Integrand size = 26, antiderivative size = 142 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=-\frac {107 \sqrt {1-2 x} (2+3 x)^3}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4487 \sqrt {1-2 x} (2+3 x)^2}{99825 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (2571547+1078860 x)}{5324000}-\frac {111321 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{4000 \sqrt {10}} \]
-111321/40000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+7/11*(2+3*x)^4/ (3+5*x)^(3/2)/(1-2*x)^(1/2)-107/1815*(2+3*x)^3*(1-2*x)^(1/2)/(3+5*x)^(3/2) -4487/99825*(2+3*x)^2*(1-2*x)^(1/2)/(3+5*x)^(1/2)+7/5324000*(2571547+10788 60*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.61 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {10 \left (632498543+1785872944 x+612106475 x^2-1128781170 x^3-194059800 x^4\right )+444504753 \sqrt {10-20 x} (3+5 x)^{3/2} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{159720000 \sqrt {1-2 x} (3+5 x)^{3/2}} \]
(10*(632498543 + 1785872944*x + 612106475*x^2 - 1128781170*x^3 - 194059800 *x^4) + 444504753*Sqrt[10 - 20*x]*(3 + 5*x)^(3/2)*ArcTan[Sqrt[5/2 - 5*x]/S qrt[3 + 5*x]])/(159720000*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))
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, 167, 27, 167, 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.
| \(\displaystyle \int \frac {(3 x+2)^5}{(1-2 x)^{3/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {1}{11} \int \frac {(3 x+2)^3 (519 x+290)}{2 \sqrt {1-2 x} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {1}{22} \int \frac {(3 x+2)^3 (519 x+290)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (-\frac {2}{165} \int \frac {7 (3 x+2)^2 (7707 x+4496)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {214 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (-\frac {7}{165} \int \frac {(3 x+2)^2 (7707 x+4496)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {214 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (-\frac {7}{165} \left (\frac {2}{55} \int \frac {3 (3 x+2) (89905 x+53954)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1282 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (-\frac {7}{165} \left (\frac {3}{55} \int \frac {(3 x+2) (89905 x+53954)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1282 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{22} \left (-\frac {7}{165} \left (\frac {3}{55} \left (\frac {21166893}{160} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (1078860 x+2571547)\right )+\frac {1282 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{22} \left (-\frac {7}{165} \left (\frac {3}{55} \left (\frac {21166893}{400} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (1078860 x+2571547)\right )+\frac {1282 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{22} \left (-\frac {7}{165} \left (\frac {3}{55} \left (\frac {21166893 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80 \sqrt {10}}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (1078860 x+2571547)\right )+\frac {1282 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
(7*(2 + 3*x)^4)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((-214*Sqrt[1 - 2*x]* (2 + 3*x)^3)/(165*(3 + 5*x)^(3/2)) - (7*((1282*Sqrt[1 - 2*x]*(2 + 3*x)^2)/ (55*Sqrt[3 + 5*x]) + (3*(-1/80*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2571547 + 107 8860*x)) + (21166893*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80*Sqrt[10])))/55) )/165)/22
3.26.71.3.1 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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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 = 1.22 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.18
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \left (22225237650 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-3881196000 x^{4} \sqrt {-10 x^{2}-x +3}+15557666355 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-22575623400 x^{3} \sqrt {-10 x^{2}-x +3}-5334057036 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +12242129500 x^{2} \sqrt {-10 x^{2}-x +3}-4000542777 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+35717458880 x \sqrt {-10 x^{2}-x +3}+12649970860 \sqrt {-10 x^{2}-x +3}\right )}{319440000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(168\) |
-1/319440000*(1-2*x)^(1/2)*(22225237650*10^(1/2)*arcsin(20/11*x+1/11)*x^3- 3881196000*x^4*(-10*x^2-x+3)^(1/2)+15557666355*10^(1/2)*arcsin(20/11*x+1/1 1)*x^2-22575623400*x^3*(-10*x^2-x+3)^(1/2)-5334057036*10^(1/2)*arcsin(20/1 1*x+1/11)*x+12242129500*x^2*(-10*x^2-x+3)^(1/2)-4000542777*10^(1/2)*arcsin (20/11*x+1/11)+35717458880*x*(-10*x^2-x+3)^(1/2)+12649970860*(-10*x^2-x+3) ^(1/2))/(-1+2*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)
Time = 0.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.78 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {444504753 \, \sqrt {10} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (194059800 \, x^{4} + 1128781170 \, x^{3} - 612106475 \, x^{2} - 1785872944 \, x - 632498543\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{319440000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
1/319440000*(444504753*sqrt(10)*(50*x^3 + 35*x^2 - 12*x - 9)*arctan(1/20*s qrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(19 4059800*x^4 + 1128781170*x^3 - 612106475*x^2 - 1785872944*x - 632498543)*s qrt(5*x + 3)*sqrt(-2*x + 1))/(50*x^3 + 35*x^2 - 12*x - 9)
\[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{5}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=-\frac {243 \, x^{3}}{100 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {111321}{80000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {25353 \, x^{2}}{2000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1219513649 \, x}{79860000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {5270823773}{399300000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{103125 \, {\left (5 \, \sqrt {-10 \, x^{2} - x + 3} x + 3 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
-243/100*x^3/sqrt(-10*x^2 - x + 3) - 111321/80000*sqrt(5)*sqrt(2)*arcsin(2 0/11*x + 1/11) - 25353/2000*x^2/sqrt(-10*x^2 - x + 3) + 1219513649/7986000 0*x/sqrt(-10*x^2 - x + 3) + 5270823773/399300000/sqrt(-10*x^2 - x + 3) - 2 /103125/(5*sqrt(-10*x^2 - x + 3)*x + 3*sqrt(-10*x^2 - x + 3))
Time = 0.35 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.35 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{199650000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} - \frac {111321}{40000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (215622 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 205 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 741559591 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{665500000 \, {\left (2 \, x - 1\right )}} - \frac {337 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{16637500 \, \sqrt {5 \, x + 3}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1011 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{12478125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
-1/199650000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/ 2) - 111321/40000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/6655000 00*(215622*(12*sqrt(5)*(5*x + 3) + 205*sqrt(5))*(5*x + 3) - 741559591*sqrt (5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 337/16637500*sqrt(10)*(sqrt (2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/12478125*sqrt(10)*(5*x + 3)^(3/2)*(1011*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqr t(2)*sqrt(-10*x + 5) - sqrt(22))^3
Timed out. \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^5}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]