Integrand size = 24, antiderivative size = 127 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=-\frac {3 (544568-333311 x)}{732050 \sqrt {1-2 x}}-\frac {73 (2+3 x)^3}{3630 \sqrt {1-2 x} (3+5 x)^2}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac {317 (2+3 x)^2}{19965 \sqrt {1-2 x} (3+5 x)}-\frac {4693 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{366025 \sqrt {55}} \]
7/33*(2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^2-4693/20131375*arctanh(1/11*55^(1/2) *(1-2*x)^(1/2))*55^(1/2)-3/732050*(544568-333311*x)/(1-2*x)^(1/2)-73/3630* (2+3*x)^3/(3+5*x)^2/(1-2*x)^(1/2)-317/19965*(2+3*x)^2/(3+5*x)/(1-2*x)^(1/2 )
Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.54 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {-\frac {55 \left (37428168-10907307 x-309826828 x^2-248761830 x^3+106732890 x^4\right )}{(1-2 x)^{3/2} (3+5 x)^2}-28158 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{120788250} \]
((-55*(37428168 - 10907307*x - 309826828*x^2 - 248761830*x^3 + 106732890*x ^4))/((1 - 2*x)^(3/2)*(3 + 5*x)^2) - 28158*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqr t[1 - 2*x]])/120788250
Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {109, 166, 166, 27, 163, 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.
| \(\displaystyle \int \frac {(3 x+2)^5}{(1-2 x)^{5/2} (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac {1}{33} \int \frac {(3 x+2)^3 (201 x+106)}{(1-2 x)^{3/2} (5 x+3)^3}dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{33} \left (-\frac {1}{110} \int \frac {(3 x+2)^2 (13485 x+7457)}{(1-2 x)^{3/2} (5 x+3)^2}dx-\frac {73 (3 x+2)^3}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{110} \left (-\frac {1}{55} \int \frac {15 (3 x+2) (30301 x+17242)}{(1-2 x)^{3/2} (5 x+3)}dx-\frac {634 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^3}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{110} \left (-\frac {3}{11} \int \frac {(3 x+2) (30301 x+17242)}{(1-2 x)^{3/2} (5 x+3)}dx-\frac {634 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^3}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 163 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{110} \left (-\frac {3}{11} \left (\frac {3 (544568-333311 x)}{55 \sqrt {1-2 x}}-\frac {4693}{55} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {634 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^3}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{110} \left (-\frac {3}{11} \left (\frac {4693}{55} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {3 (544568-333311 x)}{55 \sqrt {1-2 x}}\right )-\frac {634 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^3}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{110} \left (-\frac {3}{11} \left (\frac {9386 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}}+\frac {3 (544568-333311 x)}{55 \sqrt {1-2 x}}\right )-\frac {634 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^3}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
(7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + ((-73*(2 + 3*x)^3)/(110 *Sqrt[1 - 2*x]*(3 + 5*x)^2) + ((-634*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*(3 + 5 *x)) - (3*((3*(544568 - 333311*x))/(55*Sqrt[1 - 2*x]) + (9386*ArcTanh[Sqrt [5/11]*Sqrt[1 - 2*x]])/(55*Sqrt[55])))/11)/110)/33
3.22.91.3.1 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] :> 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^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* (m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), 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*(b*c - a*d)*(m + 1)*(m + n + 3)) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && 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_)^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 = 3.56 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.50
| method | result | size |
| risch | \(\frac {106732890 x^{4}-248761830 x^{3}-309826828 x^{2}-10907307 x +37428168}{2196150 \sqrt {1-2 x}\, \left (3+5 x \right )^{2} \left (-1+2 x \right )}-\frac {4693 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{20131375}\) | \(63\) |
| pseudoelliptic | \(\frac {\frac {4693 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (-1+2 x \right ) \left (3+5 x \right )^{2} \sqrt {55}}{20131375}-\frac {243 x^{4}}{5}+\frac {8292061 x^{3}}{73205}+\frac {154913414 x^{2}}{1098075}+\frac {3635769 x}{732050}-\frac {6238028}{366025}}{\left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{2}}\) | \(74\) |
| derivativedivides | \(-\frac {243 \sqrt {1-2 x}}{500}+\frac {\frac {31 \left (1-2 x \right )^{\frac {3}{2}}}{33275}-\frac {343 \sqrt {1-2 x}}{166375}}{\left (-6-10 x \right )^{2}}-\frac {4693 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{20131375}+\frac {16807}{15972 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {36015}{14641 \sqrt {1-2 x}}\) | \(75\) |
| default | \(-\frac {243 \sqrt {1-2 x}}{500}+\frac {\frac {31 \left (1-2 x \right )^{\frac {3}{2}}}{33275}-\frac {343 \sqrt {1-2 x}}{166375}}{\left (-6-10 x \right )^{2}}-\frac {4693 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{20131375}+\frac {16807}{15972 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {36015}{14641 \sqrt {1-2 x}}\) | \(75\) |
| trager | \(-\frac {\left (106732890 x^{4}-248761830 x^{3}-309826828 x^{2}-10907307 x +37428168\right ) \sqrt {1-2 x}}{2196150 \left (10 x^{2}+x -3\right )^{2}}+\frac {4693 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{40262750}\) | \(85\) |
1/2196150*(106732890*x^4-248761830*x^3-309826828*x^2-10907307*x+37428168)/ (1-2*x)^(1/2)/(3+5*x)^2/(-1+2*x)-4693/20131375*arctanh(1/11*55^(1/2)*(1-2* x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.82 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {14079 \, \sqrt {55} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (106732890 \, x^{4} - 248761830 \, x^{3} - 309826828 \, x^{2} - 10907307 \, x + 37428168\right )} \sqrt {-2 \, x + 1}}{120788250 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
1/120788250*(14079*sqrt(55)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(106732890*x^4 - 248761830 *x^3 - 309826828*x^2 - 10907307*x + 37428168)*sqrt(-2*x + 1))/(100*x^4 + 2 0*x^3 - 59*x^2 - 6*x + 9)
Timed out. \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {4693}{40262750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {243}{500} \, \sqrt {-2 \, x + 1} + \frac {1350542040 \, {\left (2 \, x - 1\right )}^{3} + 6520170349 \, {\left (2 \, x - 1\right )}^{2} + 18157562500 \, x - 6282516625}{21961500 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
4693/40262750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sq rt(-2*x + 1))) - 243/500*sqrt(-2*x + 1) + 1/21961500*(1350542040*(2*x - 1) ^3 + 6520170349*(2*x - 1)^2 + 18157562500*x - 6282516625)/(25*(-2*x + 1)^( 7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.77 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {4693}{40262750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {243}{500} \, \sqrt {-2 \, x + 1} - \frac {2401 \, {\left (360 \, x - 103\right )}}{175692 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {155 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}}{665500 \, {\left (5 \, x + 3\right )}^{2}} \]
4693/40262750*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt( 55) + 5*sqrt(-2*x + 1))) - 243/500*sqrt(-2*x + 1) - 2401/175692*(360*x - 1 03)/((2*x - 1)*sqrt(-2*x + 1)) + 1/665500*(155*(-2*x + 1)^(3/2) - 343*sqrt (-2*x + 1))/(5*x + 3)^2
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.63 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {\frac {12005\,x}{363}+\frac {592742759\,{\left (2\,x-1\right )}^2}{49912500}+\frac {22509034\,{\left (2\,x-1\right )}^3}{9150625}-\frac {415373}{36300}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}}-\frac {243\,\sqrt {1-2\,x}}{500}-\frac {4693\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{20131375} \]