Integrand size = 26, antiderivative size = 128 \[ \int \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x} \, dx=\frac {3558401 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000}-\frac {323491 (1-2 x)^{3/2} \sqrt {3+5 x}}{128000}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {39142411 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1280000 \sqrt {10}} \]
-3/50*(1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(3/2)-21/16000*(1-2*x)^(3/2)*(3+5*x) ^(3/2)*(731+444*x)+39142411/12800000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*1 0^(1/2)-323491/128000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+3558401/1280000*(1-2*x)^ (1/2)*(3+5*x)^(1/2)
Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x} \, dx=\frac {10 \sqrt {1-2 x} \left (-12847047-12404125 x+59852860 x^2+126832800 x^3+107568000 x^4+34560000 x^5\right )-39142411 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{12800000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(-12847047 - 12404125*x + 59852860*x^2 + 126832800*x^3 + 107568000*x^4 + 34560000*x^5) - 39142411*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(12800000*Sqrt[3 + 5*x])
Time = 0.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {111, 27, 164, 60, 60, 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.
| \(\displaystyle \int \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {1}{50} \int -\frac {7}{2} \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3} (111 x+70)dx-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{100} \int \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3} (111 x+70)dx-\frac {3}{50} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {7}{100} \left (\frac {46213}{320} \int \sqrt {1-2 x} \sqrt {5 x+3}dx-\frac {3}{160} (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {7}{100} \left (\frac {46213}{320} \left (\frac {11}{8} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {3}{160} (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {7}{100} \left (\frac {46213}{320} \left (\frac {11}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {3}{160} (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {7}{100} \left (\frac {46213}{320} \left (\frac {11}{8} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {3}{160} (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {7}{100} \left (\frac {46213}{320} \left (\frac {11}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {3}{160} (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
(-3*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/50 + (7*((-3*(1 - 2*x)^(3 /2)*(3 + 5*x)^(3/2)*(731 + 444*x))/160 + (46213*(-1/4*((1 - 2*x)^(3/2)*Sqr t[3 + 5*x]) + (11*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11] *Sqrt[3 + 5*x]])/(5*Sqrt[10])))/8))/320))/100
3.23.63.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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 = 1.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {\left (6912000 x^{4}+17366400 x^{3}+14946720 x^{2}+3002540 x -4282349\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1280000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {39142411 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{25600000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(108\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (138240000 x^{4} \sqrt {-10 x^{2}-x +3}+347328000 x^{3} \sqrt {-10 x^{2}-x +3}+298934400 x^{2} \sqrt {-10 x^{2}-x +3}+39142411 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+60050800 x \sqrt {-10 x^{2}-x +3}-85646980 \sqrt {-10 x^{2}-x +3}\right )}{25600000 \sqrt {-10 x^{2}-x +3}}\) | \(121\) |
-1/1280000*(6912000*x^4+17366400*x^3+14946720*x^2+3002540*x-4282349)*(-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)+39142411/25600000*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 = 77, normalized size of antiderivative = 0.60 \[ \int \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x} \, dx=\frac {1}{1280000} \, {\left (6912000 \, x^{4} + 17366400 \, x^{3} + 14946720 \, x^{2} + 3002540 \, x - 4282349\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {39142411}{25600000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/1280000*(6912000*x^4 + 17366400*x^3 + 14946720*x^2 + 3002540*x - 4282349 )*sqrt(5*x + 3)*sqrt(-2*x + 1) - 39142411/25600000*sqrt(10)*arctan(1/20*sq rt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right )^{3} \sqrt {5 x + 3}\, dx \]
Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.68 \[ \int \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x} \, dx=-\frac {27}{50} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {5211}{4000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {19191}{16000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {323491}{64000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {39142411}{25600000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {323491}{1280000} \, \sqrt {-10 \, x^{2} - x + 3} \]
-27/50*(-10*x^2 - x + 3)^(3/2)*x^2 - 5211/4000*(-10*x^2 - x + 3)^(3/2)*x - 19191/16000*(-10*x^2 - x + 3)^(3/2) + 323491/64000*sqrt(-10*x^2 - x + 3)* x - 39142411/25600000*sqrt(10)*arcsin(-20/11*x - 1/11) + 323491/1280000*sq rt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (95) = 190\).
Time = 0.34 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.15 \[ \int \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x} \, dx=\frac {9}{64000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {117}{3200000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {57}{20000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {37}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {12}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
9/64000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sq rt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 117/3200000*sqrt(5)*(2*(4*(8* (60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 57/20000*sqr t(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4 785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 37/500*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sq rt(5*x + 3))) + 12/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
Time = 14.16 (sec) , antiderivative size = 881, normalized size of antiderivative = 6.88 \[ \int \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x} \, dx=\text {Too large to display} \]
((22297589*((1 - 2*x)^(1/2) - 1))/(12207031250*(3^(1/2) - (5*x + 3)^(1/2)) ) + (369826027*((1 - 2*x)^(1/2) - 1)^3)/(4882812500*(3^(1/2) - (5*x + 3)^( 1/2))^3) - (4945417109*((1 - 2*x)^(1/2) - 1)^5)/(2441406250*(3^(1/2) - (5* x + 3)^(1/2))^5) + (1598593169*((1 - 2*x)^(1/2) - 1)^7)/(195312500*(3^(1/2 ) - (5*x + 3)^(1/2))^7) - (914901953*((1 - 2*x)^(1/2) - 1)^9)/(156250000*( 3^(1/2) - (5*x + 3)^(1/2))^9) + (914901953*((1 - 2*x)^(1/2) - 1)^11)/(6250 0000*(3^(1/2) - (5*x + 3)^(1/2))^11) - (1598593169*((1 - 2*x)^(1/2) - 1)^1 3)/(12500000*(3^(1/2) - (5*x + 3)^(1/2))^13) + (4945417109*((1 - 2*x)^(1/2 ) - 1)^15)/(25000000*(3^(1/2) - (5*x + 3)^(1/2))^15) - (369826027*((1 - 2* x)^(1/2) - 1)^17)/(8000000*(3^(1/2) - (5*x + 3)^(1/2))^17) - (22297589*((1 - 2*x)^(1/2) - 1)^19)/(3200000*(3^(1/2) - (5*x + 3)^(1/2))^19) - (8192*3^ (1/2)*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (9 0112*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(9765625*(3^(1/2) - (5*x + 3)^(1/2)) ^4) + (8316928*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(9765625*(3^(1/2) - (5*x + 3)^(1/2))^6) - (216457216*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(9765625*(3^(1 /2) - (5*x + 3)^(1/2))^8) + (58587136*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(1 953125*(3^(1/2) - (5*x + 3)^(1/2))^10) - (54114304*3^(1/2)*((1 - 2*x)^(1/2 ) - 1)^12)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^12) + (519808*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^14) + (1408*3^(1/2 )*((1 - 2*x)^(1/2) - 1)^16)/(625*(3^(1/2) - (5*x + 3)^(1/2))^16) - (32*...