Integrand size = 26, antiderivative size = 77 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {333}{400} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}+\frac {3827 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400 \sqrt {10}} \]
3827/4000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-333/400*(1-2*x)^(1/ 2)*(3+5*x)^(1/2)-3/20*(2+3*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {-30 \sqrt {1-2 x} \left (453+935 x+300 x^2\right )-3827 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{4000 \sqrt {3+5 x}} \]
(-30*Sqrt[1 - 2*x]*(453 + 935*x + 300*x^2) - 3827*Sqrt[30 + 50*x]*ArcTan[S qrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(4000*Sqrt[3 + 5*x])
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {101, 27, 90, 64, 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.
\(\displaystyle \int \frac {(3 x+2)^2}{\sqrt {1-2 x} \sqrt {5 x+3}} \, dx\) |
\(\Big \downarrow \) 101 |
\(\displaystyle -\frac {1}{20} \int -\frac {333 x+208}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {3}{20} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{40} \int \frac {333 x+208}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {3}{20} \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{40} \left (\frac {3827}{20} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {333}{10} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{20} \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {1}{40} \left (\frac {3827}{50} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {333}{10} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{20} \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{40} \left (\frac {3827 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10 \sqrt {10}}-\frac {333}{10} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{20} \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}\) |
(-3*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x])/20 + ((-333*Sqrt[1 - 2*x]*Sqrt[ 3 + 5*x])/10 + (3827*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10*Sqrt[10]))/40
3.25.91.3.1 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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 = 70, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (3827 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3600 x \sqrt {-10 x^{2}-x +3}-9060 \sqrt {-10 x^{2}-x +3}\right )}{8000 \sqrt {-10 x^{2}-x +3}}\) | \(70\) |
risch | \(\frac {3 \left (151+60 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{400 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {3827 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{8000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(93\) |
1/8000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(3827*10^(1/2)*arcsin(20/11*x+1/11)-360 0*x*(-10*x^2-x+3)^(1/2)-9060*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {3}{400} \, {\left (60 \, x + 151\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3827}{8000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
-3/400*(60*x + 151)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3827/8000*sqrt(10)*arct an(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{2}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.53 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {9}{20} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3827}{8000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {453}{400} \, \sqrt {-10 \, x^{2} - x + 3} \]
-9/20*sqrt(-10*x^2 - x + 3)*x - 3827/8000*sqrt(10)*arcsin(-20/11*x - 1/11) - 453/400*sqrt(-10*x^2 - x + 3)
Time = 0.31 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.58 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {1}{4000} \, \sqrt {5} {\left (6 \, {\left (60 \, x + 151\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 3827 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
-1/4000*sqrt(5)*(6*(60*x + 151)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 3827*sqrt( 2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
Time = 11.64 (sec) , antiderivative size = 360, normalized size of antiderivative = 4.68 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {3827\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{2000}-\frac {\frac {627\,\left (\sqrt {1-2\,x}-1\right )}{15625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {7941\,{\left (\sqrt {1-2\,x}-1\right )}^3}{6250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}+\frac {7941\,{\left (\sqrt {1-2\,x}-1\right )}^5}{2500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {627\,{\left (\sqrt {1-2\,x}-1\right )}^7}{1000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {384\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1632\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {96\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}}{\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^2}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {24\,{\left (\sqrt {1-2\,x}-1\right )}^4}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {8\,{\left (\sqrt {1-2\,x}-1\right )}^6}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^8}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {16}{625}} \]
(3827*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/2000 - ((627*((1 - 2*x)^(1/2) - 1))/(15625*(3^(1/2) - (5*x + 3)^(1/2))) - (7941*((1 - 2*x)^(1/2) - 1)^3)/(6250*(3^(1/2) - (5*x + 3)^(1/ 2))^3) + (7941*((1 - 2*x)^(1/2) - 1)^5)/(2500*(3^(1/2) - (5*x + 3)^(1/2))^ 5) - (627*((1 - 2*x)^(1/2) - 1)^7)/(1000*(3^(1/2) - (5*x + 3)^(1/2))^7) + (384*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (1632*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(625*(3^(1/2) - (5*x + 3)^(1/2))^ 4) + (96*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(25*(3^(1/2) - (5*x + 3)^(1/2))^ 6))/((32*((1 - 2*x)^(1/2) - 1)^2)/(125*(3^(1/2) - (5*x + 3)^(1/2))^2) + (2 4*((1 - 2*x)^(1/2) - 1)^4)/(25*(3^(1/2) - (5*x + 3)^(1/2))^4) + (8*((1 - 2 *x)^(1/2) - 1)^6)/(5*(3^(1/2) - (5*x + 3)^(1/2))^6) + ((1 - 2*x)^(1/2) - 1 )^8/(3^(1/2) - (5*x + 3)^(1/2))^8 + 16/625)