Integrand size = 27, antiderivative size = 98 \[ \int \frac {(4+3 x)^5}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {6418499}{23814 (5-2 x)^{3/2}}-\frac {58527757}{500094 \sqrt {5-2 x}}+\frac {206+215 x}{63 (5-2 x)^{3/2} \left (2+3 x+x^2\right )}-\frac {2848 \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )}{2187}-\frac {192 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{343 \sqrt {7}} \] Output:
6418499/23814/(5-2*x)^(3/2)-58527757/500094/(5-2*x)^(1/2)+1/63*(206+215*x) /(5-2*x)^(3/2)/(x^2+3*x+2)-2848/2187*arctanh(1/3*(5-2*x)^(1/2))-192/2401*a rctanh(1/7*(5-2*x)^(1/2)*7^(1/2))*7^(1/2)
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84 \[ \int \frac {(4+3 x)^5}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {-157032692-118866610 x+96658118 x^2+58527757 x^3}{250047 (5-2 x)^{3/2} (1+x) (2+x)}-\frac {2848 \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )}{2187}-\frac {192 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{343 \sqrt {7}} \] Input:
Integrate[(4 + 3*x)^5/((5 - 2*x)^(5/2)*(2 + 3*x + x^2)^2),x]
Output:
(-157032692 - 118866610*x + 96658118*x^2 + 58527757*x^3)/(250047*(5 - 2*x) ^(3/2)*(1 + x)*(2 + x)) - (2848*ArcTanh[Sqrt[5 - 2*x]/3])/2187 - (192*ArcT anh[Sqrt[5 - 2*x]/Sqrt[7]])/(343*Sqrt[7])
Time = 0.54 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1265, 25, 2159, 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.
| \(\displaystyle \int \frac {(3 x+4)^5}{(5-2 x)^{5/2} \left (x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1265 |
| \(\displaystyle \frac {215 x+206}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}-\frac {1}{63} \int -\frac {15309 x^3+56133 x^2+72068 x+32108}{(5-2 x)^{5/2} \left (x^2+3 x+2\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{63} \int \frac {15309 x^3+56133 x^2+72068 x+32108}{(5-2 x)^{5/2} \left (x^2+3 x+2\right )}dx+\frac {215 x+206}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \frac {1}{63} \int \left (\frac {19936}{(5-2 x)^{5/2} (2 x+4)}-\frac {15309}{2 (5-2 x)^{3/2}}+\frac {96957}{2 (5-2 x)^{5/2}}+\frac {1728}{(5-2 x)^{5/2} (2 x+2)}\right )dx+\frac {215 x+206}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{63} \left (-\frac {19936}{243} \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )-\frac {1728 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{49 \sqrt {7}}-\frac {58527757}{7938 \sqrt {5-2 x}}+\frac {6418499}{378 (5-2 x)^{3/2}}\right )+\frac {215 x+206}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\) |
Input:
Int[(4 + 3*x)^5/((5 - 2*x)^(5/2)*(2 + 3*x + x^2)^2),x]
Output:
(206 + 215*x)/(63*(5 - 2*x)^(3/2)*(2 + 3*x + x^2)) + (6418499/(378*(5 - 2* x)^(3/2)) - 58527757/(7938*Sqrt[5 - 2*x]) - (19936*ArcTanh[Sqrt[5 - 2*x]/3 ])/243 - (1728*ArcTanh[Sqrt[5 - 2*x]/Sqrt[7]])/(49*Sqrt[7]))/63
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x) ^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[(f + g*x)^n, a + b*x + c *x^2, x], x, 1]}, Simp[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)*((R*(b*c *d - b^2*e + 2*a*c*e) - a*S*(2*c*d - b*e) + c*(R*(2*c*d - b*e) - S*(b*d - 2 *a*e))*x)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*(a + b*x + c *x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Q + R*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a *c*e^2*(m + 2*p + 3)) - S*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(S*(b*d - 2*a*e) - R*(2*c*d - b*e))*(m + 2*p + 4)* x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && IGtQ[n, 1] && LtQ[p , -1] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 1.94 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87
| method | result | size |
| risch | \(-\frac {58527757 x^{3}+96658118 x^{2}-118866610 x -157032692}{250047 \left (x^{2}+3 x +2\right ) \sqrt {5-2 x}\, \left (-5+2 x \right )}-\frac {1424 \ln \left (\sqrt {5-2 x}+3\right )}{2187}-\frac {192 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{2401}+\frac {1424 \ln \left (\sqrt {5-2 x}-3\right )}{2187}\) | \(85\) |
| derivativedivides | \(\frac {2 \sqrt {5-2 x}}{343 \left (-2 x -2\right )}-\frac {192 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{2401}-\frac {32}{729 \left (\sqrt {5-2 x}-3\right )}+\frac {1424 \ln \left (\sqrt {5-2 x}-3\right )}{2187}-\frac {32}{729 \left (\sqrt {5-2 x}+3\right )}-\frac {1424 \ln \left (\sqrt {5-2 x}+3\right )}{2187}+\frac {6436343}{23814 \left (5-2 x \right )^{\frac {3}{2}}}-\frac {58486769}{500094 \sqrt {5-2 x}}\) | \(104\) |
| default | \(\frac {2 \sqrt {5-2 x}}{343 \left (-2 x -2\right )}-\frac {192 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{2401}-\frac {32}{729 \left (\sqrt {5-2 x}-3\right )}+\frac {1424 \ln \left (\sqrt {5-2 x}-3\right )}{2187}-\frac {32}{729 \left (\sqrt {5-2 x}+3\right )}-\frac {1424 \ln \left (\sqrt {5-2 x}+3\right )}{2187}+\frac {6436343}{23814 \left (5-2 x \right )^{\frac {3}{2}}}-\frac {58486769}{500094 \sqrt {5-2 x}}\) | \(104\) |
| trager | \(\frac {\left (58527757 x^{3}+96658118 x^{2}-118866610 x -157032692\right ) \sqrt {5-2 x}}{250047 \left (-5+2 x \right )^{2} \left (x^{2}+3 x +2\right )}-\frac {96 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x +7 \sqrt {5-2 x}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right )}{x +1}\right )}{2401}+\frac {1424 \ln \left (\frac {-7+x +3 \sqrt {5-2 x}}{2+x}\right )}{2187}\) | \(106\) |
| pseudoelliptic | \(\frac {\frac {384 \sqrt {5-2 x}\, \left (2+x \right ) \sqrt {7}\, \left (x +1\right ) \left (x -\frac {5}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right )}{2401}-\frac {2848 \sqrt {5-2 x}\, \left (2+x \right ) \left (x +1\right ) \left (x -\frac {5}{2}\right ) \ln \left (\sqrt {5-2 x}-3\right )}{2187}+\frac {2848 \sqrt {5-2 x}\, \left (2+x \right ) \left (x +1\right ) \left (x -\frac {5}{2}\right ) \ln \left (\sqrt {5-2 x}+3\right )}{2187}+\frac {58527757 x^{3}}{250047}+\frac {96658118 x^{2}}{250047}-\frac {118866610 x}{250047}-\frac {157032692}{250047}}{\left (x +1\right ) \left (5-2 x \right )^{\frac {3}{2}} \left (2+x \right )}\) | \(124\) |
Input:
int((3*x+4)^5/(5-2*x)^(5/2)/(x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
-1/250047*(58527757*x^3+96658118*x^2-118866610*x-157032692)/(x^2+3*x+2)/(5 -2*x)^(1/2)/(-5+2*x)-1424/2187*ln((5-2*x)^(1/2)+3)-192/2401*arctanh(1/7*(5 -2*x)^(1/2)*7^(1/2))*7^(1/2)+1424/2187*ln((5-2*x)^(1/2)-3)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (73) = 146\).
Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.62 \[ \int \frac {(4+3 x)^5}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {209952 \, \sqrt {7} {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )} \log \left (\frac {x + \sqrt {7} \sqrt {-2 \, x + 5} - 6}{x + 1}\right ) - 3419024 \, {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )} \log \left (\sqrt {-2 \, x + 5} + 3\right ) + 3419024 \, {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )} \log \left (\sqrt {-2 \, x + 5} - 3\right ) + 21 \, {\left (58527757 \, x^{3} + 96658118 \, x^{2} - 118866610 \, x - 157032692\right )} \sqrt {-2 \, x + 5}}{5250987 \, {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )}} \] Input:
integrate((4+3*x)^5/(5-2*x)^(5/2)/(x^2+3*x+2)^2,x, algorithm="fricas")
Output:
1/5250987*(209952*sqrt(7)*(4*x^4 - 8*x^3 - 27*x^2 + 35*x + 50)*log((x + sq rt(7)*sqrt(-2*x + 5) - 6)/(x + 1)) - 3419024*(4*x^4 - 8*x^3 - 27*x^2 + 35* x + 50)*log(sqrt(-2*x + 5) + 3) + 3419024*(4*x^4 - 8*x^3 - 27*x^2 + 35*x + 50)*log(sqrt(-2*x + 5) - 3) + 21*(58527757*x^3 + 96658118*x^2 - 118866610 *x - 157032692)*sqrt(-2*x + 5))/(4*x^4 - 8*x^3 - 27*x^2 + 35*x + 50)
Time = 108.42 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.41 \[ \int \frac {(4+3 x)^5}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {95 \sqrt {7} \left (\log {\left (\sqrt {5 - 2 x} - \sqrt {7} \right )} - \log {\left (\sqrt {5 - 2 x} + \sqrt {7} \right )}\right )}{2401} - \frac {4 \left (\begin {cases} \frac {\sqrt {7} \left (- \frac {\log {\left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} - 1\right )}\right )}{49} & \text {for}\: \sqrt {5 - 2 x} > - \sqrt {7} \wedge \sqrt {5 - 2 x} < \sqrt {7} \end {cases}\right )}{49} + \frac {1424 \log {\left (\sqrt {5 - 2 x} - 3 \right )}}{2187} - \frac {1424 \log {\left (\sqrt {5 - 2 x} + 3 \right )}}{2187} - \frac {32}{729 \left (\sqrt {5 - 2 x} + 3\right )} - \frac {32}{729 \left (\sqrt {5 - 2 x} - 3\right )} - \frac {58486769}{500094 \sqrt {5 - 2 x}} + \frac {6436343}{23814 \left (5 - 2 x\right )^{\frac {3}{2}}} \] Input:
integrate((4+3*x)**5/(5-2*x)**(5/2)/(x**2+3*x+2)**2,x)
Output:
95*sqrt(7)*(log(sqrt(5 - 2*x) - sqrt(7)) - log(sqrt(5 - 2*x) + sqrt(7)))/2 401 - 4*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*(s qrt(7)*sqrt(5 - 2*x)/7 - 1)))/49, (sqrt(5 - 2*x) < sqrt(7)) & (sqrt(5 - 2* x) > -sqrt(7))))/49 + 1424*log(sqrt(5 - 2*x) - 3)/2187 - 1424*log(sqrt(5 - 2*x) + 3)/2187 - 32/(729*(sqrt(5 - 2*x) + 3)) - 32/(729*(sqrt(5 - 2*x) - 3)) - 58486769/(500094*sqrt(5 - 2*x)) + 6436343/(23814*(5 - 2*x)**(3/2))
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int \frac {(4+3 x)^5}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {96}{2401} \, \sqrt {7} \log \left (-\frac {\sqrt {7} - \sqrt {-2 \, x + 5}}{\sqrt {7} + \sqrt {-2 \, x + 5}}\right ) + \frac {58527757 \, {\left (2 \, x - 5\right )}^{3} + 1071232591 \, {\left (2 \, x - 5\right )}^{2} + 11694555390 \, x - 20721106686}{500094 \, {\left ({\left (-2 \, x + 5\right )}^{\frac {7}{2}} - 16 \, {\left (-2 \, x + 5\right )}^{\frac {5}{2}} + 63 \, {\left (-2 \, x + 5\right )}^{\frac {3}{2}}\right )}} - \frac {1424}{2187} \, \log \left (\sqrt {-2 \, x + 5} + 3\right ) + \frac {1424}{2187} \, \log \left (\sqrt {-2 \, x + 5} - 3\right ) \] Input:
integrate((4+3*x)^5/(5-2*x)^(5/2)/(x^2+3*x+2)^2,x, algorithm="maxima")
Output:
96/2401*sqrt(7)*log(-(sqrt(7) - sqrt(-2*x + 5))/(sqrt(7) + sqrt(-2*x + 5)) ) + 1/500094*(58527757*(2*x - 5)^3 + 1071232591*(2*x - 5)^2 + 11694555390* x - 20721106686)/((-2*x + 5)^(7/2) - 16*(-2*x + 5)^(5/2) + 63*(-2*x + 5)^( 3/2)) - 1424/2187*log(sqrt(-2*x + 5) + 3) + 1424/2187*log(sqrt(-2*x + 5) - 3)
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.21 \[ \int \frac {(4+3 x)^5}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {96}{2401} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + 2 \, \sqrt {-2 \, x + 5} \right |}}{2 \, {\left (\sqrt {7} + \sqrt {-2 \, x + 5}\right )}}\right ) - \frac {2 \, {\left (10247 \, {\left (-2 \, x + 5\right )}^{\frac {3}{2}} - 70271 \, \sqrt {-2 \, x + 5}\right )}}{250047 \, {\left ({\left (2 \, x - 5\right )}^{2} + 32 \, x - 17\right )}} - \frac {279841 \, {\left (209 \, x - 281\right )}}{250047 \, {\left (2 \, x - 5\right )} \sqrt {-2 \, x + 5}} - \frac {1424}{2187} \, \log \left (\sqrt {-2 \, x + 5} + 3\right ) + \frac {1424}{2187} \, \log \left ({\left | \sqrt {-2 \, x + 5} - 3 \right |}\right ) \] Input:
integrate((4+3*x)^5/(5-2*x)^(5/2)/(x^2+3*x+2)^2,x, algorithm="giac")
Output:
96/2401*sqrt(7)*log(1/2*abs(-2*sqrt(7) + 2*sqrt(-2*x + 5))/(sqrt(7) + sqrt (-2*x + 5))) - 2/250047*(10247*(-2*x + 5)^(3/2) - 70271*sqrt(-2*x + 5))/(( 2*x - 5)^2 + 32*x - 17) - 279841/250047*(209*x - 281)/((2*x - 5)*sqrt(-2*x + 5)) - 1424/2187*log(sqrt(-2*x + 5) + 3) + 1424/2187*log(abs(sqrt(-2*x + 5) - 3))
Time = 13.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85 \[ \int \frac {(4+3 x)^5}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {\frac {278441795\,x}{11907}+\frac {1071232591\,{\left (2\,x-5\right )}^2}{500094}+\frac {58527757\,{\left (2\,x-5\right )}^3}{500094}-\frac {493359683}{11907}}{63\,{\left (5-2\,x\right )}^{3/2}-16\,{\left (5-2\,x\right )}^{5/2}+{\left (5-2\,x\right )}^{7/2}}-\frac {2848\,\mathrm {atanh}\left (\frac {\sqrt {5-2\,x}}{3}\right )}{2187}-\frac {192\,\sqrt {7}\,\mathrm {atanh}\left (\frac {\sqrt {7}\,\sqrt {5-2\,x}}{7}\right )}{2401} \] Input:
int((3*x + 4)^5/((5 - 2*x)^(5/2)*(3*x + x^2 + 2)^2),x)
Output:
((278441795*x)/11907 + (1071232591*(2*x - 5)^2)/500094 + (58527757*(2*x - 5)^3)/500094 - 493359683/11907)/(63*(5 - 2*x)^(3/2) - 16*(5 - 2*x)^(5/2) + (5 - 2*x)^(7/2)) - (2848*atanh((5 - 2*x)^(1/2)/3))/2187 - (192*7^(1/2)*at anh((7^(1/2)*(5 - 2*x)^(1/2))/7))/2401
Time = 0.18 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.80 \[ \int \frac {(4+3 x)^5}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {419904 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right ) x^{3}+209952 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right ) x^{2}-2309472 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right ) x -2099520 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right )-419904 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right ) x^{3}-209952 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right ) x^{2}+2309472 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right ) x +2099520 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right )+6838048 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right ) x^{3}+3419024 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right ) x^{2}-37609264 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right ) x -34190240 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right )-6838048 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right ) x^{3}-3419024 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right ) x^{2}+37609264 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right ) x +34190240 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right )-1229082897 x^{3}-2029820478 x^{2}+2496198810 x +3297686532}{5250987 \sqrt {-2 x +5}\, \left (2 x^{3}+x^{2}-11 x -10\right )} \] Input:
int((4+3*x)^5/(5-2*x)^(5/2)/(x^2+3*x+2)^2,x)
Output:
(419904*sqrt( - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7))*x**3 + 20 9952*sqrt( - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7))*x**2 - 23094 72*sqrt( - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7))*x - 2099520*sq rt( - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7)) - 419904*sqrt( - 2* x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x**3 - 209952*sqrt( - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x**2 + 2309472*sqrt( - 2*x + 5 )*sqrt(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x + 2099520*sqrt( - 2*x + 5)*sqr t(7)*log(sqrt( - 2*x + 5) + sqrt(7)) + 6838048*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) - 3)*x**3 + 3419024*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) - 3)* x**2 - 37609264*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) - 3)*x - 34190240*sq rt( - 2*x + 5)*log(sqrt( - 2*x + 5) - 3) - 6838048*sqrt( - 2*x + 5)*log(sq rt( - 2*x + 5) + 3)*x**3 - 3419024*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) + 3)*x**2 + 37609264*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) + 3)*x + 3419024 0*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) + 3) - 1229082897*x**3 - 202982047 8*x**2 + 2496198810*x + 3297686532)/(5250987*sqrt( - 2*x + 5)*(2*x**3 + x* *2 - 11*x - 10))