Integrand size = 26, antiderivative size = 122 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^2}-\frac {15 \sqrt {1-2 x} \sqrt {3+5 x}}{98 (2+3 x)^2}+\frac {15 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}-\frac {1585 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \]
-1585/9604*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+2/7*(3+ 5*x)^(1/2)/(2+3*x)^2/(1-2*x)^(1/2)-15/98*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3* x)^2+15/1372*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=\frac {7 \sqrt {3+5 x} \left (212+405 x-90 x^2\right )-1585 \sqrt {7-14 x} (2+3 x)^2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{9604 \sqrt {1-2 x} (2+3 x)^2} \]
(7*Sqrt[3 + 5*x]*(212 + 405*x - 90*x^2) - 1585*Sqrt[7 - 14*x]*(2 + 3*x)^2* ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(9604*Sqrt[1 - 2*x]*(2 + 3* x)^2)
Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.08, 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, 168, 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.
\(\displaystyle \int \frac {\sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)^3} \, dx\) |
\(\Big \downarrow \) 110 |
\(\displaystyle \frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {2}{7} \int -\frac {5 (12 x+7)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {5}{7} \int \frac {12 x+7}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^2}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{14} \int \frac {60 x+41}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{28} \int \frac {60 x+41}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^2}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {317}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{28} \left (\frac {317}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^2}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{28} \left (\frac {317}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{28} \left (\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {317 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^2}\) |
(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*((-3*Sqrt[1 - 2*x]*Sq rt[3 + 5*x])/(14*(2 + 3*x)^2) + ((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3 *x)) - (317*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28 ))/7
3.26.26.3.1 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_)^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. \(2(95)=190\).
Time = 1.20 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.71
method | result | size |
default | \(\frac {\left (28530 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+23775 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-6340 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +1260 x^{2} \sqrt {-10 x^{2}-x +3}-6340 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-5670 x \sqrt {-10 x^{2}-x +3}-2968 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{19208 \left (2+3 x \right )^{2} \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(209\) |
1/19208*(28530*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))* x^3+23775*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-6 340*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+1260*x^2* (-10*x^2-x+3)^(1/2)-6340*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+ 3)^(1/2))-5670*x*(-10*x^2-x+3)^(1/2)-2968*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/ 2)*(3+5*x)^(1/2)/(2+3*x)^2/(-1+2*x)/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=-\frac {1585 \, \sqrt {7} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (90 \, x^{2} - 405 \, x - 212\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{19208 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]
-1/19208*(1585*sqrt(7)*(18*x^3 + 15*x^2 - 4*x - 4)*arctan(1/14*sqrt(7)*(37 *x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(90*x^2 - 405 *x - 212)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(18*x^3 + 15*x^2 - 4*x - 4)
\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=\int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{3}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=\frac {1585}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {25 \, x}{686 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {785}{4116 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{42 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {95}{588 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
1585/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 25/ 686*x/sqrt(-10*x^2 - x + 3) + 785/4116/sqrt(-10*x^2 - x + 3) + 1/42/(9*sqr t(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) - 95/588/(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. 278 vs. \(2 (95) = 190\).
Time = 0.46 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.28 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=\frac {317}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {8 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1715 \, {\left (2 \, x - 1\right )}} - \frac {33 \, \sqrt {10} {\left (7 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {680 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {2720 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
317/38416*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)))) - 8/1715*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1 ) - 33/98*sqrt(10)*(7*((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 - 680*(sqrt(2)*s qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2720*sqrt(5*x + 3)/(sqrt(2)*sqr t(-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)^2
Timed out. \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=\int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3} \,d x \]