Integrand size = 26, antiderivative size = 168 \[ \int \frac {(2+3 x)^5 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {847637 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{32000}+\frac {10389 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}+\frac {(2+3 x)^5 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {49 \sqrt {1-2 x} \sqrt {3+5 x} (87394471+36265980 x)}{5120000}-\frac {35439958001 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5120000 \sqrt {10}} \]
-35439958001/51200000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+(2+3*x) ^5*(3+5*x)^(1/2)/(1-2*x)^(1/2)+847637/32000*(2+3*x)^2*(1-2*x)^(1/2)*(3+5*x )^(1/2)+10389/1600*(2+3*x)^3*(1-2*x)^(1/2)*(3+5*x)^(1/2)+33/20*(2+3*x)^4*( 1-2*x)^(1/2)*(3+5*x)^(1/2)+49/5120000*(87394471+36265980*x)*(1-2*x)^(1/2)* (3+5*x)^(1/2)
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.49 \[ \int \frac {(2+3 x)^5 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {-10 \sqrt {3+5 x} \left (-5389783159+3810769458 x+2297649240 x^2+1429191360 x^3+613267200 x^4+124416000 x^5\right )+35439958001 \sqrt {10-20 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{51200000 \sqrt {1-2 x}} \]
(-10*Sqrt[3 + 5*x]*(-5389783159 + 3810769458*x + 2297649240*x^2 + 14291913 60*x^3 + 613267200*x^4 + 124416000*x^5) + 35439958001*Sqrt[10 - 20*x]*ArcT an[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(51200000*Sqrt[1 - 2*x])
Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.12, 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, 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.
| \(\displaystyle \int \frac {(3 x+2)^5 \sqrt {5 x+3}}{(1-2 x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\int \frac {5 (3 x+2)^4 (33 x+20)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \int \frac {(3 x+2)^4 (33 x+20)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \left (-\frac {1}{50} \int -\frac {(3 x+2)^3 (10389 x+6310)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {33}{50} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^4\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \left (\frac {1}{100} \int \frac {(3 x+2)^3 (10389 x+6310)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {33}{50} \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \left (\frac {1}{100} \left (-\frac {1}{40} \int -\frac {7 (3 x+2)^2 (363273 x+221404)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {10389}{40} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {33}{50} \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \left (\frac {1}{100} \left (\frac {7}{80} \int \frac {(3 x+2)^2 (363273 x+221404)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {10389}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )-\frac {33}{50} \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \left (\frac {1}{100} \left (\frac {7}{80} \left (-\frac {1}{30} \int -\frac {21 (3 x+2) (3022165 x+1853322)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {121091}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {10389}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )-\frac {33}{50} \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \left (\frac {1}{100} \left (\frac {7}{80} \left (\frac {7}{20} \int \frac {(3 x+2) (3022165 x+1853322)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {121091}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {10389}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )-\frac {33}{50} \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \left (\frac {1}{100} \left (\frac {7}{80} \left (\frac {7}{20} \left (\frac {723264449}{160} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (36265980 x+87394471)\right )-\frac {121091}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {10389}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )-\frac {33}{50} \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \left (\frac {1}{100} \left (\frac {7}{80} \left (\frac {7}{20} \left (\frac {723264449}{400} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (36265980 x+87394471)\right )-\frac {121091}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {10389}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )-\frac {33}{50} \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {(3 x+2)^5 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {5}{2} \left (\frac {1}{100} \left (\frac {7}{80} \left (\frac {7}{20} \left (\frac {723264449 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80 \sqrt {10}}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (36265980 x+87394471)\right )-\frac {121091}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {10389}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\right )-\frac {33}{50} \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}\right )\) |
((2 + 3*x)^5*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] - (5*((-33*Sqrt[1 - 2*x]*(2 + 3* x)^4*Sqrt[3 + 5*x])/50 + ((-10389*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x]) /40 + (7*((-121091*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/10 + (7*(-1/80 *(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(87394471 + 36265980*x)) + (723264449*ArcSin [Sqrt[2/11]*Sqrt[3 + 5*x]])/(80*Sqrt[10])))/20))/80)/100))/2
3.26.18.3.1 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[(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 = 1.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(-\frac {\left (-2488320000 x^{5} \sqrt {-10 x^{2}-x +3}-12265344000 x^{4} \sqrt {-10 x^{2}-x +3}-28583827200 x^{3} \sqrt {-10 x^{2}-x +3}+70879916002 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -45952984800 x^{2} \sqrt {-10 x^{2}-x +3}-35439958001 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-76215389160 x \sqrt {-10 x^{2}-x +3}+107795663180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{102400000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(157\) |
-1/102400000*(-2488320000*x^5*(-10*x^2-x+3)^(1/2)-12265344000*x^4*(-10*x^2 -x+3)^(1/2)-28583827200*x^3*(-10*x^2-x+3)^(1/2)+70879916002*10^(1/2)*arcsi n(20/11*x+1/11)*x-45952984800*x^2*(-10*x^2-x+3)^(1/2)-35439958001*10^(1/2) *arcsin(20/11*x+1/11)-76215389160*x*(-10*x^2-x+3)^(1/2)+107795663180*(-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 = 0.57 \[ \int \frac {(2+3 x)^5 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {35439958001 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (124416000 \, x^{5} + 613267200 \, x^{4} + 1429191360 \, x^{3} + 2297649240 \, x^{2} + 3810769458 \, x - 5389783159\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{102400000 \, {\left (2 \, x - 1\right )}} \]
1/102400000*(35439958001*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*(124416000*x^5 + 613 267200*x^4 + 1429191360*x^3 + 2297649240*x^2 + 3810769458*x - 5389783159)* sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)
\[ \int \frac {(2+3 x)^5 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int \frac {\left (3 x + 2\right )^{5} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^5 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=-\frac {243}{200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {103599}{16000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {35439958001}{102400000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1086219}{64000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {80155719}{256000} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {2961355719}{5120000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {16807 \, \sqrt {-10 \, x^{2} - x + 3}}{32 \, {\left (2 \, x - 1\right )}} \]
-243/200*(-10*x^2 - x + 3)^(3/2)*x^2 - 103599/16000*(-10*x^2 - x + 3)^(3/2 )*x - 35439958001/102400000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 10862 19/64000*(-10*x^2 - x + 3)^(3/2) + 80155719/256000*sqrt(-10*x^2 - x + 3)*x + 2961355719/5120000*sqrt(-10*x^2 - x + 3) - 16807/32*sqrt(-10*x^2 - x + 3)/(2*x - 1)
Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.65 \[ \int \frac {(2+3 x)^5 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=-\frac {35439958001}{51200000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (6 \, {\left (12 \, {\left (8 \, {\left (36 \, {\left (48 \, \sqrt {5} {\left (5 \, x + 3\right )} + 463 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 140711 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 10847547 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1789896455 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 177199790005 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{640000000 \, {\left (2 \, x - 1\right )}} \]
-35439958001/51200000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/640 000000*(6*(12*(8*(36*(48*sqrt(5)*(5*x + 3) + 463*sqrt(5))*(5*x + 3) + 1407 11*sqrt(5))*(5*x + 3) + 10847547*sqrt(5))*(5*x + 3) + 1789896455*sqrt(5))* (5*x + 3) - 177199790005*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)
Timed out. \[ \int \frac {(2+3 x)^5 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^5\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]