Integrand size = 26, antiderivative size = 154 \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {321709971 \sqrt {1-2 x} \sqrt {3+5 x}}{204800}+\frac {9748787 \sqrt {1-2 x} (3+5 x)^{3/2}}{51200}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac {3538809681 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{204800 \sqrt {10}} \]
-3538809681/2048000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+(2+3*x)^3 *(3+5*x)^(5/2)/(1-2*x)^(1/2)+9748787/51200*(3+5*x)^(3/2)*(1-2*x)^(1/2)+33/ 20*(2+3*x)^2*(3+5*x)^(5/2)*(1-2*x)^(1/2)+9/6400*(3+5*x)^(5/2)*(27937+13820 *x)*(1-2*x)^(1/2)+321709971/204800*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.54 \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {-10 \sqrt {3+5 x} \left (-538018839+381820658 x+233394520 x^2+148751040 x^3+65836800 x^4+13824000 x^5\right )+3538809681 \sqrt {10-20 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{2048000 \sqrt {1-2 x}} \]
(-10*Sqrt[3 + 5*x]*(-538018839 + 381820658*x + 233394520*x^2 + 148751040*x ^3 + 65836800*x^4 + 13824000*x^5) + 3538809681*Sqrt[10 - 20*x]*ArcTan[Sqrt [5/2 - 5*x]/Sqrt[3 + 5*x]])/(2048000*Sqrt[1 - 2*x])
Time = 0.23 (sec) , antiderivative size = 174, 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, 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.
| \(\displaystyle \int \frac {(3 x+2)^3 (5 x+3)^{5/2}}{(1-2 x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{5/2}}{\sqrt {1-2 x}}-\int \frac {(3 x+2)^2 (5 x+3)^{3/2} (165 x+104)}{2 \sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{5/2}}{\sqrt {1-2 x}}-\frac {1}{2} \int \frac {(3 x+2)^2 (5 x+3)^{3/2} (165 x+104)}{\sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{50} \int -\frac {5 (3 x+2) (5 x+3)^{3/2} (10365 x+6602)}{2 \sqrt {1-2 x}}dx+\frac {33}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {33}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}-\frac {1}{20} \int \frac {(3 x+2) (5 x+3)^{3/2} (10365 x+6602)}{\sqrt {1-2 x}}dx\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{20} \left (\frac {9}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (13820 x+27937)-\frac {9748787}{320} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x}}dx\right )+\frac {33}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{20} \left (\frac {9}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (13820 x+27937)-\frac {9748787}{320} \left (\frac {33}{8} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )\right )+\frac {33}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{20} \left (\frac {9}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (13820 x+27937)-\frac {9748787}{320} \left (\frac {33}{8} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )\right )+\frac {33}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{20} \left (\frac {9}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (13820 x+27937)-\frac {9748787}{320} \left (\frac {33}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )\right )+\frac {33}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{20} \left (\frac {9}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (13820 x+27937)-\frac {9748787}{320} \left (\frac {33}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )\right )+\frac {33}{10} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^3}{\sqrt {1-2 x}}\) |
((2 + 3*x)^3*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + ((33*Sqrt[1 - 2*x]*(2 + 3*x) ^2*(3 + 5*x)^(5/2))/10 + ((9*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(27937 + 13820* x))/160 - (9748787*(-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[1 0])))/8))/320)/20)/2
3.26.40.3.1 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 = 1.20 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.02
| method | result | size |
| default | \(-\frac {\left (-276480000 x^{5} \sqrt {-10 x^{2}-x +3}-1316736000 x^{4} \sqrt {-10 x^{2}-x +3}-2975020800 x^{3} \sqrt {-10 x^{2}-x +3}+7077619362 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -4667890400 x^{2} \sqrt {-10 x^{2}-x +3}-3538809681 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-7636413160 x \sqrt {-10 x^{2}-x +3}+10760376780 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{4096000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(157\) |
-1/4096000*(-276480000*x^5*(-10*x^2-x+3)^(1/2)-1316736000*x^4*(-10*x^2-x+3 )^(1/2)-2975020800*x^3*(-10*x^2-x+3)^(1/2)+7077619362*10^(1/2)*arcsin(20/1 1*x+1/11)*x-4667890400*x^2*(-10*x^2-x+3)^(1/2)-3538809681*10^(1/2)*arcsin( 20/11*x+1/11)-7636413160*x*(-10*x^2-x+3)^(1/2)+10760376780*(-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 = 96, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {3538809681 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (13824000 \, x^{5} + 65836800 \, x^{4} + 148751040 \, x^{3} + 233394520 \, x^{2} + 381820658 \, x - 538018839\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4096000 \, {\left (2 \, x - 1\right )}} \]
1/4096000*(3538809681*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*s qrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(13824000*x^5 + 6583680 0*x^4 + 148751040*x^3 + 233394520*x^2 + 381820658*x - 538018839)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)
\[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.82 \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=-\frac {675 \, x^{6}}{2 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {57915 \, x^{5}}{32 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {588291 \, x^{4}}{128 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {40330643 \, x^{3}}{5120 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {52185737 \, x^{2}}{4096 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3538809681}{4096000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1544632221 \, x}{204800 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1614056517}{204800 \, \sqrt {-10 \, x^{2} - x + 3}} \]
-675/2*x^6/sqrt(-10*x^2 - x + 3) - 57915/32*x^5/sqrt(-10*x^2 - x + 3) - 58 8291/128*x^4/sqrt(-10*x^2 - x + 3) - 40330643/5120*x^3/sqrt(-10*x^2 - x + 3) - 52185737/4096*x^2/sqrt(-10*x^2 - x + 3) + 3538809681/4096000*sqrt(10) *arcsin(-20/11*x - 1/11) + 1544632221/204800*x/sqrt(-10*x^2 - x + 3) + 161 4056517/204800/sqrt(-10*x^2 - x + 3)
Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=-\frac {3538809681}{2048000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, {\left (24 \, {\left (36 \, {\left (16 \, \sqrt {5} {\left (5 \, x + 3\right )} + 141 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 42197 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9748787 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 536183285 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 17694048405 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{25600000 \, {\left (2 \, x - 1\right )}} \]
-3538809681/2048000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/25600 000*(2*(4*(24*(36*(16*sqrt(5)*(5*x + 3) + 141*sqrt(5))*(5*x + 3) + 42197*s qrt(5))*(5*x + 3) + 9748787*sqrt(5))*(5*x + 3) + 536183285*sqrt(5))*(5*x + 3) - 17694048405*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)
Timed out. \[ \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]