Integrand size = 26, antiderivative size = 62 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=\frac {1}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3 \sqrt {7}} \]
2/21*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+1/3*arcsin(1/ 11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(138\) vs. \(2(62)=124\).
Time = 0.84 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=-\frac {2}{21} \left (\sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+7 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )+\sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right ) \]
(-2*(Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 7*Sqrt[10]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10*x])] + Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))]))/21
Time = 0.18 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {140, 27, 64, 104, 217, 223}
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.
\(\displaystyle \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)} \, dx\) |
\(\Big \downarrow \) 140 |
\(\displaystyle \frac {5}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\int -\frac {1}{3 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {5}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {2}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {2}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {2}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {2}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\) |
(Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 + (2*ArcTan[Sqrt[1 - 2*x]/(S qrt[7]*Sqrt[3 + 5*x])])/(3*Sqrt[7])
3.25.63.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [2/b Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] || PosQ[b])
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*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -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])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Time = 1.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (7 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )\right )}{42 \sqrt {-10 x^{2}-x +3}}\) | \(69\) |
1/42*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(7*10^(1/2)*arcsin(20/11*x+1/11)-2*7^(1/2 )*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)))/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=\frac {1}{21} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {1}{6} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/21*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/ (10*x^2 + x - 3)) - 1/6*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=\int \frac {\sqrt {5 x + 3}}{\sqrt {1 - 2 x} \left (3 x + 2\right )}\, dx \]
Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=\frac {1}{6} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{21} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) \]
1/6*sqrt(10)*arcsin(20/11*x + 1/11) - 1/21*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2))
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (44) = 88\).
Time = 0.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.27 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=-\frac {1}{210} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{6} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \]
-1/210*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5 ) - sqrt(22)))) + 1/6*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2) *sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - s qrt(22))))
Time = 4.60 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx=-\frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {5580\,\sqrt {21}\,x+2699\,\sqrt {21}-5489\,\sqrt {35\,x+21}+649\,\sqrt {21-42\,x}+2141\,\sqrt {7}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}}{7400\,x-5489\,\sqrt {1-2\,x}-4543\,\sqrt {15\,x+9}+3063\,\sqrt {3}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}+9929}\right )}{21}-\frac {2\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}-\sqrt {10-20\,x}}{2\,\sqrt {3}-2\,\sqrt {5\,x+3}}\right )}{3} \]
- (2*7^(1/2)*atan((5580*21^(1/2)*x + 2699*21^(1/2) - 5489*(35*x + 21)^(1/2 ) + 649*(21 - 42*x)^(1/2) + 2141*7^(1/2)*(1 - 2*x)^(1/2)*(5*x + 3)^(1/2))/ (7400*x - 5489*(1 - 2*x)^(1/2) - 4543*(15*x + 9)^(1/2) + 3063*3^(1/2)*(1 - 2*x)^(1/2)*(5*x + 3)^(1/2) + 9929)))/21 - (2*10^(1/2)*atan((10^(1/2) - (1 0 - 20*x)^(1/2))/(2*3^(1/2) - 2*(5*x + 3)^(1/2))))/3