Integrand size = 26, antiderivative size = 93 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=-\frac {29 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)}+\frac {4 (3+5 x)^{3/2}}{77 \sqrt {1-2 x} (2+3 x)}-\frac {29 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}} \]
-29/343*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+4/77*(3+5* x)^(3/2)/(2+3*x)/(1-2*x)^(1/2)-29/539*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {7 \sqrt {3+5 x} (5+18 x)-29 \sqrt {7-14 x} (2+3 x) \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {1-2 x} (2+3 x)} \]
(7*Sqrt[3 + 5*x]*(5 + 18*x) - 29*Sqrt[7 - 14*x]*(2 + 3*x)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[1 - 2*x]*(2 + 3*x))
Time = 0.18 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {107, 105, 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.
\(\displaystyle \int \frac {\sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 107 |
\(\displaystyle \frac {29}{77} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {4 (5 x+3)^{3/2}}{77 \sqrt {1-2 x} (3 x+2)}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {29}{77} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {4 (5 x+3)^{3/2}}{77 \sqrt {1-2 x} (3 x+2)}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {29}{77} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {4 (5 x+3)^{3/2}}{77 \sqrt {1-2 x} (3 x+2)}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {29}{77} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {4 (5 x+3)^{3/2}}{77 \sqrt {1-2 x} (3 x+2)}\) |
(4*(3 + 5*x)^(3/2))/(77*Sqrt[1 - 2*x]*(2 + 3*x)) + (29*(-1/7*(Sqrt[1 - 2*x ]*Sqrt[3 + 5*x])/(2 + 3*x) - (11*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5* x])])/(7*Sqrt[7])))/77
3.26.25.3.1 Defintions of rubi rules used
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[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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. \(160\) vs. \(2(72)=144\).
Time = 1.20 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.73
method | result | size |
default | \(\frac {\left (174 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+29 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -58 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-252 x \sqrt {-10 x^{2}-x +3}-70 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{686 \left (2+3 x \right ) \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(161\) |
1/686*(174*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+ 29*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-58*7^(1/2) *arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-252*x*(-10*x^2-x+3)^(1 /2)-70*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)/(-1+2*x)/( -10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=-\frac {29 \, \sqrt {7} {\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (18 \, x + 5\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{686 \, {\left (6 \, x^{2} + x - 2\right )}} \]
-1/686*(29*sqrt(7)*(6*x^2 + x - 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5* x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(18*x + 5)*sqrt(5*x + 3)*sqrt (-2*x + 1))/(6*x^2 + x - 2)
\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {29}{686} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {30 \, x}{49 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {19}{147 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{21 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
29/686*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 30/49*x /sqrt(-10*x^2 - x + 3) + 19/147/sqrt(-10*x^2 - x + 3) + 1/21/(3*sqrt(-10*x ^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (72) = 144\).
Time = 0.40 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {29}{6860} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {4 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{245 \, {\left (2 \, x - 1\right )}} - \frac {66 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{49 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
29/6860*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)))) - 4/245*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 66/49*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sq rt(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. \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2} \,d x \]