Integrand size = 26, antiderivative size = 137 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {207895 \sqrt {1-2 x}}{6468 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {753 \sqrt {1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac {20743985 \sqrt {1-2 x}}{71148 \sqrt {3+5 x}}-\frac {392283 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{196 \sqrt {7}} \]
-392283/1372*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-20789 5/6468*(1-2*x)^(1/2)/(3+5*x)^(3/2)+3/14*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(3 /2)+753/196*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(3/2)+20743985/71148*(1-2*x)^(1/ 2)/(3+5*x)^(1/2)
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {\sqrt {1-2 x} \left (240342364+1135041037 x+1784145090 x^2+933479325 x^3\right )}{71148 (2+3 x)^2 (3+5 x)^{3/2}}-\frac {392283 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{196 \sqrt {7}} \]
(Sqrt[1 - 2*x]*(240342364 + 1135041037*x + 1784145090*x^2 + 933479325*x^3) )/(71148*(2 + 3*x)^2*(3 + 5*x)^(3/2)) - (392283*ArcTan[Sqrt[1 - 2*x]/(Sqrt [7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])
Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {114, 27, 168, 27, 169, 27, 169, 27, 104, 217}
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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{14} \int \frac {131-180 x}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \int \frac {131-180 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{7} \int \frac {23507-30120 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {753 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{14} \int \frac {23507-30120 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {753 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{14} \left (-\frac {2}{33} \int \frac {2651953-2494740 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {415790 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {753 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{14} \left (-\frac {1}{33} \int \frac {2651953-2494740 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {415790 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {753 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{14} \left (\frac {1}{33} \left (\frac {2}{11} \int \frac {142398729}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {41487970 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {415790 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {753 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{14} \left (\frac {1}{33} \left (12945339 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {41487970 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {415790 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {753 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{14} \left (\frac {1}{33} \left (25890678 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {41487970 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {415790 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {753 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{14} \left (\frac {1}{33} \left (\frac {41487970 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {25890678 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\right )-\frac {415790 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {753 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\) |
(3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + ((753*Sqrt[1 - 2*x])/ (7*(2 + 3*x)*(3 + 5*x)^(3/2)) + ((-415790*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/ 2)) + ((41487970*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) - (25890678*ArcTan[Sqrt [1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7])/33)/14)/28
3.26.15.3.1 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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(104)=208\).
Time = 3.97 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.82
| method | result | size |
| default | \(\frac {\left (32039714025 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+81167275530 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+77037712389 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+13068710550 x^{3} \sqrt {-10 x^{2}-x +3}+32466910212 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +24978031260 x^{2} \sqrt {-10 x^{2}-x +3}+5126354244 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+15890574518 x \sqrt {-10 x^{2}-x +3}+3364793096 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{996072 \left (2+3 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(250\) |
1/996072*(32039714025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^ (1/2))*x^4+81167275530*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3) ^(1/2))*x^3+77037712389*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3 )^(1/2))*x^2+13068710550*x^3*(-10*x^2-x+3)^(1/2)+32466910212*7^(1/2)*arcta n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+24978031260*x^2*(-10*x^2-x +3)^(1/2)+5126354244*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^( 1/2))+15890574518*x*(-10*x^2-x+3)^(1/2)+3364793096*(-10*x^2-x+3)^(1/2))*(1 -2*x)^(1/2)/(2+3*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {142398729 \, \sqrt {7} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (933479325 \, x^{3} + 1784145090 \, x^{2} + 1135041037 \, x + 240342364\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{996072 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
-1/996072*(142398729*sqrt(7)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*ar ctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3 )) - 14*(933479325*x^3 + 1784145090*x^2 + 1135041037*x + 240342364)*sqrt(5 *x + 3)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{3} \sqrt {-2 \, x + 1}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (104) = 208\).
Time = 0.43 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.72 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {25}{5808} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {392283}{27440} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {2425}{242} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {297 \, \sqrt {10} {\left (461 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {110600 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {442400 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
-25/5808*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4* sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 392283/27440*sqrt( 70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(- 10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22) ))) + 2425/242*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3 ) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 297/98*sqrt(10 )*(461*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 110600*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 442400*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2
Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}} \,d x \]