Integrand size = 27, antiderivative size = 73 \[ \int \frac {(5-2 x)^2 (4+3 x)^{3/2}}{2+3 x+x^2} \, dx=422 \sqrt {4+3 x}-\frac {64}{3} (4+3 x)^{3/2}+\frac {8}{15} (4+3 x)^{5/2}-324 \sqrt {2} \arctan \left (\frac {\sqrt {4+3 x}}{\sqrt {2}}\right )-98 \text {arctanh}\left (\sqrt {4+3 x}\right ) \] Output:
422*(4+3*x)^(1/2)-64/3*(4+3*x)^(3/2)+8/15*(4+3*x)^(5/2)-324*2^(1/2)*arctan (1/2*2^(1/2)*(4+3*x)^(1/2))-98*arctanh((4+3*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \frac {(5-2 x)^2 (4+3 x)^{3/2}}{2+3 x+x^2} \, dx=\frac {2}{5} \sqrt {4+3 x} \left (863-128 x+12 x^2\right )-324 \sqrt {2} \arctan \left (\sqrt {2+\frac {3 x}{2}}\right )-98 \text {arctanh}\left (\sqrt {4+3 x}\right ) \] Input:
Integrate[((5 - 2*x)^2*(4 + 3*x)^(3/2))/(2 + 3*x + x^2),x]
Output:
(2*Sqrt[4 + 3*x]*(863 - 128*x + 12*x^2))/5 - 324*Sqrt[2]*ArcTan[Sqrt[2 + ( 3*x)/2]] - 98*ArcTanh[Sqrt[4 + 3*x]]
Time = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03, 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.
\(\displaystyle \int \frac {(5-2 x)^2 (3 x+4)^{3/2}}{x^2+3 x+2} \, dx\) |
\(\Big \downarrow \) 1199 |
\(\displaystyle \frac {2}{3} \int \left (4 (3 x+4)^2-96 (3 x+4)-\frac {3 (422-275 (3 x+4))}{-(3 x+4)^2-3 x-2}+633\right )d\sqrt {3 x+4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} \left (-486 \sqrt {2} \arctan \left (\frac {\sqrt {3 x+4}}{\sqrt {2}}\right )-147 \text {arctanh}\left (\sqrt {3 x+4}\right )+\frac {4}{5} (3 x+4)^{5/2}-32 (3 x+4)^{3/2}+633 \sqrt {3 x+4}\right )\) |
Input:
Int[((5 - 2*x)^2*(4 + 3*x)^(3/2))/(2 + 3*x + x^2),x]
Output:
(2*(633*Sqrt[4 + 3*x] - 32*(4 + 3*x)^(3/2) + (4*(4 + 3*x)^(5/2))/5 - 486*S qrt[2]*ArcTan[Sqrt[4 + 3*x]/Sqrt[2]] - 147*ArcTanh[Sqrt[4 + 3*x]]))/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 = 1.46 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\frac {2 \left (12 x^{2}-128 x +863\right ) \sqrt {3 x +4}}{5}-49 \ln \left (\sqrt {3 x +4}+1\right )-324 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )+49 \ln \left (\sqrt {3 x +4}-1\right )\) | \(63\) |
pseudoelliptic | \(\frac {2 \left (12 x^{2}-128 x +863\right ) \sqrt {3 x +4}}{5}-49 \ln \left (\sqrt {3 x +4}+1\right )-324 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )+49 \ln \left (\sqrt {3 x +4}-1\right )\) | \(63\) |
derivativedivides | \(\frac {8 \left (3 x +4\right )^{\frac {5}{2}}}{15}-\frac {64 \left (3 x +4\right )^{\frac {3}{2}}}{3}+422 \sqrt {3 x +4}-49 \ln \left (\sqrt {3 x +4}+1\right )+49 \ln \left (\sqrt {3 x +4}-1\right )-324 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )\) | \(71\) |
default | \(\frac {8 \left (3 x +4\right )^{\frac {5}{2}}}{15}-\frac {64 \left (3 x +4\right )^{\frac {3}{2}}}{3}+422 \sqrt {3 x +4}-49 \ln \left (\sqrt {3 x +4}+1\right )+49 \ln \left (\sqrt {3 x +4}-1\right )-324 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )\) | \(71\) |
trager | \(\left (\frac {24}{5} x^{2}-\frac {256}{5} x +\frac {1726}{5}\right ) \sqrt {3 x +4}-162 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {3 x \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )-4 \sqrt {3 x +4}}{2+x}\right )-49 \ln \left (\frac {2 \sqrt {3 x +4}+5+3 x}{x +1}\right )\) | \(86\) |
Input:
int((5-2*x)^2*(3*x+4)^(3/2)/(x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
2/5*(12*x^2-128*x+863)*(3*x+4)^(1/2)-49*ln((3*x+4)^(1/2)+1)-324*2^(1/2)*ar ctan(1/2*2^(1/2)*(3*x+4)^(1/2))+49*ln((3*x+4)^(1/2)-1)
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.85 \[ \int \frac {(5-2 x)^2 (4+3 x)^{3/2}}{2+3 x+x^2} \, dx=\frac {2}{5} \, {\left (12 \, x^{2} - 128 \, x + 863\right )} \sqrt {3 \, x + 4} - 324 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) - 49 \, \log \left (\sqrt {3 \, x + 4} + 1\right ) + 49 \, \log \left (\sqrt {3 \, x + 4} - 1\right ) \] Input:
integrate((5-2*x)^2*(4+3*x)^(3/2)/(x^2+3*x+2),x, algorithm="fricas")
Output:
2/5*(12*x^2 - 128*x + 863)*sqrt(3*x + 4) - 324*sqrt(2)*arctan(1/2*sqrt(2)* sqrt(3*x + 4)) - 49*log(sqrt(3*x + 4) + 1) + 49*log(sqrt(3*x + 4) - 1)
Time = 1.82 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \frac {(5-2 x)^2 (4+3 x)^{3/2}}{2+3 x+x^2} \, dx=\frac {8 \left (3 x + 4\right )^{\frac {5}{2}}}{15} - \frac {64 \left (3 x + 4\right )^{\frac {3}{2}}}{3} + 422 \sqrt {3 x + 4} + 49 \log {\left (\sqrt {3 x + 4} - 1 \right )} - 49 \log {\left (\sqrt {3 x + 4} + 1 \right )} - 324 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {3 x + 4}}{2} \right )} \] Input:
integrate((5-2*x)**2*(4+3*x)**(3/2)/(x**2+3*x+2),x)
Output:
8*(3*x + 4)**(5/2)/15 - 64*(3*x + 4)**(3/2)/3 + 422*sqrt(3*x + 4) + 49*log (sqrt(3*x + 4) - 1) - 49*log(sqrt(3*x + 4) + 1) - 324*sqrt(2)*atan(sqrt(2) *sqrt(3*x + 4)/2)
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int \frac {(5-2 x)^2 (4+3 x)^{3/2}}{2+3 x+x^2} \, dx=\frac {8}{15} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {64}{3} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - 324 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) + 422 \, \sqrt {3 \, x + 4} - 49 \, \log \left (\sqrt {3 \, x + 4} + 1\right ) + 49 \, \log \left (\sqrt {3 \, x + 4} - 1\right ) \] Input:
integrate((5-2*x)^2*(4+3*x)^(3/2)/(x^2+3*x+2),x, algorithm="maxima")
Output:
8/15*(3*x + 4)^(5/2) - 64/3*(3*x + 4)^(3/2) - 324*sqrt(2)*arctan(1/2*sqrt( 2)*sqrt(3*x + 4)) + 422*sqrt(3*x + 4) - 49*log(sqrt(3*x + 4) + 1) + 49*log (sqrt(3*x + 4) - 1)
Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97 \[ \int \frac {(5-2 x)^2 (4+3 x)^{3/2}}{2+3 x+x^2} \, dx=\frac {8}{15} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {64}{3} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - 324 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) + 422 \, \sqrt {3 \, x + 4} - 49 \, \log \left (\sqrt {3 \, x + 4} + 1\right ) + 49 \, \log \left ({\left | \sqrt {3 \, x + 4} - 1 \right |}\right ) \] Input:
integrate((5-2*x)^2*(4+3*x)^(3/2)/(x^2+3*x+2),x, algorithm="giac")
Output:
8/15*(3*x + 4)^(5/2) - 64/3*(3*x + 4)^(3/2) - 324*sqrt(2)*arctan(1/2*sqrt( 2)*sqrt(3*x + 4)) + 422*sqrt(3*x + 4) - 49*log(sqrt(3*x + 4) + 1) + 49*log (abs(sqrt(3*x + 4) - 1))
Time = 13.56 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {(5-2 x)^2 (4+3 x)^{3/2}}{2+3 x+x^2} \, dx=422\,\sqrt {3\,x+4}-324\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {3\,x+4}}{2}\right )-\frac {64\,{\left (3\,x+4\right )}^{3/2}}{3}+\frac {8\,{\left (3\,x+4\right )}^{5/2}}{15}+\mathrm {atan}\left (\sqrt {3\,x+4}\,1{}\mathrm {i}\right )\,98{}\mathrm {i} \] Input:
int(((2*x - 5)^2*(3*x + 4)^(3/2))/(3*x + x^2 + 2),x)
Output:
atan((3*x + 4)^(1/2)*1i)*98i - 324*2^(1/2)*atan((2^(1/2)*(3*x + 4)^(1/2))/ 2) + 422*(3*x + 4)^(1/2) - (64*(3*x + 4)^(3/2))/3 + (8*(3*x + 4)^(5/2))/15
Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {(5-2 x)^2 (4+3 x)^{3/2}}{2+3 x+x^2} \, dx=-324 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {3 x +4}}{\sqrt {2}}\right )+\frac {24 \sqrt {3 x +4}\, x^{2}}{5}-\frac {256 \sqrt {3 x +4}\, x}{5}+\frac {1726 \sqrt {3 x +4}}{5}+49 \,\mathrm {log}\left (\sqrt {3 x +4}-1\right )-49 \,\mathrm {log}\left (\sqrt {3 x +4}+1\right ) \] Input:
int((5-2*x)^2*(4+3*x)^(3/2)/(x^2+3*x+2),x)
Output:
( - 1620*sqrt(2)*atan(sqrt(3*x + 4)/sqrt(2)) + 24*sqrt(3*x + 4)*x**2 - 256 *sqrt(3*x + 4)*x + 1726*sqrt(3*x + 4) + 245*log(sqrt(3*x + 4) - 1) - 245*l og(sqrt(3*x + 4) + 1))/5