Integrand size = 26, antiderivative size = 115 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {850 \sqrt {3+5 x}}{11319 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {5 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)}-\frac {75 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}} \]
-75/2401*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+2/21*(3+5 *x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)+850/11319*(3+5*x)^(1/2)/(1-2*x)^(1/2)-5/49 *(3+5*x)^(1/2)/(2+3*x)/(1-2*x)^(1/2)
Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=-\frac {7 \sqrt {3+5 x} \left (-1623-1460 x+5100 x^2\right )-2475 \sqrt {7-14 x} \left (-2+x+6 x^2\right ) \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{79233 (1-2 x)^{3/2} (2+3 x)} \]
-1/79233*(7*Sqrt[3 + 5*x]*(-1623 - 1460*x + 5100*x^2) - 2475*Sqrt[7 - 14*x ]*(-2 + x + 6*x^2)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/((1 - 2* x)^(3/2)*(2 + 3*x))
Time = 0.21 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {110, 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.
| \(\displaystyle \int \frac {\sqrt {5 x+3}}{(1-2 x)^{5/2} (3 x+2)^2} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}-\frac {2}{21} \int -\frac {5 (12 x+7)}{2 (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{21} \int \frac {12 x+7}{(1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{21} \left (\frac {1}{7} \int \frac {5 (12 x+11)}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{21} \left (\frac {5}{14} \int \frac {12 x+11}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{21} \left (\frac {5}{14} \left (\frac {68 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {2}{77} \int -\frac {99}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{21} \left (\frac {5}{14} \left (\frac {9}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {68 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )-\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5}{21} \left (\frac {5}{14} \left (\frac {18}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {68 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )-\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{21} \left (\frac {5}{14} \left (\frac {68 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {18 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )-\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}\) |
(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) + (5*((-3*Sqrt[3 + 5*x])/ (7*Sqrt[1 - 2*x]*(2 + 3*x)) + (5*((68*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) - (18*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])))/14))/21
3.26.86.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (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_) )^(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. \(208\) vs. \(2(88)=176\).
Time = 4.06 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.82
| method | result | size |
| default | \(\frac {\left (29700 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-9900 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-12375 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -71400 x^{2} \sqrt {-10 x^{2}-x +3}+4950 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+20440 x \sqrt {-10 x^{2}-x +3}+22722 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{158466 \left (2+3 x \right ) \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(209\) |
1/158466*(29700*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) *x^3-9900*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-1 2375*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-71400*x^ 2*(-10*x^2-x+3)^(1/2)+4950*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2- x+3)^(1/2))+20440*x*(-10*x^2-x+3)^(1/2)+22722*(-10*x^2-x+3)^(1/2))*(1-2*x) ^(1/2)*(3+5*x)^(1/2)/(2+3*x)/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=-\frac {2475 \, \sqrt {7} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (5100 \, x^{2} - 1460 \, x - 1623\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{158466 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
-1/158466*(2475*sqrt(7)*(12*x^3 - 4*x^2 - 5*x + 2)*arctan(1/14*sqrt(7)*(37 *x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(5100*x^2 - 1 460*x - 1623)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(12*x^3 - 4*x^2 - 5*x + 2)
\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{2}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {75}{4802} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {4250 \, x}{11319 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {625}{11319 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {100 \, x}{147 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{63 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {215}{441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
75/4802*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 4250/1 1319*x/sqrt(-10*x^2 - x + 3) + 625/11319/sqrt(-10*x^2 - x + 3) + 100/147*x /(-10*x^2 - x + 3)^(3/2) - 1/63/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 215/441/(-10*x^2 - x + 3)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (88) = 176\).
Time = 0.42 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {15}{9604} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {198 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{343 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} - \frac {8 \, {\left (163 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1089 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{282975 \, {\left (2 \, x - 1\right )}^{2}} \]
15/9604*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 198/343*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)))/(((s qrt(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) - 8/282975*(163*sqrt(5)*(5*x + 3) - 1089*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2
Timed out. \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2} \,d x \]