Integrand size = 26, antiderivative size = 144 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {7396875 \sqrt {1-2 x}}{30184 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 \sqrt {3+5 x}}+\frac {255 \sqrt {1-2 x}}{196 (2+3 x)^2 \sqrt {3+5 x}}+\frac {44475 \sqrt {1-2 x}}{2744 (2+3 x) \sqrt {3+5 x}}+\frac {4616025 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]
4616025/19208*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-7396 875/30184*(1-2*x)^(1/2)/(3+5*x)^(1/2)+1/7*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^ (1/2)+255/196*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(1/2)+44475/2744*(1-2*x)^(1/ 2)/(2+3*x)/(3+5*x)^(1/2)
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=\frac {-\frac {7 \sqrt {1-2 x} \left (57135248+260298990 x+395028225 x^2+199715625 x^3\right )}{(2+3 x)^3 \sqrt {3+5 x}}+50776275 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{211288} \]
((-7*Sqrt[1 - 2*x]*(57135248 + 260298990*x + 395028225*x^2 + 199715625*x^3 ))/((2 + 3*x)^3*Sqrt[3 + 5*x]) + 50776275*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sq rt[7]*Sqrt[3 + 5*x])])/211288
Time = 0.23 (sec) , antiderivative size = 157, 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, 168, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{21} \int \frac {45 (3-4 x)}{2 \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{3/2}}dx+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {15}{14} \int \frac {3-4 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{3/2}}dx+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {15}{14} \left (\frac {1}{14} \int \frac {5 (107-136 x)}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {17 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {15}{14} \left (\frac {5}{28} \int \frac {107-136 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {17 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {15}{14} \left (\frac {5}{28} \left (\frac {1}{7} \int \frac {12609-11860 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx+\frac {593 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {17 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {15}{14} \left (\frac {5}{28} \left (\frac {1}{14} \int \frac {12609-11860 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx+\frac {593 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {17 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {15}{14} \left (\frac {5}{28} \left (\frac {1}{14} \left (-\frac {2}{11} \int \frac {677017}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {197250 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {593 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {17 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {15}{14} \left (\frac {5}{28} \left (\frac {1}{14} \left (-61547 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {197250 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {593 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {17 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {15}{14} \left (\frac {5}{28} \left (\frac {1}{14} \left (-123094 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {197250 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {593 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {17 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {15}{14} \left (\frac {5}{28} \left (\frac {1}{14} \left (\frac {123094 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}-\frac {197250 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {593 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {17 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (15*((17*Sqrt[1 - 2*x])/(14* (2 + 3*x)^2*Sqrt[3 + 5*x]) + (5*((593*Sqrt[1 - 2*x])/(7*(2 + 3*x)*Sqrt[3 + 5*x]) + ((-197250*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) + (123094*ArcTan[Sqrt [1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7])/14))/28))/14
3.26.6.3.1 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. \(2(111)=222\).
Time = 1.18 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.74
| method | result | size |
| default | \(-\frac {\left (6854797125 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+17822472525 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+17365486050 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+2796018750 x^{3} \sqrt {-10 x^{2}-x +3}+7514888700 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +5530395150 x^{2} \sqrt {-10 x^{2}-x +3}+1218630600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3644185860 x \sqrt {-10 x^{2}-x +3}+799893472 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{422576 \left (2+3 x \right )^{3} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(250\) |
-1/422576*(6854797125*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^ (1/2))*x^4+17822472525*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3) ^(1/2))*x^3+17365486050*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3 )^(1/2))*x^2+2796018750*x^3*(-10*x^2-x+3)^(1/2)+7514888700*7^(1/2)*arctan( 1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+5530395150*x^2*(-10*x^2-x+3) ^(1/2)+1218630600*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2 ))+3644185860*x*(-10*x^2-x+3)^(1/2)+799893472*(-10*x^2-x+3)^(1/2))*(1-2*x) ^(1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=\frac {50776275 \, \sqrt {7} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (199715625 \, x^{3} + 395028225 \, x^{2} + 260298990 \, x + 57135248\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{422576 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
1/422576*(50776275*sqrt(7)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*arct an(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(199715625*x^3 + 395028225*x^2 + 260298990*x + 57135248)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=\int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{4} \sqrt {-2 \, x + 1}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (111) = 222\).
Time = 0.43 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.56 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {923205}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {125}{22} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {7425 \, \sqrt {10} {\left (487 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 217280 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {25693248 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {102772992 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1372 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
-923205/76832*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(-1 0*x + 5) - sqrt(22)))) - 125/22*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))) - 7425/1372*sqrt(10)*(487*((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)))^5 + 217280*(( 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 + 25693248*(sqrt(2)*sqrt(-10*x + 5) - sq rt(22))/sqrt(5*x + 3) - 102772992*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*sqrt( 5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3
Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}} \,d x \]