Integrand size = 26, antiderivative size = 183 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {9738340821 \sqrt {1-2 x} \sqrt {3+5 x}}{1638400}+\frac {295101237 \sqrt {1-2 x} (3+5 x)^{3/2}}{409600}+\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {107121749031 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1638400 \sqrt {10}} \]
-107121749031/16384000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+(2+3*x )^4*(3+5*x)^(5/2)/(1-2*x)^(1/2)+295101237/409600*(3+5*x)^(3/2)*(1-2*x)^(1/ 2)+999/160*(2+3*x)^2*(3+5*x)^(5/2)*(1-2*x)^(1/2)+13/8*(2+3*x)^3*(3+5*x)^(5 /2)*(1-2*x)^(1/2)+1/51200*(3+5*x)^(5/2)*(7611023+3765060*x)*(1-2*x)^(1/2)+ 9738340821/1638400*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {-10 \sqrt {3+5 x} \left (-16267424049+11734056318 x+7755469800 x^2+5945485120 x^3+3687379200 x^4+1479168000 x^5+276480000 x^6\right )+107121749031 \sqrt {10-20 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{16384000 \sqrt {1-2 x}} \]
(-10*Sqrt[3 + 5*x]*(-16267424049 + 11734056318*x + 7755469800*x^2 + 594548 5120*x^3 + 3687379200*x^4 + 1479168000*x^5 + 276480000*x^6) + 107121749031 *Sqrt[10 - 20*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(16384000*Sqrt[1 - 2*x])
Time = 0.26 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {108, 27, 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.
| \(\displaystyle \int \frac {(3 x+2)^4 (5 x+3)^{5/2}}{(1-2 x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^4 (5 x+3)^{5/2}}{\sqrt {1-2 x}}-\int \frac {(3 x+2)^3 (5 x+3)^{3/2} (195 x+122)}{2 \sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^4 (5 x+3)^{5/2}}{\sqrt {1-2 x}}-\frac {1}{2} \int \frac {(3 x+2)^3 (5 x+3)^{3/2} (195 x+122)}{\sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{60} \int -\frac {15 (3 x+2)^2 (5 x+3)^{3/2} (4995 x+3148)}{2 \sqrt {1-2 x}}dx+\frac {13}{4} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {13}{4} \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{5/2}-\frac {1}{8} \int \frac {(3 x+2)^2 (5 x+3)^{3/2} (4995 x+3148)}{\sqrt {1-2 x}}dx\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{50} \int -\frac {5 (3 x+2) (5 x+3)^{3/2} (313755 x+199846)}{2 \sqrt {1-2 x}}dx+\frac {999}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {13}{4} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{8} \left (\frac {999}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}-\frac {1}{20} \int \frac {(3 x+2) (5 x+3)^{3/2} (313755 x+199846)}{\sqrt {1-2 x}}dx\right )+\frac {13}{4} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)-\frac {295101237}{320} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x}}dx\right )+\frac {999}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {13}{4} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)-\frac {295101237}{320} \left (\frac {33}{8} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )\right )+\frac {999}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {13}{4} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)-\frac {295101237}{320} \left (\frac {33}{8} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )\right )+\frac {999}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {13}{4} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)-\frac {295101237}{320} \left (\frac {33}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )\right )+\frac {999}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {13}{4} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)-\frac {295101237}{320} \left (\frac {33}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )\right )+\frac {999}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {13}{4} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3\right )+\frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}\) |
((2 + 3*x)^4*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + ((13*Sqrt[1 - 2*x]*(2 + 3*x) ^3*(3 + 5*x)^(5/2))/4 + ((999*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/1 0 + ((Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(7611023 + 3765060*x))/160 - (29510123 7*(-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))/320)/20 )/8)/2
3.26.39.3.1 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 = 5.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(-\frac {\left (-5529600000 \sqrt {-10 x^{2}-x +3}\, x^{6}-29583360000 x^{5} \sqrt {-10 x^{2}-x +3}-73747584000 x^{4} \sqrt {-10 x^{2}-x +3}-118909702400 x^{3} \sqrt {-10 x^{2}-x +3}+214243498062 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -155109396000 x^{2} \sqrt {-10 x^{2}-x +3}-107121749031 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-234681126360 x \sqrt {-10 x^{2}-x +3}+325348480980 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{32768000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(174\) |
-1/32768000*(-5529600000*(-10*x^2-x+3)^(1/2)*x^6-29583360000*x^5*(-10*x^2- x+3)^(1/2)-73747584000*x^4*(-10*x^2-x+3)^(1/2)-118909702400*x^3*(-10*x^2-x +3)^(1/2)+214243498062*10^(1/2)*arcsin(20/11*x+1/11)*x-155109396000*x^2*(- 10*x^2-x+3)^(1/2)-107121749031*10^(1/2)*arcsin(20/11*x+1/11)-234681126360* x*(-10*x^2-x+3)^(1/2)+325348480980*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5 *x)^(1/2)/(-1+2*x)/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.55 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {107121749031 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (276480000 \, x^{6} + 1479168000 \, x^{5} + 3687379200 \, x^{4} + 5945485120 \, x^{3} + 7755469800 \, x^{2} + 11734056318 \, x - 16267424049\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{32768000 \, {\left (2 \, x - 1\right )}} \]
1/32768000*(107121749031*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1 )*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(276480000*x^6 + 147 9168000*x^5 + 3687379200*x^4 + 5945485120*x^3 + 7755469800*x^2 + 117340563 18*x - 16267424049)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)
\[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {\left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.78 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=-\frac {3375 \, x^{7}}{4 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {80325 \, x^{6}}{16 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3574125 \, x^{5}}{256 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {25493477 \, x^{4}}{1024 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1415345109 \, x^{3}}{40960 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {8193669099 \, x^{2}}{163840 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {107121749031}{32768000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {46134951291 \, x}{1638400 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {48802272147}{1638400 \, \sqrt {-10 \, x^{2} - x + 3}} \]
-3375/4*x^7/sqrt(-10*x^2 - x + 3) - 80325/16*x^6/sqrt(-10*x^2 - x + 3) - 3 574125/256*x^5/sqrt(-10*x^2 - x + 3) - 25493477/1024*x^4/sqrt(-10*x^2 - x + 3) - 1415345109/40960*x^3/sqrt(-10*x^2 - x + 3) - 8193669099/163840*x^2/ sqrt(-10*x^2 - x + 3) + 107121749031/32768000*sqrt(10)*arcsin(-20/11*x - 1 /11) + 46134951291/1638400*x/sqrt(-10*x^2 - x + 3) + 48802272147/1638400/s qrt(-10*x^2 - x + 3)
Time = 0.32 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=-\frac {107121749031}{16384000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, {\left (8 \, {\left (108 \, {\left (16 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 35 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4299 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 3832457 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 295101237 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 16230568035 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 535608745155 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{204800000 \, {\left (2 \, x - 1\right )}} \]
-107121749031/16384000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/20 4800000*(2*(4*(8*(108*(16*(4*sqrt(5)*(5*x + 3) + 35*sqrt(5))*(5*x + 3) + 4 299*sqrt(5))*(5*x + 3) + 3832457*sqrt(5))*(5*x + 3) + 295101237*sqrt(5))*( 5*x + 3) + 16230568035*sqrt(5))*(5*x + 3) - 535608745155*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)
Timed out. \[ \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]