Integrand size = 27, antiderivative size = 139 \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=-\frac {27 \sqrt {5-2 x}}{8 (4+3 x)^2}-\frac {1215 \sqrt {5-2 x}}{184 (4+3 x)}-\frac {\sqrt {5-2 x} (15+7 x)}{8 \left (2+3 x+x^2\right )}+\frac {115}{24} \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )-\frac {32481}{184} \sqrt {\frac {3}{23}} \text {arctanh}\left (\sqrt {\frac {3}{23}} \sqrt {5-2 x}\right )+\frac {156 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{\sqrt {7}} \] Output:
-27/8*(5-2*x)^(1/2)/(4+3*x)^2-1215*(5-2*x)^(1/2)/(736+552*x)-(5-2*x)^(1/2) *(15+7*x)/(8*x^2+24*x+16)+115/24*arctanh(1/3*(5-2*x)^(1/2))-32481/4232*69^ (1/2)*arctanh(1/23*69^(1/2)*(5-2*x)^(1/2))+156/7*arctanh(1/7*(5-2*x)^(1/2) *7^(1/2))*7^(1/2)
Time = 0.40 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=-\frac {\sqrt {5-2 x} \left (16482+34589 x+23385 x^2+5094 x^3\right )}{184 (4+3 x)^2 \left (2+3 x+x^2\right )}+\frac {115}{24} \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )-\frac {32481}{184} \sqrt {\frac {3}{23}} \text {arctanh}\left (\sqrt {\frac {3}{23}} \sqrt {5-2 x}\right )+\frac {156 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{\sqrt {7}} \] Input:
Integrate[Sqrt[5 - 2*x]/((4 + 3*x)^3*(2 + 3*x + x^2)^2),x]
Output:
-1/184*(Sqrt[5 - 2*x]*(16482 + 34589*x + 23385*x^2 + 5094*x^3))/((4 + 3*x) ^2*(2 + 3*x + x^2)) + (115*ArcTanh[Sqrt[5 - 2*x]/3])/24 - (32481*Sqrt[3/23 ]*ArcTanh[Sqrt[3/23]*Sqrt[5 - 2*x]])/184 + (156*ArcTanh[Sqrt[5 - 2*x]/Sqrt [7]])/Sqrt[7]
Time = 0.58 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.40, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1289, 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.
| \(\displaystyle \int \frac {\sqrt {5-2 x}}{(3 x+4)^3 \left (x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle \int \left (-\frac {11 \sqrt {5-2 x}}{x+1}-\frac {13 \sqrt {5-2 x}}{16 (x+2)}+\frac {567 \sqrt {5-2 x}}{16 (3 x+4)}+\frac {\sqrt {5-2 x}}{(x+1)^2}-\frac {\sqrt {5-2 x}}{8 (x+2)^2}+\frac {81 \sqrt {5-2 x}}{4 (3 x+4)^2}+\frac {81 \sqrt {5-2 x}}{4 (3 x+4)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {115}{24} \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )-\frac {63}{8} \sqrt {69} \text {arctanh}\left (\sqrt {\frac {3}{23}} \sqrt {5-2 x}\right )+\frac {423}{92} \sqrt {\frac {3}{23}} \text {arctanh}\left (\sqrt {\frac {3}{23}} \sqrt {5-2 x}\right )+22 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )+\frac {2 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{\sqrt {7}}-\frac {\sqrt {5-2 x}}{x+1}+\frac {\sqrt {5-2 x}}{8 (x+2)}-\frac {1215 \sqrt {5-2 x}}{184 (3 x+4)}-\frac {27 \sqrt {5-2 x}}{8 (3 x+4)^2}\) |
Input:
Int[Sqrt[5 - 2*x]/((4 + 3*x)^3*(2 + 3*x + x^2)^2),x]
Output:
-(Sqrt[5 - 2*x]/(1 + x)) + Sqrt[5 - 2*x]/(8*(2 + x)) - (27*Sqrt[5 - 2*x])/ (8*(4 + 3*x)^2) - (1215*Sqrt[5 - 2*x])/(184*(4 + 3*x)) + (115*ArcTanh[Sqrt [5 - 2*x]/3])/24 + (423*Sqrt[3/23]*ArcTanh[Sqrt[3/23]*Sqrt[5 - 2*x]])/92 - (63*Sqrt[69]*ArcTanh[Sqrt[3/23]*Sqrt[5 - 2*x]])/8 + (2*ArcTanh[Sqrt[5 - 2 *x]/Sqrt[7]])/Sqrt[7] + 22*Sqrt[7]*ArcTanh[Sqrt[5 - 2*x]/Sqrt[7]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(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}, x] && ( IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
Time = 2.34 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(\frac {10188 x^{4}+21300 x^{3}-47747 x^{2}-139981 x -82410}{184 \left (3 x +4\right )^{2} \sqrt {5-2 x}\, \left (x^{2}+3 x +2\right )}+\frac {115 \ln \left (\sqrt {5-2 x}+3\right )}{48}+\frac {156 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{7}-\frac {32481 \sqrt {69}\, \operatorname {arctanh}\left (\frac {\sqrt {69}\, \sqrt {5-2 x}}{23}\right )}{4232}-\frac {115 \ln \left (\sqrt {5-2 x}-3\right )}{48}\) | \(108\) |
| derivativedivides | \(\frac {2 \sqrt {5-2 x}}{-2 x -2}+\frac {156 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{7}-\frac {1}{8 \left (\sqrt {5-2 x}+3\right )}+\frac {115 \ln \left (\sqrt {5-2 x}+3\right )}{48}-\frac {1}{8 \left (\sqrt {5-2 x}-3\right )}-\frac {115 \ln \left (\sqrt {5-2 x}-3\right )}{48}+\frac {\frac {3645 \left (5-2 x \right )^{\frac {3}{2}}}{92}-\frac {1269 \sqrt {5-2 x}}{4}}{\left (-8-6 x \right )^{2}}-\frac {32481 \sqrt {69}\, \operatorname {arctanh}\left (\frac {\sqrt {69}\, \sqrt {5-2 x}}{23}\right )}{4232}\) | \(132\) |
| default | \(\frac {2 \sqrt {5-2 x}}{-2 x -2}+\frac {156 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{7}-\frac {1}{8 \left (\sqrt {5-2 x}+3\right )}+\frac {115 \ln \left (\sqrt {5-2 x}+3\right )}{48}-\frac {1}{8 \left (\sqrt {5-2 x}-3\right )}-\frac {115 \ln \left (\sqrt {5-2 x}-3\right )}{48}+\frac {\frac {3645 \left (5-2 x \right )^{\frac {3}{2}}}{92}-\frac {1269 \sqrt {5-2 x}}{4}}{\left (-8-6 x \right )^{2}}-\frac {32481 \sqrt {69}\, \operatorname {arctanh}\left (\frac {\sqrt {69}\, \sqrt {5-2 x}}{23}\right )}{4232}\) | \(132\) |
| pseudoelliptic | \(\frac {-1364202 \left (x +\frac {4}{3}\right )^{2} \left (2+x \right ) \left (x +1\right ) \sqrt {69}\, \operatorname {arctanh}\left (\frac {\sqrt {69}\, \sqrt {5-2 x}}{23}\right )+3961152 \left (x +\frac {4}{3}\right )^{2} \left (2+x \right ) \sqrt {7}\, \left (x +1\right ) \operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right )-425845 \left (x +\frac {4}{3}\right )^{2} \left (2+x \right ) \left (x +1\right ) \ln \left (\sqrt {5-2 x}-3\right )+425845 \left (x +\frac {4}{3}\right )^{2} \left (2+x \right ) \left (x +1\right ) \ln \left (\sqrt {5-2 x}+3\right )-546756 \sqrt {5-2 x}\, \left (x^{3}+\frac {7795}{1698} x^{2}+\frac {34589}{5094} x +\frac {2747}{849}\right )}{177744 \left (x +\frac {4}{3}\right )^{2} \left (2+x \right ) \left (x +1\right )}\) | \(145\) |
| trager | \(-\frac {\left (5094 x^{3}+23385 x^{2}+34589 x +16482\right ) \sqrt {5-2 x}}{184 \left (3 x +4\right )^{2} \left (x^{2}+3 x +2\right )}+\frac {115 \ln \left (\frac {3 \sqrt {5-2 x}+7-x}{2+x}\right )}{48}-\frac {78 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x -6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right )+7 \sqrt {5-2 x}}{x +1}\right )}{7}-\frac {81 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11095269\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11095269\right ) x +27669 \sqrt {5-2 x}+19 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11095269\right )}{3 x +4}\right )}{8464}\) | \(151\) |
Input:
int((5-2*x)^(1/2)/(3*x+4)^3/(x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
1/184*(10188*x^4+21300*x^3-47747*x^2-139981*x-82410)/(3*x+4)^2/(5-2*x)^(1/ 2)/(x^2+3*x+2)+115/48*ln((5-2*x)^(1/2)+3)+156/7*arctanh(1/7*(5-2*x)^(1/2)* 7^(1/2))*7^(1/2)-32481/4232*69^(1/2)*arctanh(1/23*69^(1/2)*(5-2*x)^(1/2))- 115/48*ln((5-2*x)^(1/2)-3)
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (105) = 210\).
Time = 0.08 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=\frac {682101 \, \sqrt {\frac {3}{23}} {\left (9 \, x^{4} + 51 \, x^{3} + 106 \, x^{2} + 96 \, x + 32\right )} \log \left (\frac {3 \, x + 23 \, \sqrt {\frac {3}{23}} \sqrt {-2 \, x + 5} - 19}{3 \, x + 4}\right ) + 86112 \, \sqrt {7} {\left (9 \, x^{4} + 51 \, x^{3} + 106 \, x^{2} + 96 \, x + 32\right )} \log \left (\frac {x - \sqrt {7} \sqrt {-2 \, x + 5} - 6}{x + 1}\right ) + 18515 \, {\left (9 \, x^{4} + 51 \, x^{3} + 106 \, x^{2} + 96 \, x + 32\right )} \log \left (\sqrt {-2 \, x + 5} + 3\right ) - 18515 \, {\left (9 \, x^{4} + 51 \, x^{3} + 106 \, x^{2} + 96 \, x + 32\right )} \log \left (\sqrt {-2 \, x + 5} - 3\right ) - 42 \, {\left (5094 \, x^{3} + 23385 \, x^{2} + 34589 \, x + 16482\right )} \sqrt {-2 \, x + 5}}{7728 \, {\left (9 \, x^{4} + 51 \, x^{3} + 106 \, x^{2} + 96 \, x + 32\right )}} \] Input:
integrate((5-2*x)^(1/2)/(4+3*x)^3/(x^2+3*x+2)^2,x, algorithm="fricas")
Output:
1/7728*(682101*sqrt(3/23)*(9*x^4 + 51*x^3 + 106*x^2 + 96*x + 32)*log((3*x + 23*sqrt(3/23)*sqrt(-2*x + 5) - 19)/(3*x + 4)) + 86112*sqrt(7)*(9*x^4 + 5 1*x^3 + 106*x^2 + 96*x + 32)*log((x - sqrt(7)*sqrt(-2*x + 5) - 6)/(x + 1)) + 18515*(9*x^4 + 51*x^3 + 106*x^2 + 96*x + 32)*log(sqrt(-2*x + 5) + 3) - 18515*(9*x^4 + 51*x^3 + 106*x^2 + 96*x + 32)*log(sqrt(-2*x + 5) - 3) - 42* (5094*x^3 + 23385*x^2 + 34589*x + 16482)*sqrt(-2*x + 5))/(9*x^4 + 51*x^3 + 106*x^2 + 96*x + 32)
Time = 56.94 (sec) , antiderivative size = 539, normalized size of antiderivative = 3.88 \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=\frac {1377 \sqrt {69} \left (\log {\left (\sqrt {5 - 2 x} - \frac {\sqrt {69}}{3} \right )} - \log {\left (\sqrt {5 - 2 x} + \frac {\sqrt {69}}{3} \right )}\right )}{368} - \frac {79 \sqrt {7} \left (\log {\left (\sqrt {5 - 2 x} - \sqrt {7} \right )} - \log {\left (\sqrt {5 - 2 x} + \sqrt {7} \right )}\right )}{7} - 567 \left (\begin {cases} \frac {\sqrt {69} \left (- \frac {\log {\left (\frac {\sqrt {69} \sqrt {5 - 2 x}}{23} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {69} \sqrt {5 - 2 x}}{23} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {69} \sqrt {5 - 2 x}}{23} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {69} \sqrt {5 - 2 x}}{23} - 1\right )}\right )}{1587} & \text {for}\: \sqrt {5 - 2 x} > - \frac {\sqrt {69}}{3} \wedge \sqrt {5 - 2 x} < \frac {\sqrt {69}}{3} \end {cases}\right ) + 1242 \left (\begin {cases} \frac {\sqrt {69} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {69} \sqrt {5 - 2 x}}{23} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {69} \sqrt {5 - 2 x}}{23} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {69} \sqrt {5 - 2 x}}{23} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {69} \sqrt {5 - 2 x}}{23} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {69} \sqrt {5 - 2 x}}{23} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {69} \sqrt {5 - 2 x}}{23} - 1\right )^{2}}\right )}{36501} & \text {for}\: \sqrt {5 - 2 x} > - \frac {\sqrt {69}}{3} \wedge \sqrt {5 - 2 x} < \frac {\sqrt {69}}{3} \end {cases}\right ) - 28 \left (\begin {cases} \frac {\sqrt {7} \left (- \frac {\log {\left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} - 1\right )}\right )}{49} & \text {for}\: \sqrt {5 - 2 x} > - \sqrt {7} \wedge \sqrt {5 - 2 x} < \sqrt {7} \end {cases}\right ) - \frac {115 \log {\left (\sqrt {5 - 2 x} - 3 \right )}}{48} + \frac {115 \log {\left (\sqrt {5 - 2 x} + 3 \right )}}{48} - \frac {1}{8 \left (\sqrt {5 - 2 x} + 3\right )} - \frac {1}{8 \left (\sqrt {5 - 2 x} - 3\right )} \] Input:
integrate((5-2*x)**(1/2)/(4+3*x)**3/(x**2+3*x+2)**2,x)
Output:
1377*sqrt(69)*(log(sqrt(5 - 2*x) - sqrt(69)/3) - log(sqrt(5 - 2*x) + sqrt( 69)/3))/368 - 79*sqrt(7)*(log(sqrt(5 - 2*x) - sqrt(7)) - log(sqrt(5 - 2*x) + sqrt(7)))/7 - 567*Piecewise((sqrt(69)*(-log(sqrt(69)*sqrt(5 - 2*x)/23 - 1)/4 + log(sqrt(69)*sqrt(5 - 2*x)/23 + 1)/4 - 1/(4*(sqrt(69)*sqrt(5 - 2*x )/23 + 1)) - 1/(4*(sqrt(69)*sqrt(5 - 2*x)/23 - 1)))/1587, (sqrt(5 - 2*x) > -sqrt(69)/3) & (sqrt(5 - 2*x) < sqrt(69)/3))) + 1242*Piecewise((sqrt(69)* (3*log(sqrt(69)*sqrt(5 - 2*x)/23 - 1)/16 - 3*log(sqrt(69)*sqrt(5 - 2*x)/23 + 1)/16 + 3/(16*(sqrt(69)*sqrt(5 - 2*x)/23 + 1)) + 1/(16*(sqrt(69)*sqrt(5 - 2*x)/23 + 1)**2) + 3/(16*(sqrt(69)*sqrt(5 - 2*x)/23 - 1)) - 1/(16*(sqrt (69)*sqrt(5 - 2*x)/23 - 1)**2))/36501, (sqrt(5 - 2*x) > -sqrt(69)/3) & (sq rt(5 - 2*x) < sqrt(69)/3))) - 28*Piecewise((sqrt(7)*(-log(sqrt(7)*sqrt(5 - 2*x)/7 - 1)/4 + log(sqrt(7)*sqrt(5 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(7)*sqrt(5 - 2*x)/7 + 1)) - 1/(4*(sqrt(7)*sqrt(5 - 2*x)/7 - 1)))/49, (sqrt(5 - 2*x) < sqrt(7)) & (sqrt(5 - 2*x) > -sqrt(7)))) - 115*log(sqrt(5 - 2*x) - 3)/48 + 115*log(sqrt(5 - 2*x) + 3)/48 - 1/(8*(sqrt(5 - 2*x) + 3)) - 1/(8*(sqrt(5 - 2*x) - 3))
Time = 0.16 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=\frac {32481}{8464} \, \sqrt {69} \log \left (-\frac {\sqrt {69} - 3 \, \sqrt {-2 \, x + 5}}{\sqrt {69} + 3 \, \sqrt {-2 \, x + 5}}\right ) - \frac {78}{7} \, \sqrt {7} \log \left (-\frac {\sqrt {7} - \sqrt {-2 \, x + 5}}{\sqrt {7} + \sqrt {-2 \, x + 5}}\right ) + \frac {2547 \, {\left (-2 \, x + 5\right )}^{\frac {7}{2}} - 61590 \, {\left (-2 \, x + 5\right )}^{\frac {5}{2}} + 494053 \, {\left (-2 \, x + 5\right )}^{\frac {3}{2}} - 1314818 \, \sqrt {-2 \, x + 5}}{46 \, {\left (9 \, {\left (2 \, x - 5\right )}^{4} + 282 \, {\left (2 \, x - 5\right )}^{3} + 3304 \, {\left (2 \, x - 5\right )}^{2} + 34316 \, x - 52463\right )}} + \frac {115}{48} \, \log \left (\sqrt {-2 \, x + 5} + 3\right ) - \frac {115}{48} \, \log \left (\sqrt {-2 \, x + 5} - 3\right ) \] Input:
integrate((5-2*x)^(1/2)/(4+3*x)^3/(x^2+3*x+2)^2,x, algorithm="maxima")
Output:
32481/8464*sqrt(69)*log(-(sqrt(69) - 3*sqrt(-2*x + 5))/(sqrt(69) + 3*sqrt( -2*x + 5))) - 78/7*sqrt(7)*log(-(sqrt(7) - sqrt(-2*x + 5))/(sqrt(7) + sqrt (-2*x + 5))) + 1/46*(2547*(-2*x + 5)^(7/2) - 61590*(-2*x + 5)^(5/2) + 4940 53*(-2*x + 5)^(3/2) - 1314818*sqrt(-2*x + 5))/(9*(2*x - 5)^4 + 282*(2*x - 5)^3 + 3304*(2*x - 5)^2 + 34316*x - 52463) + 115/48*log(sqrt(-2*x + 5) + 3 ) - 115/48*log(sqrt(-2*x + 5) - 3)
Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=\frac {32481}{8464} \, \sqrt {69} \log \left (\frac {{\left | -2 \, \sqrt {69} + 6 \, \sqrt {-2 \, x + 5} \right |}}{2 \, {\left (\sqrt {69} + 3 \, \sqrt {-2 \, x + 5}\right )}}\right ) - \frac {78}{7} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + 2 \, \sqrt {-2 \, x + 5} \right |}}{2 \, {\left (\sqrt {7} + \sqrt {-2 \, x + 5}\right )}}\right ) + \frac {7 \, {\left (-2 \, x + 5\right )}^{\frac {3}{2}} - 65 \, \sqrt {-2 \, x + 5}}{4 \, {\left ({\left (2 \, x - 5\right )}^{2} + 32 \, x - 17\right )}} + \frac {27 \, {\left (135 \, {\left (-2 \, x + 5\right )}^{\frac {3}{2}} - 1081 \, \sqrt {-2 \, x + 5}\right )}}{368 \, {\left (3 \, x + 4\right )}^{2}} + \frac {115}{48} \, \log \left (\sqrt {-2 \, x + 5} + 3\right ) - \frac {115}{48} \, \log \left ({\left | \sqrt {-2 \, x + 5} - 3 \right |}\right ) \] Input:
integrate((5-2*x)^(1/2)/(4+3*x)^3/(x^2+3*x+2)^2,x, algorithm="giac")
Output:
32481/8464*sqrt(69)*log(1/2*abs(-2*sqrt(69) + 6*sqrt(-2*x + 5))/(sqrt(69) + 3*sqrt(-2*x + 5))) - 78/7*sqrt(7)*log(1/2*abs(-2*sqrt(7) + 2*sqrt(-2*x + 5))/(sqrt(7) + sqrt(-2*x + 5))) + 1/4*(7*(-2*x + 5)^(3/2) - 65*sqrt(-2*x + 5))/((2*x - 5)^2 + 32*x - 17) + 27/368*(135*(-2*x + 5)^(3/2) - 1081*sqrt (-2*x + 5))/(3*x + 4)^2 + 115/48*log(sqrt(-2*x + 5) + 3) - 115/48*log(abs( sqrt(-2*x + 5) - 3))
Time = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx=-\frac {\frac {28583\,\sqrt {5-2\,x}}{9}-\frac {494053\,{\left (5-2\,x\right )}^{3/2}}{414}+\frac {10265\,{\left (5-2\,x\right )}^{5/2}}{69}-\frac {283\,{\left (5-2\,x\right )}^{7/2}}{46}}{\frac {34316\,x}{9}+\frac {3304\,{\left (2\,x-5\right )}^2}{9}+\frac {94\,{\left (2\,x-5\right )}^3}{3}+{\left (2\,x-5\right )}^4-\frac {52463}{9}}-\frac {\mathrm {atan}\left (\frac {\sqrt {5-2\,x}\,1{}\mathrm {i}}{3}\right )\,115{}\mathrm {i}}{24}-\frac {\sqrt {7}\,\mathrm {atan}\left (\frac {\sqrt {7}\,\sqrt {5-2\,x}\,1{}\mathrm {i}}{7}\right )\,156{}\mathrm {i}}{7}+\frac {\sqrt {69}\,\mathrm {atan}\left (\frac {\sqrt {69}\,\sqrt {5-2\,x}\,1{}\mathrm {i}}{23}\right )\,32481{}\mathrm {i}}{4232} \] Input:
int((5 - 2*x)^(1/2)/((3*x + 4)^3*(3*x + x^2 + 2)^2),x)
Output:
(69^(1/2)*atan((69^(1/2)*(5 - 2*x)^(1/2)*1i)/23)*32481i)/4232 - (7^(1/2)*a tan((7^(1/2)*(5 - 2*x)^(1/2)*1i)/7)*156i)/7 - (atan(((5 - 2*x)^(1/2)*1i)/3 )*115i)/24 - ((28583*(5 - 2*x)^(1/2))/9 - (494053*(5 - 2*x)^(3/2))/414 + ( 10265*(5 - 2*x)^(5/2))/69 - (283*(5 - 2*x)^(7/2))/46)/((34316*x)/9 + (3304 *(2*x - 5)^2)/9 + (94*(2*x - 5)^3)/3 + (2*x - 5)^4 - 52463/9)
Time = 0.19 (sec) , antiderivative size = 553, normalized size of antiderivative = 3.98 \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^3 \left (2+3 x+x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((5-2*x)^(1/2)/(4+3*x)^3/(x^2+3*x+2)^2,x)
Output:
( - 4920804*sqrt( - 2*x + 5)*x**3 - 22589910*sqrt( - 2*x + 5)*x**2 - 33412 974*sqrt( - 2*x + 5)*x - 15921612*sqrt( - 2*x + 5) + 6138909*sqrt(69)*log( 3*sqrt( - 2*x + 5) - sqrt(69))*x**4 + 34787151*sqrt(69)*log(3*sqrt( - 2*x + 5) - sqrt(69))*x**3 + 72302706*sqrt(69)*log(3*sqrt( - 2*x + 5) - sqrt(69 ))*x**2 + 65481696*sqrt(69)*log(3*sqrt( - 2*x + 5) - sqrt(69))*x + 2182723 2*sqrt(69)*log(3*sqrt( - 2*x + 5) - sqrt(69)) - 6138909*sqrt(69)*log(3*sqr t( - 2*x + 5) + sqrt(69))*x**4 - 34787151*sqrt(69)*log(3*sqrt( - 2*x + 5) + sqrt(69))*x**3 - 72302706*sqrt(69)*log(3*sqrt( - 2*x + 5) + sqrt(69))*x* *2 - 65481696*sqrt(69)*log(3*sqrt( - 2*x + 5) + sqrt(69))*x - 21827232*sqr t(69)*log(3*sqrt( - 2*x + 5) + sqrt(69)) - 17825184*sqrt(7)*log(sqrt( - 2* x + 5) - sqrt(7))*x**4 - 101009376*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7)) *x**3 - 209941056*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7))*x**2 - 190135296 *sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7))*x - 63378432*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7)) + 17825184*sqrt(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x** 4 + 101009376*sqrt(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x**3 + 209941056*sqr t(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x**2 + 190135296*sqrt(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x + 63378432*sqrt(7)*log(sqrt( - 2*x + 5) + sqrt(7)) - 3832605*log(sqrt( - 2*x + 5) - 3)*x**4 - 21718095*log(sqrt( - 2*x + 5) - 3)*x**3 - 45139570*log(sqrt( - 2*x + 5) - 3)*x**2 - 40881120*log(sqrt( - 2 *x + 5) - 3)*x - 13627040*log(sqrt( - 2*x + 5) - 3) + 3832605*log(sqrt(...