Integrand size = 26, antiderivative size = 113 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{128000 \sqrt {10}} \]
10866247/1280000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-259/800*(2+3 *x)^2*(1-2*x)^(1/2)*(3+5*x)^(1/2)-3/40*(2+3*x)^3*(1-2*x)^(1/2)*(3+5*x)^(1/ 2)-7/128000*(187559+77820*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {-30 \sqrt {1-2 x} \left (1555473+3980075 x+3204060 x^2+1744800 x^3+432000 x^4\right )-10866247 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1280000 \sqrt {3+5 x}} \]
(-30*Sqrt[1 - 2*x]*(1555473 + 3980075*x + 3204060*x^2 + 1744800*x^3 + 4320 00*x^4) - 10866247*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/ (1280000*Sqrt[3 + 5*x])
Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {111, 27, 170, 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.
| \(\displaystyle \int \frac {(3 x+2)^4}{\sqrt {1-2 x} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {1}{40} \int -\frac {7 (3 x+2)^2 (111 x+68)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {3}{40} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{80} \int \frac {(3 x+2)^2 (111 x+68)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {7}{80} \left (-\frac {1}{30} \int -\frac {3 (3 x+2) (6485 x+3978)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {37}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{80} \left (\frac {1}{20} \int \frac {(3 x+2) (6485 x+3978)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {37}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {7}{80} \left (\frac {1}{20} \left (\frac {1552321}{160} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (77820 x+187559)\right )-\frac {37}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {7}{80} \left (\frac {1}{20} \left (\frac {1552321}{400} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (77820 x+187559)\right )-\frac {37}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {7}{80} \left (\frac {1}{20} \left (\frac {1552321 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80 \sqrt {10}}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (77820 x+187559)\right )-\frac {37}{10} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}\) |
(-3*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x])/40 + (7*((-37*Sqrt[1 - 2*x]*( 2 + 3*x)^2*Sqrt[3 + 5*x])/10 + (-1/80*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(187559 + 77820*x)) + (1552321*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80*Sqrt[10]))/2 0))/80
3.25.89.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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 = 1.17 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(\frac {3 \left (86400 x^{3}+297120 x^{2}+462540 x +518491\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{128000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {10866247 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2560000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-5184000 x^{3} \sqrt {-10 x^{2}-x +3}-17827200 x^{2} \sqrt {-10 x^{2}-x +3}+10866247 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-27752400 x \sqrt {-10 x^{2}-x +3}-31109460 \sqrt {-10 x^{2}-x +3}\right )}{2560000 \sqrt {-10 x^{2}-x +3}}\) | \(104\) |
3/128000*(86400*x^3+297120*x^2+462540*x+518491)*(-1+2*x)*(3+5*x)^(1/2)/(-( -1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+10866247/2560 000*10^(1/2)*arcsin(20/11*x+1/11)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3 +5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {3}{128000} \, {\left (86400 \, x^{3} + 297120 \, x^{2} + 462540 \, x + 518491\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {10866247}{2560000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
-3/128000*(86400*x^3 + 297120*x^2 + 462540*x + 518491)*sqrt(5*x + 3)*sqrt( -2*x + 1) - 10866247/2560000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt (5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{4}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {81}{40} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {5571}{800} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {69381}{6400} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {10866247}{2560000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {1555473}{128000} \, \sqrt {-10 \, x^{2} - x + 3} \]
-81/40*sqrt(-10*x^2 - x + 3)*x^3 - 5571/800*sqrt(-10*x^2 - x + 3)*x^2 - 69 381/6400*sqrt(-10*x^2 - x + 3)*x - 10866247/2560000*sqrt(10)*arcsin(-20/11 *x - 1/11) - 1555473/128000*sqrt(-10*x^2 - x + 3)
Time = 0.33 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.56 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {1}{6400000} \, \sqrt {5} {\left (6 \, {\left (12 \, {\left (8 \, {\left (180 \, x + 403\right )} {\left (5 \, x + 3\right )} + 16609\right )} {\left (5 \, x + 3\right )} + 1646339\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 54331235 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
-1/6400000*sqrt(5)*(6*(12*(8*(180*x + 403)*(5*x + 3) + 16609)*(5*x + 3) + 1646339)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 54331235*sqrt(2)*arcsin(1/11*sqrt (22)*sqrt(5*x + 3)))
Time = 12.98 (sec) , antiderivative size = 708, normalized size of antiderivative = 6.27 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {10866247\,\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 )}{640000}-\frac {\frac {6770247\,\left (\sqrt {1-2\,x}-1\right )}{195312500\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {33291573\,{\left (\sqrt {1-2\,x}-1\right )}^3}{78125000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {883182573\,{\left (\sqrt {1-2\,x}-1\right )}^5}{156250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {451883391\,{\left (\sqrt {1-2\,x}-1\right )}^7}{62500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {451883391\,{\left (\sqrt {1-2\,x}-1\right )}^9}{25000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {883182573\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{10000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {33291573\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{800000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}-\frac {6770247\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{320000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {49152\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {258048\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1032192\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {16147968\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {258048\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {16128\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {768\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}}{\frac {1024\,{\left (\sqrt {1-2\,x}-1\right )}^2}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^6}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {224\,{\left (\sqrt {1-2\,x}-1\right )}^8}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {448\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {112\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {16\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{16}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}+\frac {256}{390625}} \]
(10866247*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5* x + 3)^(1/2)))))/640000 - ((6770247*((1 - 2*x)^(1/2) - 1))/(195312500*(3^( 1/2) - (5*x + 3)^(1/2))) - (33291573*((1 - 2*x)^(1/2) - 1)^3)/(78125000*(3 ^(1/2) - (5*x + 3)^(1/2))^3) - (883182573*((1 - 2*x)^(1/2) - 1)^5)/(156250 000*(3^(1/2) - (5*x + 3)^(1/2))^5) + (451883391*((1 - 2*x)^(1/2) - 1)^7)/( 62500000*(3^(1/2) - (5*x + 3)^(1/2))^7) - (451883391*((1 - 2*x)^(1/2) - 1) ^9)/(25000000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (883182573*((1 - 2*x)^(1/2) - 1)^11)/(10000000*(3^(1/2) - (5*x + 3)^(1/2))^11) + (33291573*((1 - 2*x) ^(1/2) - 1)^13)/(800000*(3^(1/2) - (5*x + 3)^(1/2))^13) - (6770247*((1 - 2 *x)^(1/2) - 1)^15)/(320000*(3^(1/2) - (5*x + 3)^(1/2))^15) + (49152*3^(1/2 )*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (25804 8*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^4) + (1032192*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(78125*(3^(1/2) - (5*x + 3)^(1/ 2))^6) + (16147968*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(390625*(3^(1/2) - (5* x + 3)^(1/2))^8) + (258048*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(3125*(3^(1/2 ) - (5*x + 3)^(1/2))^10) + (16128*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(125*( 3^(1/2) - (5*x + 3)^(1/2))^12) + (768*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(2 5*(3^(1/2) - (5*x + 3)^(1/2))^14))/((1024*((1 - 2*x)^(1/2) - 1)^2)/(78125* (3^(1/2) - (5*x + 3)^(1/2))^2) + (1792*((1 - 2*x)^(1/2) - 1)^4)/(15625*(3^ (1/2) - (5*x + 3)^(1/2))^4) + (1792*((1 - 2*x)^(1/2) - 1)^6)/(3125*(3^(...