Integrand size = 26, antiderivative size = 142 \[ \int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac {734 \sqrt {1-2 x} (2+3 x)^3}{9075 \sqrt {3+5 x}}+\frac {511 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{30250}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (938509+366420 x)}{4840000}+\frac {462357 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40000 \sqrt {10}} \]
462357/400000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-2/165*(2+3*x)^4 *(1-2*x)^(1/2)/(3+5*x)^(3/2)-734/9075*(2+3*x)^3*(1-2*x)^(1/2)/(3+5*x)^(1/2 )+511/30250*(2+3*x)^2*(1-2*x)^(1/2)*(3+5*x)^(1/2)-7/4840000*(938509+366420 *x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {\sqrt {1-2 x} \left (199549721+795297410 x+1030526145 x^2+502791300 x^3+117612000 x^4\right )}{14520000 (3+5 x)^{3/2}}-\frac {462357 \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{40000 \sqrt {10}} \]
-1/14520000*(Sqrt[1 - 2*x]*(199549721 + 795297410*x + 1030526145*x^2 + 502 791300*x^3 + 117612000*x^4))/(3 + 5*x)^(3/2) - (462357*ArcTan[Sqrt[5/2 - 5 *x]/Sqrt[3 + 5*x]])/(40000*Sqrt[10])
Time = 0.23 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {109, 27, 167, 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.
| \(\displaystyle \int \frac {(3 x+2)^5}{\sqrt {1-2 x} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {2}{165} \int -\frac {(3 x+2)^3 (261 x+230)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{165} \int \frac {(3 x+2)^3 (261 x+230)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{165} \left (\frac {2}{55} \int \frac {63 (196-73 x) (3 x+2)^2}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {734 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{165} \left (\frac {63}{55} \int \frac {(196-73 x) (3 x+2)^2}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {734 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{165} \left (\frac {63}{55} \left (\frac {73}{30} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}-\frac {1}{30} \int -\frac {(3 x+2) (30535 x+21038)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {734 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{165} \left (\frac {63}{55} \left (\frac {1}{60} \int \frac {(3 x+2) (30535 x+21038)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {73}{30} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {734 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{165} \left (\frac {63}{55} \left (\frac {1}{60} \left (\frac {7992171}{160} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (366420 x+938509)\right )+\frac {73}{30} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {734 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{165} \left (\frac {63}{55} \left (\frac {1}{60} \left (\frac {7992171}{400} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (366420 x+938509)\right )+\frac {73}{30} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {734 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{165} \left (\frac {63}{55} \left (\frac {1}{60} \left (\frac {7992171 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80 \sqrt {10}}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (366420 x+938509)\right )+\frac {73}{30} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {734 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}\) |
(-2*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(165*(3 + 5*x)^(3/2)) + ((-734*Sqrt[1 - 2*x ]*(2 + 3*x)^3)/(55*Sqrt[3 + 5*x]) + (63*((73*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqr t[3 + 5*x])/30 + (-1/80*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(938509 + 366420*x)) + (7992171*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80*Sqrt[10]))/60))/55)/165
3.26.7.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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 = 1.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.04
| method | result | size |
| default | \(\frac {\left (-2352240000 x^{4} \sqrt {-10 x^{2}-x +3}+4195889775 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-10055826000 x^{3} \sqrt {-10 x^{2}-x +3}+5035067730 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -20610522900 x^{2} \sqrt {-10 x^{2}-x +3}+1510520319 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-15905948200 x \sqrt {-10 x^{2}-x +3}-3990994420 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{290400000 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(147\) |
1/290400000*(-2352240000*x^4*(-10*x^2-x+3)^(1/2)+4195889775*10^(1/2)*arcsi n(20/11*x+1/11)*x^2-10055826000*x^3*(-10*x^2-x+3)^(1/2)+5035067730*10^(1/2 )*arcsin(20/11*x+1/11)*x-20610522900*x^2*(-10*x^2-x+3)^(1/2)+1510520319*10 ^(1/2)*arcsin(20/11*x+1/11)-15905948200*x*(-10*x^2-x+3)^(1/2)-3990994420*( -10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {167835591 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (117612000 \, x^{4} + 502791300 \, x^{3} + 1030526145 \, x^{2} + 795297410 \, x + 199549721\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{290400000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
-1/290400000*(167835591*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)* (20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(117612000* x^4 + 502791300*x^3 + 1030526145*x^2 + 795297410*x + 199549721)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)
\[ \int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{5}}{\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {81}{250} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {462357}{800000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {9963}{10000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {305343}{200000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{103125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {998 \, \sqrt {-10 \, x^{2} - x + 3}}{1134375 \, {\left (5 \, x + 3\right )}} \]
-81/250*sqrt(-10*x^2 - x + 3)*x^2 + 462357/800000*sqrt(5)*sqrt(2)*arcsin(2 0/11*x + 1/11) - 9963/10000*sqrt(-10*x^2 - x + 3)*x - 305343/200000*sqrt(- 10*x^2 - x + 3) - 2/103125*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 998 /1134375*sqrt(-10*x^2 - x + 3)/(5*x + 3)
Time = 0.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.30 \[ \int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {27}{1000000} \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 75 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 7745 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{90750000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {462357}{400000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {333 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{7562500 \, \sqrt {5 \, x + 3}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {999 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{5671875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
-27/1000000*(12*(8*sqrt(5)*(5*x + 3) + 75*sqrt(5))*(5*x + 3) + 7745*sqrt(5 ))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1/90750000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 462357/400000*sqrt(10)*arcsin(1/11*s qrt(22)*sqrt(5*x + 3)) - 333/7562500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - s qrt(22))/sqrt(5*x + 3) + 1/5671875*sqrt(10)*(5*x + 3)^(3/2)*(999*(sqrt(2)* sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sq rt(22))^3
Timed out. \[ \int \frac {(2+3 x)^5}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^5}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]