Integrand size = 26, antiderivative size = 142 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {76587 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{17600}+\frac {939}{880} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (18424549+7645620 x)}{2816000}-\frac {291096141 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000 \sqrt {10}} \]
-291096141/2560000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+7/11*(2+3* x)^4*(3+5*x)^(1/2)/(1-2*x)^(1/2)+76587/17600*(2+3*x)^2*(1-2*x)^(1/2)*(3+5* x)^(1/2)+939/880*(2+3*x)^3*(1-2*x)^(1/2)*(3+5*x)^(1/2)+21/2816000*(1842454 9+7645620*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.55 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {-10 \sqrt {3+5 x} \left (-488641609+332129358 x+171939240 x^2+76887360 x^3+17107200 x^4\right )+3202057551 \sqrt {10-20 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{28160000 \sqrt {1-2 x}} \]
(-10*Sqrt[3 + 5*x]*(-488641609 + 332129358*x + 171939240*x^2 + 76887360*x^ 3 + 17107200*x^4) + 3202057551*Sqrt[10 - 20*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt [3 + 5*x]])/(28160000*Sqrt[1 - 2*x])
Time = 0.23 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {109, 27, 170, 27, 170, 27, 164, 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.
| \(\displaystyle \int \frac {(3 x+2)^5}{(1-2 x)^{3/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^4 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {3 (3 x+2)^3 (313 x+190)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^4 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {3}{22} \int \frac {(3 x+2)^3 (313 x+190)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {7 (3 x+2)^4 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {3}{22} \left (-\frac {1}{40} \int -\frac {7 (3 x+2)^2 (10941 x+6668)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {313}{40} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^4 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {3}{22} \left (\frac {7}{80} \int \frac {(3 x+2)^2 (10941 x+6668)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {313}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {7 (3 x+2)^4 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {3}{22} \left (\frac {7}{80} \left (-\frac {1}{30} \int -\frac {3 (3 x+2) (637135 x+390718)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {3647}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {313}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^4 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {3}{22} \left (\frac {7}{80} \left (\frac {1}{20} \int \frac {(3 x+2) (637135 x+390718)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {3647}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {313}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {7 (3 x+2)^4 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {3}{22} \left (\frac {7}{80} \left (\frac {1}{20} \left (\frac {152478931}{160} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (7645620 x+18424549)\right )-\frac {3647}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {313}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {7 (3 x+2)^4 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {3}{22} \left (\frac {7}{80} \left (\frac {1}{20} \left (\frac {152478931}{400} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (7645620 x+18424549)\right )-\frac {3647}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {313}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {7 (3 x+2)^4 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {3}{22} \left (\frac {7}{80} \left (\frac {1}{20} \left (\frac {152478931 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80 \sqrt {10}}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (7645620 x+18424549)\right )-\frac {3647}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {313}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )\) |
(7*(2 + 3*x)^4*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) - (3*((-313*Sqrt[1 - 2*x] *(2 + 3*x)^3*Sqrt[3 + 5*x])/40 + (7*((-3647*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt [3 + 5*x])/10 + (-1/80*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(18424549 + 7645620*x) ) + (152478931*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80*Sqrt[10]))/20))/80))/ 22
3.26.50.3.1 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[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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 = 140, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {\left (-342144000 x^{4} \sqrt {-10 x^{2}-x +3}-1537747200 x^{3} \sqrt {-10 x^{2}-x +3}+6404115102 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -3438784800 x^{2} \sqrt {-10 x^{2}-x +3}-3202057551 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-6642587160 x \sqrt {-10 x^{2}-x +3}+9772832180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{56320000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(140\) |
-1/56320000*(-342144000*x^4*(-10*x^2-x+3)^(1/2)-1537747200*x^3*(-10*x^2-x+ 3)^(1/2)+6404115102*10^(1/2)*arcsin(20/11*x+1/11)*x-3438784800*x^2*(-10*x^ 2-x+3)^(1/2)-3202057551*10^(1/2)*arcsin(20/11*x+1/11)-6642587160*x*(-10*x^ 2-x+3)^(1/2)+9772832180*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/( -1+2*x)/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {3202057551 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (17107200 \, x^{4} + 76887360 \, x^{3} + 171939240 \, x^{2} + 332129358 \, x - 488641609\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{56320000 \, {\left (2 \, x - 1\right )}} \]
1/56320000*(3202057551*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*(17107200*x^4 + 768873 60*x^3 + 171939240*x^2 + 332129358*x - 488641609)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)
\[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{5}}{\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {243}{80} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + \frac {24273}{1600} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {291096141}{5120000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {487863}{12800} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {19975419}{256000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {16807 \, \sqrt {-10 \, x^{2} - x + 3}}{176 \, {\left (2 \, x - 1\right )}} \]
243/80*sqrt(-10*x^2 - x + 3)*x^3 + 24273/1600*sqrt(-10*x^2 - x + 3)*x^2 - 291096141/5120000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 487863/12800*sq rt(-10*x^2 - x + 3)*x + 19975419/256000*sqrt(-10*x^2 - x + 3) - 16807/176* sqrt(-10*x^2 - x + 3)/(2*x - 1)
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=-\frac {291096141}{2560000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (198 \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} + 377 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 29669 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4900505 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 16010291851 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{352000000 \, {\left (2 \, x - 1\right )}} \]
-291096141/2560000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/352000 000*(198*(12*(8*(36*sqrt(5)*(5*x + 3) + 377*sqrt(5))*(5*x + 3) + 29669*sqr t(5))*(5*x + 3) + 4900505*sqrt(5))*(5*x + 3) - 16010291851*sqrt(5))*sqrt(5 *x + 3)*sqrt(-10*x + 5)/(2*x - 1)
Timed out. \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^5}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \]