4)^(1/2)
Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.46 \[ \int \frac {1}{(4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {\sqrt {3 x +4}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {3 x +4}}{\sqrt {2}}\right )+2 \sqrt {3 x +4}\, \mathrm {log}\left (\sqrt {3 x +4}-1\right )-2 \sqrt {3 x +4}\, \mathrm {log}\left (\sqrt {3 x +4}+1\right )+6}{2 \sqrt {3 x +4}} \] Input:
int(1/(4+3*x)^(3/2)/(x^2+3*x+2),x)
Output:
(sqrt(3*x + 4)*sqrt(2)*atan(sqrt(3*x + 4)/sqrt(2)) + 2*sqrt(3*x + 4)*log(s qrt(3*x + 4) - 1) - 2*sqrt(3*x + 4)*log(sqrt(3*x + 4) + 1) + 6)/(2*sqrt(3* x + 4))