Integrand size = 26, antiderivative size = 137 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {4433 \sqrt {1-2 x}}{12 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {539 \sqrt {1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac {40213 \sqrt {1-2 x}}{12 \sqrt {3+5 x}}-\frac {13145}{4} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \] Output:
-4433/12*(1-2*x)^(1/2)/(3+5*x)^(3/2)+7/6*(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^( 3/2)+539/12*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(3/2)+40213/12*(1-2*x)^(1/2)/(3+ 5*x)^(1/2)-13145/4*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
Time = 1.94 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {1}{12} \left (\frac {\sqrt {1-2 x} \left (465916+2200321 x+3458634 x^2+1809585 x^3\right )}{(2+3 x)^2 (3+5 x)^{3/2}}+39435 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+39435 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right ) \] Input:
Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^3*(3 + 5*x)^(5/2)),x]
Output:
((Sqrt[1 - 2*x]*(465916 + 2200321*x + 3458634*x^2 + 1809585*x^3))/((2 + 3* x)^2*(3 + 5*x)^(3/2)) + 39435*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sq rt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 39435*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))])/12
Time = 0.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {107, 105, 105, 105, 104, 217}
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^3 (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {239}{28} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {239}{28} \left (\frac {55}{2} \int \frac {(1-2 x)^{3/2}}{(3 x+2) (5 x+3)^{5/2}}dx+\frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {239}{28} \left (\frac {55}{2} \left (-7 \int \frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}\right )+\frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {239}{28} \left (\frac {55}{2} \left (-7 \left (-7 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}\right )+\frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {239}{28} \left (\frac {55}{2} \left (-7 \left (-14 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}\right )+\frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {239}{28} \left (\frac {55}{2} \left (-7 \left (2 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}\right )+\frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
Input:
Int[(1 - 2*x)^(5/2)/((2 + 3*x)^3*(3 + 5*x)^(5/2)),x]
Output:
(3*(1 - 2*x)^(7/2))/(14*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (239*((1 - 2*x)^(5/ 2)/((2 + 3*x)*(3 + 5*x)^(3/2)) + (55*((-2*(1 - 2*x)^(3/2))/(3*(3 + 5*x)^(3 /2)) - 7*((-2*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] + 2*Sqrt[7]*ArcTan[Sqrt[1 - 2*x ]/(Sqrt[7]*Sqrt[3 + 5*x])])))/2))/28
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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])
Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(104)=208\).
Time = 0.23 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.82
| method | result | size |
| default | \(\frac {\left (8872875 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+22477950 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+21334335 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+3619170 x^{3} \sqrt {-10 x^{2}-x +3}+8991180 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +6917268 x^{2} \sqrt {-10 x^{2}-x +3}+1419660 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4400642 x \sqrt {-10 x^{2}-x +3}+931832 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{24 \left (2+3 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(250\) |
Input:
int((1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/24*(8872875*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x ^4+22477950*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3 +21334335*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+3 619170*x^3*(-10*x^2-x+3)^(1/2)+8991180*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/ 2)/(-10*x^2-x+3)^(1/2))*x+6917268*x^2*(-10*x^2-x+3)^(1/2)+1419660*7^(1/2)* arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4400642*x*(-10*x^2-x+3) ^(1/2)+931832*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^2/(-10*x^2-x+3)^( 1/2)/(3+5*x)^(3/2)
Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {39435 \, \sqrt {7} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 2 \, {\left (1809585 \, x^{3} + 3458634 \, x^{2} + 2200321 \, x + 465916\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{24 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
-1/24*(39435*sqrt(7)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*arctan(1/1 4*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 2*( 1809585*x^3 + 3458634*x^2 + 2200321*x + 465916)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((1-2*x)**(5/2)/(2+3*x)**3/(3+5*x)**(5/2),x)
Output:
Integral((1 - 2*x)**(5/2)/((3*x + 2)**3*(5*x + 3)**(5/2)), x)
Time = 0.12 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.26 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {13145}{8} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {40213 \, x}{6 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {69977}{20 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {454757 \, x}{270 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2401}{162 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {25039}{108 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {1473541}{1620 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
13145/8*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 40213/ 6*x/sqrt(-10*x^2 - x + 3) + 69977/20/sqrt(-10*x^2 - x + 3) + 454757/270*x/ (-10*x^2 - x + 3)^(3/2) + 2401/162/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-1 0*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 25039/108/(3*(-10*x^ 2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 1473541/1620/(-10*x^2 - x + 3)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (104) = 208\).
Time = 0.28 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.72 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {11}{240} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {2629}{16} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1133}{10} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {77 \, \sqrt {10} {\left (437 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {103880 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {415520 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{2 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="giac")
Output:
-11/240*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 2629/16*sqrt(70)*sq rt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 1133/10*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 77/2*sqrt(10)*(437*(( sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt( 2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 103880*(sqrt(2)*sqrt(-10*x + 5) - sqrt (22))/sqrt(5*x + 3) - 415520*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt (22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int((1 - 2*x)^(5/2)/((3*x + 2)^3*(5*x + 3)^(5/2)),x)
Output:
int((1 - 2*x)^(5/2)/((3*x + 2)^3*(5*x + 3)^(5/2)), x)
Time = 0.26 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.90 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {1774575 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{3}+3430845 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}+2208360 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +473220 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-1774575 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{3}-3430845 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}-2208360 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -473220 \sqrt {5 x +3}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )+1809585 \sqrt {-2 x +1}\, x^{3}+3458634 \sqrt {-2 x +1}\, x^{2}+2200321 \sqrt {-2 x +1}\, x +465916 \sqrt {-2 x +1}}{12 \sqrt {5 x +3}\, \left (45 x^{3}+87 x^{2}+56 x +12\right )} \] Input:
int((1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^(5/2),x)
Output:
(1774575*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 3430845*sqrt(5*x + 3)*sqr t(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11 ))/2))/sqrt(2))*x**2 + 2208360*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt (35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 473220 *sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1 )*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 1774575*sqrt(5*x + 3)*sqrt(7)*atan((sq rt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2 ))*x**3 - 3430845*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin ((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 2208360*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5) )/sqrt(11))/2))/sqrt(2))*x - 473220*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) + 180 9585*sqrt( - 2*x + 1)*x**3 + 3458634*sqrt( - 2*x + 1)*x**2 + 2200321*sqrt( - 2*x + 1)*x + 465916*sqrt( - 2*x + 1))/(12*sqrt(5*x + 3)*(45*x**3 + 87*x **2 + 56*x + 12))