Integrand size = 26, antiderivative size = 157 \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {9444023 \sqrt {1-2 x} \sqrt {3+5 x}}{4096}-\frac {9444023 \sqrt {1-2 x} (3+5 x)^{3/2}}{33792}-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {103884253 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{4096 \sqrt {10}} \]
1/3*(2+3*x)^3*(3+5*x)^(5/2)/(1-2*x)^(3/2)+103884253/40960*arcsin(1/11*22^( 1/2)*(3+5*x)^(1/2))*10^(1/2)-373/66*(2+3*x)^2*(3+5*x)^(5/2)/(1-2*x)^(1/2)- 9444023/33792*(3+5*x)^(3/2)*(1-2*x)^(1/2)-1/1408*(3+5*x)^(5/2)*(81191+4016 4*x)*(1-2*x)^(1/2)-9444023/4096*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.59 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.55 \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {-\frac {5 \sqrt {3+5 x} \left (47216961-129940960 x+40614996 x^2+15301008 x^3+5477760 x^4+1036800 x^5\right )}{(1-2 x)^{3/2}}-311652759 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{61440} \]
((-5*Sqrt[3 + 5*x]*(47216961 - 129940960*x + 40614996*x^2 + 15301008*x^3 + 5477760*x^4 + 1036800*x^5))/(1 - 2*x)^(3/2) - 311652759*Sqrt[10]*ArcTan[S qrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10*x])])/61440
Time = 0.24 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {108, 27, 167, 27, 164, 60, 60, 64, 223}
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.
| \(\displaystyle \int \frac {(3 x+2)^3 (5 x+3)^{5/2}}{(1-2 x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(3 x+2)^2 (5 x+3)^{3/2} (165 x+104)}{2 (1-2 x)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{6} \int \frac {(3 x+2)^2 (5 x+3)^{3/2} (165 x+104)}{(1-2 x)^{3/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{6} \left (-\frac {1}{11} \int -\frac {(3 x+2) (5 x+3)^{3/2} (50205 x+31978)}{2 \sqrt {1-2 x}}dx-\frac {373 (3 x+2)^2 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{22} \int \frac {(3 x+2) (5 x+3)^{3/2} (50205 x+31978)}{\sqrt {1-2 x}}dx-\frac {373 (3 x+2)^2 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {9444023}{64} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x}}dx-\frac {3}{32} \sqrt {1-2 x} (5 x+3)^{5/2} (40164 x+81191)\right )-\frac {373 (3 x+2)^2 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {9444023}{64} \left (\frac {33}{8} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3}{32} \sqrt {1-2 x} (5 x+3)^{5/2} (40164 x+81191)\right )-\frac {373 (3 x+2)^2 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {9444023}{64} \left (\frac {33}{8} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3}{32} \sqrt {1-2 x} (5 x+3)^{5/2} (40164 x+81191)\right )-\frac {373 (3 x+2)^2 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {9444023}{64} \left (\frac {33}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3}{32} \sqrt {1-2 x} (5 x+3)^{5/2} (40164 x+81191)\right )-\frac {373 (3 x+2)^2 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{22} \left (\frac {9444023}{64} \left (\frac {33}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3}{32} \sqrt {1-2 x} (5 x+3)^{5/2} (40164 x+81191)\right )-\frac {373 (3 x+2)^2 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}\) |
((2 + 3*x)^3*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) + ((-373*(2 + 3*x)^2*(3 + 5*x)^(5/2))/(11*Sqrt[1 - 2*x]) + ((-3*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(811 91 + 40164*x))/32 + (9444023*(-1/4*(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (33*( -1/2*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]) /(2*Sqrt[10])))/8))/64)/22)/6
3.26.99.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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]
Time = 1.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(\frac {\left (-20736000 x^{5} \sqrt {-10 x^{2}-x +3}-109555200 x^{4} \sqrt {-10 x^{2}-x +3}+1246611036 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-306020160 x^{3} \sqrt {-10 x^{2}-x +3}-1246611036 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -812299920 x^{2} \sqrt {-10 x^{2}-x +3}+311652759 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2598819200 x \sqrt {-10 x^{2}-x +3}-944339220 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{245760 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(171\) |
1/245760*(-20736000*x^5*(-10*x^2-x+3)^(1/2)-109555200*x^4*(-10*x^2-x+3)^(1 /2)+1246611036*10^(1/2)*arcsin(20/11*x+1/11)*x^2-306020160*x^3*(-10*x^2-x+ 3)^(1/2)-1246611036*10^(1/2)*arcsin(20/11*x+1/11)*x-812299920*x^2*(-10*x^2 -x+3)^(1/2)+311652759*10^(1/2)*arcsin(20/11*x+1/11)+2598819200*x*(-10*x^2- x+3)^(1/2)-944339220*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+ 2*x)^2/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {311652759 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (1036800 \, x^{5} + 5477760 \, x^{4} + 15301008 \, x^{3} + 40614996 \, x^{2} - 129940960 \, x + 47216961\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{245760 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/245760*(311652759*sqrt(10)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(1036800*x^5 + 5 477760*x^4 + 15301008*x^3 + 40614996*x^2 - 129940960*x + 47216961)*sqrt(5* x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)
\[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}}}\, dx \]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.07 \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {2606989}{2048} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {395307}{81920} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) + \frac {495}{256} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {343 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{16 \, {\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac {441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{32 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac {63 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{16 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{64 \, {\left (2 \, x - 1\right )}} - \frac {16335}{1024} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x + \frac {68607}{4096} \, \sqrt {10 \, x^{2} - 21 \, x + 8} - \frac {114345}{512} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {18865 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{192 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {24255 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{128 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {3465 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{128 \, {\left (2 \, x - 1\right )}} + \frac {207515 \, \sqrt {-10 \, x^{2} - x + 3}}{384 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {3721795 \, \sqrt {-10 \, x^{2} - x + 3}}{768 \, {\left (2 \, x - 1\right )}} \]
2606989/2048*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 395307/81920*I*sqrt( 5)*sqrt(2)*arcsin(20/11*x - 21/11) + 495/256*(-10*x^2 - x + 3)^(3/2) - 343 /16*(-10*x^2 - x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x + 1) - 441/32* (-10*x^2 - x + 3)^(5/2)/(8*x^3 - 12*x^2 + 6*x - 1) - 63/16*(-10*x^2 - x + 3)^(5/2)/(4*x^2 - 4*x + 1) - 27/64*(-10*x^2 - x + 3)^(5/2)/(2*x - 1) - 163 35/1024*sqrt(10*x^2 - 21*x + 8)*x + 68607/4096*sqrt(10*x^2 - 21*x + 8) - 1 14345/512*sqrt(-10*x^2 - x + 3) - 18865/192*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 24255/128*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 3465/128*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 207515/384*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 3721795/768*sqrt(-10*x^2 - x + 3)/(2*x - 1)
Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {103884253}{40960} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (3 \, {\left (36 \, {\left (8 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 137 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 13627 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9444023 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 1038842530 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 17140901745 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{7680000 \, {\left (2 \, x - 1\right )}^{2}} \]
103884253/40960*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/7680000*( 4*(3*(36*(8*(12*sqrt(5)*(5*x + 3) + 137*sqrt(5))*(5*x + 3) + 13627*sqrt(5) )*(5*x + 3) + 9444023*sqrt(5))*(5*x + 3) - 1038842530*sqrt(5))*(5*x + 3) + 17140901745*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2
Timed out. \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]