Integrand size = 26, antiderivative size = 130 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}-\frac {4887 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}} \]
-4887/343*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-58/539/( 3+5*x)^(3/2)/(1-2*x)^(1/2)+3/7/(2+3*x)/(3+5*x)^(3/2)/(1-2*x)^(1/2)-28705/1 7787*(1-2*x)^(1/2)/(3+5*x)^(3/2)+2841815/195657*(1-2*x)^(1/2)/(3+5*x)^(1/2 )
Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=\frac {16461125+20145298 x-63467215 x^2-85254450 x^3}{195657 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}-\frac {4887 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}} \]
(16461125 + 20145298*x - 63467215*x^2 - 85254450*x^3)/(195657*Sqrt[1 - 2*x ]*(2 + 3*x)*(3 + 5*x)^(3/2)) - (4887*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])
Time = 0.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {114, 27, 169, 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.
\(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {1}{7} \int \frac {61-180 x}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}}dx+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{14} \int \frac {61-180 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}}dx+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{14} \left (-\frac {2}{77} \int -\frac {3653-3480 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx-\frac {116}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \int \frac {3653-3480 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx-\frac {116}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \left (-\frac {2}{33} \int \frac {361687-344460 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {57410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {116}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \left (-\frac {1}{33} \int \frac {361687-344460 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {57410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {116}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \left (\frac {1}{33} \left (\frac {2}{11} \int \frac {19513791}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {5683630 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {57410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {116}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \left (\frac {1}{33} \left (1773981 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {5683630 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {57410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {116}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \left (\frac {1}{33} \left (3547962 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {5683630 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {57410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {116}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \left (\frac {1}{33} \left (\frac {5683630 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {3547962 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\right )-\frac {57410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {116}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\) |
3/(7*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(3/2)) + (-116/(77*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((-57410*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) + ((5683630* Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) - (3547962*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7] *Sqrt[3 + 5*x])])/Sqrt[7])/33)/77)/14
3.26.78.3.1 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] && 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. \(256\) vs. \(2(97)=194\).
Time = 4.06 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.98
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \left (2927068650 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+4000327155 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+663468894 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1193562300 x^{3} \sqrt {-10 x^{2}-x +3}-995203341 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +888541010 x^{2} \sqrt {-10 x^{2}-x +3}-351248238 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-282034172 x \sqrt {-10 x^{2}-x +3}-230455750 \sqrt {-10 x^{2}-x +3}\right )}{2739198 \left (2+3 x \right ) \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(257\) |
1/2739198*(1-2*x)^(1/2)*(2927068650*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/ (-10*x^2-x+3)^(1/2))*x^4+4000327155*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/ (-10*x^2-x+3)^(1/2))*x^3+663468894*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/( -10*x^2-x+3)^(1/2))*x^2+1193562300*x^3*(-10*x^2-x+3)^(1/2)-995203341*7^(1/ 2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+888541010*x^2*(-10 *x^2-x+3)^(1/2)-351248238*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x +3)^(1/2))-282034172*x*(-10*x^2-x+3)^(1/2)-230455750*(-10*x^2-x+3)^(1/2))/ (2+3*x)/(-1+2*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=-\frac {19513791 \, \sqrt {7} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (85254450 \, x^{3} + 63467215 \, x^{2} - 20145298 \, x - 16461125\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2739198 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \]
-1/2739198*(19513791*sqrt(7)*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*arct an(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(85254450*x^3 + 63467215*x^2 - 20145298*x - 16461125)*sqrt(5*x + 3)* sqrt(-2*x + 1))/(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (97) = 194\).
Time = 0.39 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.61 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=-\frac {25}{63888} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {4887}{6860} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {775}{1331} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {32 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{326095 \, {\left (2 \, x - 1\right )}} + \frac {1782 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{49 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
-25/63888*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 + 4887/6860*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))) ) + 775/1331*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))) - 32/326095*sqrt(5 )*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1782/49*sqrt(10)*((sqrt(2)*sqr t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*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)
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{5/2}} \,d x \]