Integrand size = 24, antiderivative size = 140 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^3} \, dx=-\frac {56556 \sqrt {1-2 x} (2+3 x)^2}{378125}-\frac {927 \sqrt {1-2 x} (2+3 x)^3}{211750}-\frac {\sqrt {1-2 x} (2+3 x)^5}{110 (3+5 x)^2}-\frac {117 \sqrt {1-2 x} (2+3 x)^4}{3025 (3+5 x)}-\frac {9 \sqrt {1-2 x} (2815648+934875 x)}{3781250}-\frac {33069 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1890625 \sqrt {55}} \]
-33069/103984375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-56556/37812 5*(2+3*x)^2*(1-2*x)^(1/2)-927/211750*(2+3*x)^3*(1-2*x)^(1/2)-1/110*(2+3*x) ^5*(1-2*x)^(1/2)/(3+5*x)^2-117/3025*(2+3*x)^4*(1-2*x)^(1/2)/(3+5*x)-9/3781 250*(2815648+934875*x)*(1-2*x)^(1/2)
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {-\frac {55 \sqrt {1-2 x} \left (1804176536+7254126105 x+9876010320 x^2+6078090150 x^3+2690374500 x^4+551306250 x^5\right )}{(3+5 x)^2}-462966 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1455781250} \]
((-55*Sqrt[1 - 2*x]*(1804176536 + 7254126105*x + 9876010320*x^2 + 60780901 50*x^3 + 2690374500*x^4 + 551306250*x^5))/(3 + 5*x)^2 - 462966*Sqrt[55]*Ar cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1455781250
Time = 0.24 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {109, 27, 166, 170, 27, 170, 25, 164, 73, 219}
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.
| \(\displaystyle \int \frac {(3 x+2)^6}{\sqrt {1-2 x} (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {1}{110} \int -\frac {3 (3 x+2)^4 (59 x+51)}{\sqrt {1-2 x} (5 x+3)^2}dx-\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{110} \int \frac {(3 x+2)^4 (59 x+51)}{\sqrt {1-2 x} (5 x+3)^2}dx-\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{55} \int \frac {(3 x+2)^3 (309 x+2390)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {78 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{55} \left (-\frac {1}{35} \int -\frac {7 (3 x+2)^2 (37704 x+24827)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {309}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {78 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{55} \left (\frac {1}{5} \int \frac {(3 x+2)^2 (37704 x+24827)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {309}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {78 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{55} \left (\frac {1}{5} \left (-\frac {1}{25} \int -\frac {(3 x+2) (2804625 x+1693798)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {37704}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {309}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {78 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{25} \int \frac {(3 x+2) (2804625 x+1693798)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {37704}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {309}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {78 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{25} \left (\frac {11023}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-\frac {3}{5} \sqrt {1-2 x} (934875 x+2815648)\right )-\frac {37704}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {309}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {78 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{25} \left (-\frac {11023}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {3}{5} \sqrt {1-2 x} (934875 x+2815648)\right )-\frac {37704}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {309}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {78 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{25} \left (-\frac {22046 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}-\frac {3}{5} \sqrt {1-2 x} (934875 x+2815648)\right )-\frac {37704}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {309}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {78 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}\) |
-1/110*(Sqrt[1 - 2*x]*(2 + 3*x)^5)/(3 + 5*x)^2 + (3*((-78*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(55*(3 + 5*x)) + ((-309*Sqrt[1 - 2*x]*(2 + 3*x)^3)/35 + ((-3770 4*Sqrt[1 - 2*x]*(2 + 3*x)^2)/25 + ((-3*Sqrt[1 - 2*x]*(2815648 + 934875*x)) /5 - (22046*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55]))/25)/5)/55))/1 10
3.21.58.3.1 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] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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] && ILtQ[m, -1] && GtQ[n, 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Time = 1.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.47
| method | result | size |
| risch | \(\frac {1102612500 x^{6}+4829442750 x^{5}+9465805800 x^{4}+13673930490 x^{3}+4632241890 x^{2}-3645773033 x -1804176536}{26468750 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {33069 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{103984375}\) | \(66\) |
| pseudoelliptic | \(\frac {-462966 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}-55 \sqrt {1-2 x}\, \left (551306250 x^{5}+2690374500 x^{4}+6078090150 x^{3}+9876010320 x^{2}+7254126105 x +1804176536\right )}{1455781250 \left (3+5 x \right )^{2}}\) | \(70\) |
| derivativedivides | \(\frac {729 \left (1-2 x \right )^{\frac {7}{2}}}{7000}-\frac {26973 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {111213 \left (1-2 x \right )^{\frac {3}{2}}}{25000}-\frac {276183 \sqrt {1-2 x}}{25000}+\frac {\frac {399 \left (1-2 x \right )^{\frac {3}{2}}}{378125}-\frac {401 \sqrt {1-2 x}}{171875}}{\left (-6-10 x \right )^{2}}-\frac {33069 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{103984375}\) | \(84\) |
| default | \(\frac {729 \left (1-2 x \right )^{\frac {7}{2}}}{7000}-\frac {26973 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {111213 \left (1-2 x \right )^{\frac {3}{2}}}{25000}-\frac {276183 \sqrt {1-2 x}}{25000}+\frac {\frac {399 \left (1-2 x \right )^{\frac {3}{2}}}{378125}-\frac {401 \sqrt {1-2 x}}{171875}}{\left (-6-10 x \right )^{2}}-\frac {33069 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{103984375}\) | \(84\) |
| trager | \(-\frac {\left (551306250 x^{5}+2690374500 x^{4}+6078090150 x^{3}+9876010320 x^{2}+7254126105 x +1804176536\right ) \sqrt {1-2 x}}{26468750 \left (3+5 x \right )^{2}}-\frac {33069 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{207968750}\) | \(87\) |
1/26468750*(1102612500*x^6+4829442750*x^5+9465805800*x^4+13673930490*x^3+4 632241890*x^2-3645773033*x-1804176536)/(3+5*x)^2/(1-2*x)^(1/2)-33069/10398 4375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {231483 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (551306250 \, x^{5} + 2690374500 \, x^{4} + 6078090150 \, x^{3} + 9876010320 \, x^{2} + 7254126105 \, x + 1804176536\right )} \sqrt {-2 \, x + 1}}{1455781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
1/1455781250*(231483*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt (-2*x + 1) - 8)/(5*x + 3)) - 55*(551306250*x^5 + 2690374500*x^4 + 60780901 50*x^3 + 9876010320*x^2 + 7254126105*x + 1804176536)*sqrt(-2*x + 1))/(25*x ^2 + 30*x + 9)
Timed out. \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {729}{7000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {26973}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {111213}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {33069}{207968750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {276183}{25000} \, \sqrt {-2 \, x + 1} + \frac {1995 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 4411 \, \sqrt {-2 \, x + 1}}{1890625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
729/7000*(-2*x + 1)^(7/2) - 26973/25000*(-2*x + 1)^(5/2) + 111213/25000*(- 2*x + 1)^(3/2) + 33069/207968750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1 ))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 276183/25000*sqrt(-2*x + 1) + 1/189062 5*(1995*(-2*x + 1)^(3/2) - 4411*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)
Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^3} \, dx=-\frac {729}{7000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {26973}{25000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {111213}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {33069}{207968750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {276183}{25000} \, \sqrt {-2 \, x + 1} + \frac {1995 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 4411 \, \sqrt {-2 \, x + 1}}{7562500 \, {\left (5 \, x + 3\right )}^{2}} \]
-729/7000*(2*x - 1)^3*sqrt(-2*x + 1) - 26973/25000*(2*x - 1)^2*sqrt(-2*x + 1) + 111213/25000*(-2*x + 1)^(3/2) + 33069/207968750*sqrt(55)*log(1/2*abs (-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 276183/ 25000*sqrt(-2*x + 1) + 1/7562500*(1995*(-2*x + 1)^(3/2) - 4411*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {111213\,{\left (1-2\,x\right )}^{3/2}}{25000}-\frac {276183\,\sqrt {1-2\,x}}{25000}-\frac {26973\,{\left (1-2\,x\right )}^{5/2}}{25000}+\frac {729\,{\left (1-2\,x\right )}^{7/2}}{7000}-\frac {\frac {401\,\sqrt {1-2\,x}}{4296875}-\frac {399\,{\left (1-2\,x\right )}^{3/2}}{9453125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,33069{}\mathrm {i}}{103984375} \]
(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*33069i)/103984375 - (2761 83*(1 - 2*x)^(1/2))/25000 + (111213*(1 - 2*x)^(3/2))/25000 - (26973*(1 - 2 *x)^(5/2))/25000 + (729*(1 - 2*x)^(7/2))/7000 - ((401*(1 - 2*x)^(1/2))/429 6875 - (399*(1 - 2*x)^(3/2))/9453125)/((44*x)/5 + (2*x - 1)^2 + 11/25)