Integrand size = 29, antiderivative size = 181 \[ \int \frac {(5-2 x)^{5/2} (4+3 x)^{5/2}}{2+3 x+x^2} \, dx=-\frac {196243}{384} \sqrt {5-2 x} \sqrt {4+3 x}+\frac {36571}{288} \sqrt {5-2 x} (4+3 x)^{3/2}-\frac {269}{36} \sqrt {5-2 x} (4+3 x)^{5/2}-\frac {1}{2} (5-2 x)^{3/2} (4+3 x)^{5/2}-\frac {1414603 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right )}{384 \sqrt {6}}+1944 \sqrt {2} \arctan \left (\frac {3 \sqrt {4+3 x}}{\sqrt {2} \sqrt {5-2 x}}\right )-98 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {7} \sqrt {4+3 x}}{\sqrt {5-2 x}}\right ) \] Output:
-196243/384*(5-2*x)^(1/2)*(4+3*x)^(1/2)+36571/288*(5-2*x)^(1/2)*(4+3*x)^(3 /2)-269/36*(5-2*x)^(1/2)*(4+3*x)^(5/2)-1/2*(5-2*x)^(3/2)*(4+3*x)^(5/2)-141 4603/2304*arcsin(1/23*46^(1/2)*(4+3*x)^(1/2))*6^(1/2)+1944*2^(1/2)*arctan( 3/2*(4+3*x)^(1/2)*2^(1/2)/(5-2*x)^(1/2))-98*7^(1/2)*arctanh(7^(1/2)*(4+3*x )^(1/2)/(5-2*x)^(1/2))
Time = 0.60 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.72 \[ \int \frac {(5-2 x)^{5/2} (4+3 x)^{5/2}}{2+3 x+x^2} \, dx=\frac {1}{384} \sqrt {5-2 x} \sqrt {4+3 x} \left (-62467+60524 x-25248 x^2+3456 x^3\right )-1944 \sqrt {2} \arctan \left (\frac {\sqrt {10-4 x}}{3 \sqrt {4+3 x}}\right )+\frac {1414603 \arctan \left (\frac {\sqrt {\frac {15}{2}-3 x}}{\sqrt {4+3 x}}\right )}{384 \sqrt {6}}-98 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7} \sqrt {4+3 x}}\right ) \] Input:
Integrate[((5 - 2*x)^(5/2)*(4 + 3*x)^(5/2))/(2 + 3*x + x^2),x]
Output:
(Sqrt[5 - 2*x]*Sqrt[4 + 3*x]*(-62467 + 60524*x - 25248*x^2 + 3456*x^3))/38 4 - 1944*Sqrt[2]*ArcTan[Sqrt[10 - 4*x]/(3*Sqrt[4 + 3*x])] + (1414603*ArcTa n[Sqrt[15/2 - 3*x]/Sqrt[4 + 3*x]])/(384*Sqrt[6]) - 98*Sqrt[7]*ArcTanh[Sqrt [5 - 2*x]/(Sqrt[7]*Sqrt[4 + 3*x])]
Time = 0.92 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.64, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {1201, 25, 90, 60, 60, 60, 64, 223, 2153, 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.
| \(\displaystyle \int \frac {(5-2 x)^{5/2} (3 x+4)^{5/2}}{x^2+3 x+2} \, dx\) |
| \(\Big \downarrow \) 1201 |
| \(\displaystyle \int -\frac {(5-2 x)^{3/2} \sqrt {3 x+4} (29 x+22)}{x^2+3 x+2}dx+3 \int (17-6 x) (5-2 x)^{3/2} \sqrt {3 x+4}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \int (17-6 x) (5-2 x)^{3/2} \sqrt {3 x+4}dx-\int \frac {(5-2 x)^{3/2} \sqrt {3 x+4} (29 x+22)}{x^2+3 x+2}dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle 3 \left (\frac {131}{8} \int (5-2 x)^{3/2} \sqrt {3 x+4}dx+\frac {1}{4} (3 x+4)^{3/2} (5-2 x)^{5/2}\right )-\int \frac {(5-2 x)^{3/2} \sqrt {3 x+4} (29 x+22)}{x^2+3 x+2}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle 3 \left (\frac {131}{8} \left (\frac {23}{12} \int \frac {(5-2 x)^{3/2}}{\sqrt {3 x+4}}dx-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )+\frac {1}{4} (3 x+4)^{3/2} (5-2 x)^{5/2}\right )-\int \frac {(5-2 x)^{3/2} \sqrt {3 x+4} (29 x+22)}{x^2+3 x+2}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle 3 \left (\frac {131}{8} \left (\frac {23}{12} \left (\frac {23}{4} \int \frac {\sqrt {5-2 x}}{\sqrt {3 x+4}}dx+\frac {1}{6} \sqrt {3 x+4} (5-2 x)^{3/2}\right )-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )+\frac {1}{4} (3 x+4)^{3/2} (5-2 x)^{5/2}\right )-\int \frac {(5-2 x)^{3/2} \sqrt {3 x+4} (29 x+22)}{x^2+3 x+2}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle 3 \left (\frac {131}{8} \left (\frac {23}{12} \left (\frac {23}{4} \left (\frac {23}{6} \int \frac {1}{\sqrt {5-2 x} \sqrt {3 x+4}}dx+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{6} \sqrt {3 x+4} (5-2 x)^{3/2}\right )-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )+\frac {1}{4} (3 x+4)^{3/2} (5-2 x)^{5/2}\right )-\int \frac {(5-2 x)^{3/2} \sqrt {3 x+4} (29 x+22)}{x^2+3 x+2}dx\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle 3 \left (\frac {131}{8} \left (\frac {23}{12} \left (\frac {23}{4} \left (\frac {23}{9} \int \frac {1}{\sqrt {\frac {23}{3}-\frac {2}{3} (3 x+4)}}d\sqrt {3 x+4}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{6} \sqrt {3 x+4} (5-2 x)^{3/2}\right )-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )+\frac {1}{4} (3 x+4)^{3/2} (5-2 x)^{5/2}\right )-\int \frac {(5-2 x)^{3/2} \sqrt {3 x+4} (29 x+22)}{x^2+3 x+2}dx\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle 3 \left (\frac {131}{8} \left (\frac {23}{12} \left (\frac {23}{4} \left (\frac {23 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{3 \sqrt {6}}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{6} \sqrt {3 x+4} (5-2 x)^{3/2}\right )-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )+\frac {1}{4} (3 x+4)^{3/2} (5-2 x)^{5/2}\right )-\int \frac {(5-2 x)^{3/2} \sqrt {3 x+4} (29 x+22)}{x^2+3 x+2}dx\) |
| \(\Big \downarrow \) 2153 |
| \(\displaystyle 3 \left (\frac {131}{8} \left (\frac {23}{12} \left (\frac {23}{4} \left (\frac {23 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{3 \sqrt {6}}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{6} \sqrt {3 x+4} (5-2 x)^{3/2}\right )-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )+\frac {1}{4} (3 x+4)^{3/2} (5-2 x)^{5/2}\right )-\int \left (\frac {72 (5-2 x)^{3/2} \sqrt {3 x+4}}{2 x+4}-\frac {14 (5-2 x)^{3/2} \sqrt {3 x+4}}{2 x+2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7469 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{12 \sqrt {6}}-2819 \sqrt {\frac {3}{2}} \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )+3 \left (\frac {131}{8} \left (\frac {23}{12} \left (\frac {23}{4} \left (\frac {23 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{3 \sqrt {6}}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{6} \sqrt {3 x+4} (5-2 x)^{3/2}\right )-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )+\frac {1}{4} (3 x+4)^{3/2} (5-2 x)^{5/2}\right )+1944 \sqrt {2} \arctan \left (\frac {3 \sqrt {3 x+4}}{\sqrt {2} \sqrt {5-2 x}}\right )-98 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {7} \sqrt {3 x+4}}{\sqrt {5-2 x}}\right )-\frac {29}{2} \sqrt {3 x+4} (5-2 x)^{3/2}-\frac {2633}{12} \sqrt {3 x+4} \sqrt {5-2 x}\) |
Input:
Int[((5 - 2*x)^(5/2)*(4 + 3*x)^(5/2))/(2 + 3*x + x^2),x]
Output:
(-2633*Sqrt[5 - 2*x]*Sqrt[4 + 3*x])/12 - (29*(5 - 2*x)^(3/2)*Sqrt[4 + 3*x] )/2 - 2819*Sqrt[3/2]*ArcSin[Sqrt[2/23]*Sqrt[4 + 3*x]] + (7469*ArcSin[Sqrt[ 2/23]*Sqrt[4 + 3*x]])/(12*Sqrt[6]) + 3*(((5 - 2*x)^(5/2)*(4 + 3*x)^(3/2))/ 4 + (131*(-1/6*((5 - 2*x)^(5/2)*Sqrt[4 + 3*x]) + (23*(((5 - 2*x)^(3/2)*Sqr t[4 + 3*x])/6 + (23*((Sqrt[5 - 2*x]*Sqrt[4 + 3*x])/3 + (23*ArcSin[Sqrt[2/2 3]*Sqrt[4 + 3*x]])/(3*Sqrt[6])))/4))/12))/8) + 1944*Sqrt[2]*ArcTan[(3*Sqrt [4 + 3*x])/(Sqrt[2]*Sqrt[5 - 2*x])] - 98*Sqrt[7]*ArcTanh[(Sqrt[7]*Sqrt[4 + 3*x])/Sqrt[5 - 2*x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [2/b Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] || PosQ[b])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> Simp[g/c^2 Int[Simp[2*c*e*f + c*d*g - b* e*g + c*e*g*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2), x], x] + Simp[1/c^2 Int[Simp[c^2*d*f^2 - 2*a*c*e*f*g - a*c*d*g^2 + a*b*e*g^2 + (c^2*e*f^2 + 2 *c^2*d*f*g - 2*b*c*e*f*g - b*c*d*g^2 + b^2*e*g^2 - a*c*e*g^2)*x, x]*(d + e* x)^(m - 1)*((f + g*x)^(n - 2)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 1 ]
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ [m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
Time = 1.95 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.90
| method | result | size |
| default | \(-\frac {\sqrt {5-2 x}\, \sqrt {3 x +4}\, \left (-41472 x^{3} \sqrt {-6 x^{2}+7 x +20}+1414603 \sqrt {3}\, \sqrt {2}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right )+302976 x^{2} \sqrt {-6 x^{2}+7 x +20}-4478976 \sqrt {2}\, \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 x^{2}+7 x +20}}\right )+225792 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 x^{2}+7 x +20}}\right )-726288 x \sqrt {-6 x^{2}+7 x +20}+749604 \sqrt {-6 x^{2}+7 x +20}\right )}{4608 \sqrt {-6 x^{2}+7 x +20}}\) | \(163\) |
| risch | \(-\frac {\left (3456 x^{3}-25248 x^{2}+60524 x -62467\right ) \sqrt {3 x +4}\, \left (-5+2 x \right ) \sqrt {\left (5-2 x \right ) \left (3 x +4\right )}}{384 \sqrt {-\left (3 x +4\right ) \left (-5+2 x \right )}\, \sqrt {5-2 x}}-\frac {\left (\frac {1414603 \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right )}{4608}+49 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 \left (x +1\right )^{2}+19 x +26}}\right )-972 \sqrt {2}\, \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 \left (2+x \right )^{2}+44+31 x}}\right )\right ) \sqrt {\left (5-2 x \right ) \left (3 x +4\right )}}{\sqrt {5-2 x}\, \sqrt {3 x +4}}\) | \(166\) |
Input:
int((5-2*x)^(5/2)*(3*x+4)^(5/2)/(x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
-1/4608*(5-2*x)^(1/2)*(3*x+4)^(1/2)*(-41472*x^3*(-6*x^2+7*x+20)^(1/2)+1414 603*3^(1/2)*2^(1/2)*arcsin(12/23*x-7/23)+302976*x^2*(-6*x^2+7*x+20)^(1/2)- 4478976*2^(1/2)*arctan(1/12*(26+31*x)*2^(1/2)/(-6*x^2+7*x+20)^(1/2))+22579 2*7^(1/2)*arctanh(1/14*(33+19*x)*7^(1/2)/(-6*x^2+7*x+20)^(1/2))-726288*x*( -6*x^2+7*x+20)^(1/2)+749604*(-6*x^2+7*x+20)^(1/2))/(-6*x^2+7*x+20)^(1/2)
Time = 0.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.93 \[ \int \frac {(5-2 x)^{5/2} (4+3 x)^{5/2}}{2+3 x+x^2} \, dx=\frac {1}{384} \, {\left (3456 \, x^{3} - 25248 \, x^{2} + 60524 \, x - 62467\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5} - 972 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (31 \, x + 26\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{12 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) + \frac {1414603}{4608} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} {\left (12 \, x - 7\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{12 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) + \frac {49}{2} \, \sqrt {7} \log \left (-\frac {4 \, \sqrt {7} {\left (19 \, x + 33\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5} - 193 \, x^{2} - 1450 \, x - 1649}{x^{2} + 2 \, x + 1}\right ) \] Input:
integrate((5-2*x)^(5/2)*(4+3*x)^(5/2)/(x^2+3*x+2),x, algorithm="fricas")
Output:
1/384*(3456*x^3 - 25248*x^2 + 60524*x - 62467)*sqrt(3*x + 4)*sqrt(-2*x + 5 ) - 972*sqrt(2)*arctan(1/12*sqrt(2)*(31*x + 26)*sqrt(3*x + 4)*sqrt(-2*x + 5)/(6*x^2 - 7*x - 20)) + 1414603/4608*sqrt(6)*arctan(1/12*sqrt(6)*(12*x - 7)*sqrt(3*x + 4)*sqrt(-2*x + 5)/(6*x^2 - 7*x - 20)) + 49/2*sqrt(7)*log(-(4 *sqrt(7)*(19*x + 33)*sqrt(3*x + 4)*sqrt(-2*x + 5) - 193*x^2 - 1450*x - 164 9)/(x^2 + 2*x + 1))
\[ \int \frac {(5-2 x)^{5/2} (4+3 x)^{5/2}}{2+3 x+x^2} \, dx=\int \frac {\left (5 - 2 x\right )^{\frac {5}{2}} \left (3 x + 4\right )^{\frac {5}{2}}}{\left (x + 1\right ) \left (x + 2\right )}\, dx \] Input:
integrate((5-2*x)**(5/2)*(4+3*x)**(5/2)/(x**2+3*x+2),x)
Output:
Integral((5 - 2*x)**(5/2)*(3*x + 4)**(5/2)/((x + 1)*(x + 2)), x)
\[ \int \frac {(5-2 x)^{5/2} (4+3 x)^{5/2}}{2+3 x+x^2} \, dx=\int { \frac {{\left (3 \, x + 4\right )}^{\frac {5}{2}} {\left (-2 \, x + 5\right )}^{\frac {5}{2}}}{x^{2} + 3 \, x + 2} \,d x } \] Input:
integrate((5-2*x)^(5/2)*(4+3*x)^(5/2)/(x^2+3*x+2),x, algorithm="maxima")
Output:
integrate((3*x + 4)^(5/2)*(-2*x + 5)^(5/2)/(x^2 + 3*x + 2), x)
Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (132) = 264\).
Time = 0.59 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.83 \[ \int \frac {(5-2 x)^{5/2} (4+3 x)^{5/2}}{2+3 x+x^2} \, dx=324 \, \sqrt {6} \sqrt {3} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {3} \sqrt {3 \, x + 4} {\left (\frac {{\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{2}}{3 \, x + 4} - 4\right )}}{18 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}}\right )\right )} + \frac {1}{3456} \, {\left (4 \, {\left (8 \, {\left (12 \, \sqrt {3} {\left (3 \, x + 4\right )} - 407 \, \sqrt {3}\right )} {\left (3 \, x + 4\right )} + 36571 \, \sqrt {3}\right )} {\left (3 \, x + 4\right )} - 588729 \, \sqrt {3}\right )} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15} + \frac {49}{6} \, \sqrt {42} \sqrt {6} \log \left (\frac {{\left | -2 \, \sqrt {42} + \frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}} \right |}}{{\left | 2 \, \sqrt {42} + \frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}} \right |}}\right ) - \frac {1414603}{4608} \, \sqrt {6} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {3 \, x + 4} {\left (\frac {{\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{2}}{3 \, x + 4} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}}\right )\right )} \] Input:
integrate((5-2*x)^(5/2)*(4+3*x)^(5/2)/(x^2+3*x+2),x, algorithm="giac")
Output:
324*sqrt(6)*sqrt(3)*(pi + 2*arctan(-1/18*sqrt(3)*sqrt(3*x + 4)*((sqrt(2)*s qrt(-6*x + 15) - sqrt(46))^2/(3*x + 4) - 4)/(sqrt(2)*sqrt(-6*x + 15) - sqr t(46)))) + 1/3456*(4*(8*(12*sqrt(3)*(3*x + 4) - 407*sqrt(3))*(3*x + 4) + 3 6571*sqrt(3))*(3*x + 4) - 588729*sqrt(3))*sqrt(3*x + 4)*sqrt(-6*x + 15) + 49/6*sqrt(42)*sqrt(6)*log(abs(-2*sqrt(42) + (sqrt(2)*sqrt(-6*x + 15) - sqr t(46))/sqrt(3*x + 4) - 4*sqrt(3*x + 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46) ))/abs(2*sqrt(42) + (sqrt(2)*sqrt(-6*x + 15) - sqrt(46))/sqrt(3*x + 4) - 4 *sqrt(3*x + 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46)))) - 1414603/4608*sqrt( 6)*(pi + 2*arctan(-1/4*sqrt(3*x + 4)*((sqrt(2)*sqrt(-6*x + 15) - sqrt(46)) ^2/(3*x + 4) - 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46))))
Timed out. \[ \int \frac {(5-2 x)^{5/2} (4+3 x)^{5/2}}{2+3 x+x^2} \, dx=\int \frac {{\left (5-2\,x\right )}^{5/2}\,{\left (3\,x+4\right )}^{5/2}}{x^2+3\,x+2} \,d x \] Input:
int(((5 - 2*x)^(5/2)*(3*x + 4)^(5/2))/(3*x + x^2 + 2),x)
Output:
int(((5 - 2*x)^(5/2)*(3*x + 4)^(5/2))/(3*x + x^2 + 2), x)
Time = 0.22 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.50 \[ \int \frac {(5-2 x)^{5/2} (4+3 x)^{5/2}}{2+3 x+x^2} \, dx=\frac {1414603 \sqrt {6}\, \mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2304}+1944 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {23}}{2}-\frac {3 \sqrt {3}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )}{2}\right )-1944 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {23}}{2}+\frac {3 \sqrt {3}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )}{2}\right )+9 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{3}-\frac {263 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{2}}{4}+\frac {15131 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x}{96}-\frac {62467 \sqrt {3 x +4}\, \sqrt {-2 x +5}}{384}-49 \sqrt {7}\, \mathrm {log}\left (-\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )-\sqrt {2}\right )+49 \sqrt {7}\, \mathrm {log}\left (-\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )+\sqrt {2}\right )-49 \sqrt {7}\, \mathrm {log}\left (\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )-\sqrt {2}\right )+49 \sqrt {7}\, \mathrm {log}\left (\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )+\sqrt {2}\right ) \] Input:
int((5-2*x)^(5/2)*(4+3*x)^(5/2)/(x^2+3*x+2),x)
Output:
(1414603*sqrt(6)*asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23)) + 4478976*sqrt( 2)*atan((sqrt(23) - 3*sqrt(3)*tan(asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23) )/2))/2) - 4478976*sqrt(2)*atan((sqrt(23) + 3*sqrt(3)*tan(asin((sqrt( - 2* x + 5)*sqrt(3))/sqrt(23))/2))/2) + 20736*sqrt(3*x + 4)*sqrt( - 2*x + 5)*x* *3 - 151488*sqrt(3*x + 4)*sqrt( - 2*x + 5)*x**2 + 363144*sqrt(3*x + 4)*sqr t( - 2*x + 5)*x - 374802*sqrt(3*x + 4)*sqrt( - 2*x + 5) - 112896*sqrt(7)*l og( - sqrt(23) + sqrt(21)*tan(asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23))/2) - sqrt(2)) + 112896*sqrt(7)*log( - sqrt(23) + sqrt(21)*tan(asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23))/2) + sqrt(2)) - 112896*sqrt(7)*log(sqrt(23) + sqrt(21)*tan(asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23))/2) - sqrt(2)) + 112 896*sqrt(7)*log(sqrt(23) + sqrt(21)*tan(asin((sqrt( - 2*x + 5)*sqrt(3))/sq rt(23))/2) + sqrt(2)))/2304