Integrand size = 26, antiderivative size = 122 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^4} \, dx=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {3895 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}-\frac {15235 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]
-15235/19208*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1/21* (1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+25/588*(1-2*x)^(1/2)*(3+5*x)^(1/2)/( 2+3*x)^2+3895/8232*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 2.00 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {5 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (5296+15930 x+11685 x^2\right )}{5 (2+3 x)^3}+3047 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+3047 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{19208} \]
(5*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5296 + 15930*x + 11685*x^2))/(5*(2 + 3 *x)^3) + 3047*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-S qrt[11] + Sqrt[5 - 10*x])] + 3047*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))]))/19208
Time = 0.22 (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.
| \(\displaystyle \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {1}{21} \int \frac {5 (8 x+7)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{42} \int \frac {8 x+7}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{42} \left (\frac {1}{14} \int \frac {193-100 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {5 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{42} \left (\frac {1}{28} \int \frac {193-100 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {5 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{42} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {9141}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {779 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {5 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{42} \left (\frac {1}{28} \left (\frac {9141}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {779 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {5 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5}{42} \left (\frac {1}{28} \left (\frac {9141}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {779 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {5 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{42} \left (\frac {1}{28} \left (\frac {779 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {9141 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {5 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\) |
-1/21*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^3 + (5*((5*Sqrt[1 - 2*x]*Sqr t[3 + 5*x])/(14*(2 + 3*x)^2) + ((779*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (9141*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/ 28))/42
3.25.66.3.1 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])
Time = 1.17 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (11685 x^{2}+15930 x +5296\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2744 \left (2+3 x \right )^{3} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {15235 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{38416 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(124\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (411345 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+822690 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+548460 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +163590 x^{2} \sqrt {-10 x^{2}-x +3}+121880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+223020 x \sqrt {-10 x^{2}-x +3}+74144 \sqrt {-10 x^{2}-x +3}\right )}{38416 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\) | \(202\) |
-1/2744*(-1+2*x)*(3+5*x)^(1/2)*(11685*x^2+15930*x+5296)/(2+3*x)^3/(-(-1+2* x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+15235/38416*7^(1/2 )*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2))*((1-2* x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^4} \, dx=-\frac {15235 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (11685 \, x^{2} + 15930 \, x + 5296\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{38416 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
-1/38416*(15235*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*( 37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(11685*x^2 + 15930*x + 5296)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\int \frac {\sqrt {5 x + 3}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{4}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {15235}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{21 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {25 \, \sqrt {-10 \, x^{2} - x + 3}}{588 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {3895 \, \sqrt {-10 \, x^{2} - x + 3}}{8232 \, {\left (3 \, x + 2\right )}} \]
15235/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/ 21*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 25/588*sqrt(-10*x^ 2 - x + 3)/(9*x^2 + 12*x + 4) + 3895/8232*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (95) = 190\).
Time = 0.45 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.54 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {3047}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {55 \, \sqrt {10} {\left (277 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 159040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {20713280 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {82853120 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1372 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
3047/76832*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(-10*x + 5) - sqrt(22)))) - 55/1372*sqrt(10)*(277*((sqrt(2)*sqrt(-10*x + 5) - sq rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 )))^5 - 159040*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 20713280*(sqrt(2)*sqr t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 82853120*sqrt(5*x + 3)/(sqrt(2)*s qrt(-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
Time = 18.79 (sec) , antiderivative size = 1273, normalized size of antiderivative = 10.43 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\text {Too large to display} \]
((8498458*((1 - 2*x)^(1/2) - 1)^5)/(5359375*(3^(1/2) - (5*x + 3)^(1/2))^5) - (3429372*((1 - 2*x)^(1/2) - 1)^3)/(5359375*(3^(1/2) - (5*x + 3)^(1/2))^ 3) - (52708*((1 - 2*x)^(1/2) - 1))/(5359375*(3^(1/2) - (5*x + 3)^(1/2))) - (4249229*((1 - 2*x)^(1/2) - 1)^7)/(1071875*(3^(1/2) - (5*x + 3)^(1/2))^7) + (857343*((1 - 2*x)^(1/2) - 1)^9)/(85750*(3^(1/2) - (5*x + 3)^(1/2))^9) + (13177*((1 - 2*x)^(1/2) - 1)^11)/(13720*(3^(1/2) - (5*x + 3)^(1/2))^11) + (418634*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(5359375*(3^(1/2) - (5*x + 3)^( 1/2))^2) + (399977*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(765625*(3^(1/2) - (5* x + 3)^(1/2))^4) - (12159864*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(5359375*(3^ (1/2) - (5*x + 3)^(1/2))^6) + (399977*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(12 2500*(3^(1/2) - (5*x + 3)^(1/2))^8) + (209317*3^(1/2)*((1 - 2*x)^(1/2) - 1 )^10)/(68600*(3^(1/2) - (5*x + 3)^(1/2))^10))/((5856*((1 - 2*x)^(1/2) - 1) ^2)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^2) - (4224*((1 - 2*x)^(1/2) - 1)^4) /(15625*(3^(1/2) - (5*x + 3)^(1/2))^4) - (14776*((1 - 2*x)^(1/2) - 1)^6)/( 15625*(3^(1/2) - (5*x + 3)^(1/2))^6) - (1056*((1 - 2*x)^(1/2) - 1)^8)/(625 *(3^(1/2) - (5*x + 3)^(1/2))^8) + (366*((1 - 2*x)^(1/2) - 1)^10)/(25*(3^(1 /2) - (5*x + 3)^(1/2))^10) + ((1 - 2*x)^(1/2) - 1)^12/(3^(1/2) - (5*x + 3) ^(1/2))^12 - (7776*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^3) + (34704*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^5) - (17352*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(3125*(...