Integrand size = 26, antiderivative size = 186 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {118054167 \sqrt {1-2 x} \sqrt {3+5 x}}{320000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {3+5 x}}{32000000}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {1298595837 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320000000 \sqrt {10}} \]
1298595837/3200000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-2/5*(1- 2*x)^(5/2)*(2+3*x)^4/(3+5*x)^(1/2)+3577399/32000000*(1-2*x)^(3/2)*(3+5*x)^ (1/2)+111/5000*(1-2*x)^(5/2)*(2+3*x)^2*(3+5*x)^(1/2)+13/50*(1-2*x)^(5/2)*( 2+3*x)^3*(3+5*x)^(1/2)-1/8000000*(1-2*x)^(5/2)*(2725981+1990620*x)*(3+5*x) ^(1/2)+118054167/320000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=\frac {10 \sqrt {1-2 x} \left (168414751+1366129125 x+938891620 x^2-3673002400 x^3-2530224000 x^4+4043520000 x^5+3456000000 x^6\right )-1298595837 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{3200000000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(168414751 + 1366129125*x + 938891620*x^2 - 3673002400*x ^3 - 2530224000*x^4 + 4043520000*x^5 + 3456000000*x^6) - 1298595837*Sqrt[3 0 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(3200000000*Sqrt[3 + 5*x] )
Time = 0.25 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {108, 170, 27, 170, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^4}{(5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{5} \int \frac {(2-39 x) (1-2 x)^{3/2} (3 x+2)^3}{\sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {2}{5} \left (\frac {13}{20} (1-2 x)^{5/2} (3 x+2)^3 \sqrt {5 x+3}-\frac {1}{60} \int -\frac {9 (36-37 x) (1-2 x)^{3/2} (3 x+2)^2}{2 \sqrt {5 x+3}}dx\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} \left (\frac {3}{40} \int \frac {(36-37 x) (1-2 x)^{3/2} (3 x+2)^2}{\sqrt {5 x+3}}dx+\frac {13}{20} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {2}{5} \left (\frac {3}{40} \left (\frac {37}{50} (1-2 x)^{5/2} (3 x+2)^2 \sqrt {5 x+3}-\frac {1}{50} \int -\frac {(1-2 x)^{3/2} (3 x+2) (11059 x+7718)}{2 \sqrt {5 x+3}}dx\right )+\frac {13}{20} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} \left (\frac {3}{40} \left (\frac {1}{100} \int \frac {(1-2 x)^{3/2} (3 x+2) (11059 x+7718)}{\sqrt {5 x+3}}dx+\frac {37}{50} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{5/2}\right )+\frac {13}{20} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {2}{5} \left (\frac {3}{40} \left (\frac {1}{100} \left (\frac {3577399}{960} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx-\frac {(1-2 x)^{5/2} \sqrt {5 x+3} (1990620 x+2725981)}{2400}\right )+\frac {37}{50} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{5/2}\right )+\frac {13}{20} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2}{5} \left (\frac {3}{40} \left (\frac {1}{100} \left (\frac {3577399}{960} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{5/2} \sqrt {5 x+3} (1990620 x+2725981)}{2400}\right )+\frac {37}{50} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{5/2}\right )+\frac {13}{20} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2}{5} \left (\frac {3}{40} \left (\frac {1}{100} \left (\frac {3577399}{960} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{5/2} \sqrt {5 x+3} (1990620 x+2725981)}{2400}\right )+\frac {37}{50} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{5/2}\right )+\frac {13}{20} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {2}{5} \left (\frac {3}{40} \left (\frac {1}{100} \left (\frac {3577399}{960} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{5/2} \sqrt {5 x+3} (1990620 x+2725981)}{2400}\right )+\frac {37}{50} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{5/2}\right )+\frac {13}{20} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {2}{5} \left (\frac {3}{40} \left (\frac {1}{100} \left (\frac {3577399}{960} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{5/2} \sqrt {5 x+3} (1990620 x+2725981)}{2400}\right )+\frac {37}{50} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{5/2}\right )+\frac {13}{20} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\) |
(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^4)/(5*Sqrt[3 + 5*x]) + (2*((13*(1 - 2*x)^(5/ 2)*(2 + 3*x)^3*Sqrt[3 + 5*x])/20 + (3*((37*(1 - 2*x)^(5/2)*(2 + 3*x)^2*Sqr t[3 + 5*x])/50 + (-1/2400*((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]*(2725981 + 199062 0*x)) + (3577399*(((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (33*((Sqrt[1 - 2*x] *Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/2 0))/960)/100))/40))/5
3.25.36.3.1 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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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 = 1.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.90
| method | result | size |
| default | \(\frac {\left (69120000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+80870400000 x^{5} \sqrt {-10 x^{2}-x +3}-50604480000 x^{4} \sqrt {-10 x^{2}-x +3}-73460048000 x^{3} \sqrt {-10 x^{2}-x +3}+6492979185 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +18777832400 x^{2} \sqrt {-10 x^{2}-x +3}+3895787511 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+27322582500 x \sqrt {-10 x^{2}-x +3}+3368295020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{6400000000 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(167\) |
1/6400000000*(69120000000*(-10*x^2-x+3)^(1/2)*x^6+80870400000*x^5*(-10*x^2 -x+3)^(1/2)-50604480000*x^4*(-10*x^2-x+3)^(1/2)-73460048000*x^3*(-10*x^2-x +3)^(1/2)+6492979185*10^(1/2)*arcsin(20/11*x+1/11)*x+18777832400*x^2*(-10* x^2-x+3)^(1/2)+3895787511*10^(1/2)*arcsin(20/11*x+1/11)+27322582500*x*(-10 *x^2-x+3)^(1/2)+3368295020*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3 )^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.54 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=-\frac {1298595837 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (3456000000 \, x^{6} + 4043520000 \, x^{5} - 2530224000 \, x^{4} - 3673002400 \, x^{3} + 938891620 \, x^{2} + 1366129125 \, x + 168414751\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6400000000 \, {\left (5 \, x + 3\right )}} \]
-1/6400000000*(1298595837*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 20*(3456000000*x^6 + 4 043520000*x^5 - 2530224000*x^4 - 3673002400*x^3 + 938891620*x^2 + 13661291 25*x + 168414751)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{4}}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=-\frac {108 \, x^{7}}{5 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1809 \, x^{6}}{125 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {284499 \, x^{5}}{10000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3009863 \, x^{4}}{200000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {138769641 \, x^{3}}{8000000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {179336663 \, x^{2}}{32000000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1298595837}{6400000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1029299623 \, x}{320000000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {168414751}{320000000 \, \sqrt {-10 \, x^{2} - x + 3}} \]
-108/5*x^7/sqrt(-10*x^2 - x + 3) - 1809/125*x^6/sqrt(-10*x^2 - x + 3) + 28 4499/10000*x^5/sqrt(-10*x^2 - x + 3) + 3009863/200000*x^4/sqrt(-10*x^2 - x + 3) - 138769641/8000000*x^3/sqrt(-10*x^2 - x + 3) - 179336663/32000000*x ^2/sqrt(-10*x^2 - x + 3) - 1298595837/6400000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 1029299623/320000000*x/sqrt(-10*x^2 - x + 3) + 168414751/3200000 00/sqrt(-10*x^2 - x + 3)
Time = 0.45 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=\frac {1}{8000000000} \, {\left (4 \, {\left (8 \, {\left (108 \, {\left (16 \, {\left (20 \, \sqrt {5} {\left (5 \, x + 3\right )} - 243 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9263 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 2532859 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 3473645 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 533500275 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {1298595837}{3200000000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {121 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{781250 \, \sqrt {5 \, x + 3}} + \frac {242 \, \sqrt {10} \sqrt {5 \, x + 3}}{390625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
1/8000000000*(4*(8*(108*(16*(20*sqrt(5)*(5*x + 3) - 243*sqrt(5))*(5*x + 3) + 9263*sqrt(5))*(5*x + 3) + 2532859*sqrt(5))*(5*x + 3) + 3473645*sqrt(5)) *(5*x + 3) - 533500275*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 1298595837 /3200000000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 121/781250*sqrt (10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 242/390625*sqrt( 10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4}{{\left (5\,x+3\right )}^{3/2}} \,d x \]