Integrand size = 26, antiderivative size = 142 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}} \]
7/33*(2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^(3/2)+2997/2000*arcsin(1/11*22^(1/2)* (3+5*x)^(1/2))*10^(1/2)-357/242*(2+3*x)^3/(3+5*x)^(3/2)/(1-2*x)^(1/2)+5281 /39930*(2+3*x)^2*(1-2*x)^(1/2)/(3+5*x)^(3/2)-1/8784600*(33035947+55300905* x)*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {168318961+19593966 x-1260430251 x^2-1247811640 x^3+213465780 x^4}{8784600 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2997 \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{200 \sqrt {10}} \]
-1/8784600*(168318961 + 19593966*x - 1260430251*x^2 - 1247811640*x^3 + 213 465780*x^4)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (2997*ArcTan[Sqrt[5/2 - 5* x]/Sqrt[3 + 5*x]])/(200*Sqrt[10])
Time = 0.24 (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, 160, 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.
| \(\displaystyle \int \frac {(3 x+2)^5}{(1-2 x)^{5/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{33} \int \frac {3 (3 x+2)^3 (169 x+94)}{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{22} \int \frac {(3 x+2)^3 (169 x+94)}{(1-2 x)^{3/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (-\frac {1}{11} \int -\frac {(3 x+2)^2 (16287 x+8716)}{2 \sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {357 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \int \frac {(3 x+2)^2 (16287 x+8716)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {357 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {2}{165} \int \frac {(3 x+2) (1675785 x+969322)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {10562 \sqrt {1-2 x} (3 x+2)^2}{165 (5 x+3)^{3/2}}\right )-\frac {357 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {1}{165} \int \frac {(3 x+2) (1675785 x+969322)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {10562 \sqrt {1-2 x} (3 x+2)^2}{165 (5 x+3)^{3/2}}\right )-\frac {357 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 160 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {1}{165} \left (\frac {11967021}{20} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} (55300905 x+33035947)}{110 \sqrt {5 x+3}}\right )+\frac {10562 \sqrt {1-2 x} (3 x+2)^2}{165 (5 x+3)^{3/2}}\right )-\frac {357 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {1}{165} \left (\frac {11967021}{50} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {\sqrt {1-2 x} (55300905 x+33035947)}{110 \sqrt {5 x+3}}\right )+\frac {10562 \sqrt {1-2 x} (3 x+2)^2}{165 (5 x+3)^{3/2}}\right )-\frac {357 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {1}{165} \left (\frac {11967021 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10 \sqrt {10}}-\frac {\sqrt {1-2 x} (55300905 x+33035947)}{110 \sqrt {5 x+3}}\right )+\frac {10562 \sqrt {1-2 x} (3 x+2)^2}{165 (5 x+3)^{3/2}}\right )-\frac {357 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
(7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + ((-357*(2 + 3*x)^3) /(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((10562*Sqrt[1 - 2*x]*(2 + 3*x)^2)/( 165*(3 + 5*x)^(3/2)) + (-1/110*(Sqrt[1 - 2*x]*(33035947 + 55300905*x))/Sqr t[3 + 5*x] + (11967021*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10*Sqrt[10]))/16 5)/22)/22
3.27.28.3.1 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[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* (f*g + e*h) - c*f*h*(m + 2)))/(b^2*d) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
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 = 4.17 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.28
| method | result | size |
| default | \(\frac {\sqrt {1-2 x}\, \left (13163723100 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}+2632744620 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-4269315600 x^{4} \sqrt {-10 x^{2}-x +3}-7766596629 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+24956232800 x^{3} \sqrt {-10 x^{2}-x +3}-789823386 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +25208605020 x^{2} \sqrt {-10 x^{2}-x +3}+1184735079 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-391879320 x \sqrt {-10 x^{2}-x +3}-3366379220 \sqrt {-10 x^{2}-x +3}\right )}{175692000 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(182\) |
1/175692000*(1-2*x)^(1/2)*(13163723100*10^(1/2)*arcsin(20/11*x+1/11)*x^4+2 632744620*10^(1/2)*arcsin(20/11*x+1/11)*x^3-4269315600*x^4*(-10*x^2-x+3)^( 1/2)-7766596629*10^(1/2)*arcsin(20/11*x+1/11)*x^2+24956232800*x^3*(-10*x^2 -x+3)^(1/2)-789823386*10^(1/2)*arcsin(20/11*x+1/11)*x+25208605020*x^2*(-10 *x^2-x+3)^(1/2)+1184735079*10^(1/2)*arcsin(20/11*x+1/11)-391879320*x*(-10* x^2-x+3)^(1/2)-3366379220*(-10*x^2-x+3)^(1/2))/(-1+2*x)^2/(-10*x^2-x+3)^(1 /2)/(3+5*x)^(3/2)
Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {131637231 \, \sqrt {10} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (213465780 \, x^{4} - 1247811640 \, x^{3} - 1260430251 \, x^{2} + 19593966 \, x + 168318961\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{175692000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
-1/175692000*(131637231*sqrt(10)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*arc tan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3) ) + 20*(213465780*x^4 - 1247811640*x^3 - 1260430251*x^2 + 19593966*x + 168 318961)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9 )
\[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{5}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.39 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {243 \, x^{4}}{10 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {999}{5856400} \, x {\left (\frac {7220 \, x}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {361}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} - \frac {2997}{4000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {360639}{2928200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {5842159 \, x}{878460 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3429 \, x^{2}}{25 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {947293}{21961500 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3016649 \, x}{90750 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1851167}{90750 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
-243/10*x^4/(-10*x^2 - x + 3)^(3/2) + 999/5856400*x*(7220*x/sqrt(-10*x^2 - x + 3) + 439230*x^2/(-10*x^2 - x + 3)^(3/2) + 361/sqrt(-10*x^2 - x + 3) + 21901*x/(-10*x^2 - x + 3)^(3/2) - 87483/(-10*x^2 - x + 3)^(3/2)) - 2997/4 000*sqrt(10)*arcsin(-20/11*x - 1/11) + 360639/2928200*sqrt(-10*x^2 - x + 3 ) - 5842159/878460*x/sqrt(-10*x^2 - x + 3) + 3429/25*x^2/(-10*x^2 - x + 3) ^(3/2) + 947293/21961500/sqrt(-10*x^2 - x + 3) + 3016649/90750*x/(-10*x^2 - x + 3)^(3/2) - 1851167/90750/(-10*x^2 - x + 3)^(3/2)
Time = 0.37 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.35 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{439230000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {2997}{2000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {31 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{3327500 \, \sqrt {5 \, x + 3}} - \frac {{\left (4 \, {\left (10673289 \, \sqrt {5} {\left (5 \, x + 3\right )} - 440040554 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 7233942969 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{5490375000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1023 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{27451875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
-1/439230000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/ 2) + 2997/2000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 31/3327500*s qrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/5490375000* (4*(10673289*sqrt(5)*(5*x + 3) - 440040554*sqrt(5))*(5*x + 3) + 7233942969 *sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 1/27451875*sqrt(10)* (5*x + 3)^(3/2)*(1023*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4 )/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3
Timed out. \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^5}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]