Integrand size = 26, antiderivative size = 84 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}} \]
44437/16000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-1/10*(2+3*x)^2*(1 -2*x)^(1/2)*(3+5*x)^(1/2)-1/1600*(5363+2220*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {-90 \sqrt {1-2 x} \left (2001+4715 x+2780 x^2+800 x^3\right )-44437 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{16000 \sqrt {3+5 x}} \]
(-90*Sqrt[1 - 2*x]*(2001 + 4715*x + 2780*x^2 + 800*x^3) - 44437*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(16000*Sqrt[3 + 5*x])
Time = 0.20 (sec) , antiderivative size = 89, 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 = {111, 27, 164, 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.
| \(\displaystyle \int \frac {(3 x+2)^3}{\sqrt {1-2 x} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {1}{30} \int -\frac {3 (3 x+2) (185 x+114)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{20} \int \frac {(3 x+2) (185 x+114)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{20} \left (\frac {44437}{160} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (2220 x+5363)\right )-\frac {1}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{20} \left (\frac {44437}{400} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (2220 x+5363)\right )-\frac {1}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{20} \left (\frac {44437 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80 \sqrt {10}}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (2220 x+5363)\right )-\frac {1}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\) |
-1/10*(Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (-1/80*(Sqrt[1 - 2*x]*Sq rt[3 + 5*x]*(5363 + 2220*x)) + (44437*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8 0*Sqrt[10]))/20
3.25.90.3.1 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_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 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 = 87, normalized size of antiderivative = 1.04
| method | result | size |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-28800 x^{2} \sqrt {-10 x^{2}-x +3}+44437 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-82800 x \sqrt {-10 x^{2}-x +3}-120060 \sqrt {-10 x^{2}-x +3}\right )}{32000 \sqrt {-10 x^{2}-x +3}}\) | \(87\) |
| risch | \(\frac {9 \left (160 x^{2}+460 x +667\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1600 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {44437 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{32000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(98\) |
1/32000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-28800*x^2*(-10*x^2-x+3)^(1/2)+44437* 10^(1/2)*arcsin(20/11*x+1/11)-82800*x*(-10*x^2-x+3)^(1/2)-120060*(-10*x^2- x+3)^(1/2))/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.80 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {9}{1600} \, {\left (160 \, x^{2} + 460 \, x + 667\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {44437}{32000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
-9/1600*(160*x^2 + 460*x + 667)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 44437/32000 *sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10 *x^2 + x - 3))
\[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{3}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {9}{10} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {207}{80} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {44437}{32000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {6003}{1600} \, \sqrt {-10 \, x^{2} - x + 3} \]
-9/10*sqrt(-10*x^2 - x + 3)*x^2 - 207/80*sqrt(-10*x^2 - x + 3)*x - 44437/3 2000*sqrt(10)*arcsin(-20/11*x - 1/11) - 6003/1600*sqrt(-10*x^2 - x + 3)
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {1}{80000} \, \sqrt {5} {\left (18 \, {\left (4 \, {\left (40 \, x + 91\right )} {\left (5 \, x + 3\right )} + 2243\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 222185 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
-1/80000*sqrt(5)*(18*(4*(40*x + 91)*(5*x + 3) + 2243)*sqrt(5*x + 3)*sqrt(- 10*x + 5) - 222185*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
Time = 8.79 (sec) , antiderivative size = 534, normalized size of antiderivative = 6.36 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {44437\,\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 )}{8000}-\frac {\frac {18837\,\left (\sqrt {1-2\,x}-1\right )}{390625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {154377\,{\left (\sqrt {1-2\,x}-1\right )}^3}{156250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {226251\,{\left (\sqrt {1-2\,x}-1\right )}^5}{156250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {226251\,{\left (\sqrt {1-2\,x}-1\right )}^7}{62500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {154377\,{\left (\sqrt {1-2\,x}-1\right )}^9}{10000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}-\frac {18837\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{4000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {4608\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {59904\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {107136\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {14976\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {288\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}}{\frac {192\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {48\,{\left (\sqrt {1-2\,x}-1\right )}^4}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^8}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{12}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {64}{15625}} \]
(44437*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/8000 - ((18837*((1 - 2*x)^(1/2) - 1))/(390625*(3^(1/2) - (5* x + 3)^(1/2))) - (154377*((1 - 2*x)^(1/2) - 1)^3)/(156250*(3^(1/2) - (5*x + 3)^(1/2))^3) - (226251*((1 - 2*x)^(1/2) - 1)^5)/(156250*(3^(1/2) - (5*x + 3)^(1/2))^5) + (226251*((1 - 2*x)^(1/2) - 1)^7)/(62500*(3^(1/2) - (5*x + 3)^(1/2))^7) + (154377*((1 - 2*x)^(1/2) - 1)^9)/(10000*(3^(1/2) - (5*x + 3)^(1/2))^9) - (18837*((1 - 2*x)^(1/2) - 1)^11)/(4000*(3^(1/2) - (5*x + 3) ^(1/2))^11) + (4608*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(15625*(3^(1/2) - (5* x + 3)^(1/2))^2) + (59904*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (107136*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(15625*( 3^(1/2) - (5*x + 3)^(1/2))^6) + (14976*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(6 25*(3^(1/2) - (5*x + 3)^(1/2))^8) + (288*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10) /(25*(3^(1/2) - (5*x + 3)^(1/2))^10))/((192*((1 - 2*x)^(1/2) - 1)^2)/(3125 *(3^(1/2) - (5*x + 3)^(1/2))^2) + (48*((1 - 2*x)^(1/2) - 1)^4)/(125*(3^(1/ 2) - (5*x + 3)^(1/2))^4) + (32*((1 - 2*x)^(1/2) - 1)^6)/(25*(3^(1/2) - (5* x + 3)^(1/2))^6) + (12*((1 - 2*x)^(1/2) - 1)^8)/(5*(3^(1/2) - (5*x + 3)^(1 /2))^8) + (12*((1 - 2*x)^(1/2) - 1)^10)/(5*(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 + 64/15625)