Integrand size = 26, antiderivative size = 116 \[ \int \frac {(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=-\frac {40787}{704} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {40787 \sqrt {1-2 x} (3+5 x)^{3/2}}{5808}+\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64 \sqrt {10}} \]
49/66*(3+5*x)^(5/2)/(1-2*x)^(3/2)+40787/640*arcsin(1/11*22^(1/2)*(3+5*x)^( 1/2))*10^(1/2)-938/363*(3+5*x)^(5/2)/(1-2*x)^(1/2)-40787/5808*(3+5*x)^(3/2 )*(1-2*x)^(1/2)-40787/704*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\frac {-10 \sqrt {3+5 x} \left (18351-52256 x+12780 x^2+2160 x^3\right )+122361 \sqrt {10-20 x} (-1+2 x) \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1920 (1-2 x)^{3/2}} \]
(-10*Sqrt[3 + 5*x]*(18351 - 52256*x + 12780*x^2 + 2160*x^3) + 122361*Sqrt[ 10 - 20*x]*(-1 + 2*x)*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(1920*(1 - 2* x)^(3/2))
Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {100, 27, 87, 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.
\(\displaystyle \int \frac {(3 x+2)^2 (5 x+3)^{3/2}}{(1-2 x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {1}{66} \int \frac {(5 x+3)^{3/2} (594 x+1579)}{2 (1-2 x)^{3/2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {1}{132} \int \frac {(5 x+3)^{3/2} (594 x+1579)}{(1-2 x)^{3/2}}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{132} \left (\frac {40787}{11} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x}}dx-\frac {3752 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{132} \left (\frac {40787}{11} \left (\frac {33}{8} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3752 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{132} \left (\frac {40787}{11} \left (\frac {33}{8} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3752 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {1}{132} \left (\frac {40787}{11} \left (\frac {33}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3752 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{132} \left (\frac {40787}{11} \left (\frac {33}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {3752 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}\) |
(49*(3 + 5*x)^(5/2))/(66*(1 - 2*x)^(3/2)) + ((-3752*(3 + 5*x)^(5/2))/(11*S qrt[1 - 2*x]) + (40787*(-1/4*(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (33*(-1/2*( Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2*Sq rt[10])))/8))/11)/132
3.26.91.3.1 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[((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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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 = 4.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {\left (489444 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-43200 x^{3} \sqrt {-10 x^{2}-x +3}-489444 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -255600 x^{2} \sqrt {-10 x^{2}-x +3}+122361 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1045120 x \sqrt {-10 x^{2}-x +3}-367020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{3840 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(137\) |
1/3840*(489444*10^(1/2)*arcsin(20/11*x+1/11)*x^2-43200*x^3*(-10*x^2-x+3)^( 1/2)-489444*10^(1/2)*arcsin(20/11*x+1/11)*x-255600*x^2*(-10*x^2-x+3)^(1/2) +122361*10^(1/2)*arcsin(20/11*x+1/11)+1045120*x*(-10*x^2-x+3)^(1/2)-367020 *(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3) ^(1/2)
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=-\frac {122361 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (2160 \, x^{3} + 12780 \, x^{2} - 52256 \, x + 18351\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3840 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/3840*(122361*sqrt(10)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1) *sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(2160*x^3 + 12780*x^2 - 52256*x + 18351)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)
\[ \int \frac {(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.33 \[ \int \frac {(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\frac {40787}{1280} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {297}{64} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {49 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{24 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {21 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{16 \, {\left (2 \, x - 1\right )}} + \frac {539 \, \sqrt {-10 \, x^{2} - x + 3}}{48 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {5873 \, \sqrt {-10 \, x^{2} - x + 3}}{48 \, {\left (2 \, x - 1\right )}} \]
40787/1280*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 297/64*sqrt(-10*x^2 - x + 3) - 49/24*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 21/4*( -10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 9/16*(-10*x^2 - x + 3)^(3/2)/(2 *x - 1) + 539/48*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 5873/48*sqrt(-1 0*x^2 - x + 3)/(2*x - 1)
Time = 0.33 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\frac {40787}{640} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 247 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 81574 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1345971 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{24000 \, {\left (2 \, x - 1\right )}^{2}} \]
40787/640*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/24000*(4*(9*(12 *sqrt(5)*(5*x + 3) + 247*sqrt(5))*(5*x + 3) - 81574*sqrt(5))*(5*x + 3) + 1 345971*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2
Timed out. \[ \int \frac {(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]