Integrand size = 24, antiderivative size = 120 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=-\frac {39185 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {39185 \sqrt {1-2 x}}{57624 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}-\frac {39185 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}} \]
-39185/605052*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+11/7*(3+5*x)^2/ (2+3*x)^4/(1-2*x)^(1/2)-39185/24696*(1-2*x)^(1/2)/(2+3*x)^2-39185/57624*(1 -2*x)^(1/2)/(2+3*x)+1/1764*(2395+3789*x)*(1-2*x)^(1/2)/(2+3*x)^4
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {\frac {21 \left (213998+1534434 x+4093057 x^2+4819755 x^3+2115990 x^4\right )}{2 \sqrt {1-2 x} (2+3 x)^4}-39185 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{605052} \]
((21*(213998 + 1534434*x + 4093057*x^2 + 4819755*x^3 + 2115990*x^4))/(2*Sq rt[1 - 2*x]*(2 + 3*x)^4) - 39185*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] )/605052
Time = 0.21 (sec) , antiderivative size = 135, 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, 25, 162, 52, 52, 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.
| \(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^{3/2} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^4}-\frac {1}{7} \int -\frac {(5 x+3) (325 x+173)}{\sqrt {1-2 x} (3 x+2)^5}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \int \frac {(5 x+3) (325 x+173)}{\sqrt {1-2 x} (3 x+2)^5}dx+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {1}{7} \left (\frac {39185}{252} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x} (3789 x+2395)}{252 (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{7} \left (\frac {39185}{252} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (3789 x+2395)}{252 (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{7} \left (\frac {39185}{252} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (3789 x+2395)}{252 (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{7} \left (\frac {39185}{252} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (3789 x+2395)}{252 (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{7} \left (\frac {39185}{252} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (3789 x+2395)}{252 (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)^4) + ((Sqrt[1 - 2*x]*(2395 + 3 789*x))/(252*(2 + 3*x)^4) + (39185*(-1/14*Sqrt[1 - 2*x]/(2 + 3*x)^2 + (3*( -1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqr t[21])))/14))/252)/7
3.22.3.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
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 = 3.42 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.47
| method | result | size |
| risch | \(\frac {2115990 x^{4}+4819755 x^{3}+4093057 x^{2}+1534434 x +213998}{57624 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {39185 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{605052}\) | \(56\) |
| pseudoelliptic | \(-\frac {352665 \left (\sqrt {21}\, \left (\frac {2}{3}+x \right )^{4} \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )-7 x^{4}-\frac {287 x^{3}}{18}-\frac {28651399 x^{2}}{2115990}-\frac {1790173 x}{352665}-\frac {748993}{1057995}\right )}{67228 \sqrt {1-2 x}\, \left (2+3 x \right )^{4}}\) | \(66\) |
| derivativedivides | \(\frac {\frac {743679 \left (1-2 x \right )^{\frac {7}{2}}}{67228}-\frac {717277 \left (1-2 x \right )^{\frac {5}{2}}}{9604}+\frac {691955 \left (1-2 x \right )^{\frac {3}{2}}}{4116}-\frac {74185 \sqrt {1-2 x}}{588}}{\left (-4-6 x \right )^{4}}-\frac {39185 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{605052}+\frac {5324}{16807 \sqrt {1-2 x}}\) | \(75\) |
| default | \(\frac {\frac {743679 \left (1-2 x \right )^{\frac {7}{2}}}{67228}-\frac {717277 \left (1-2 x \right )^{\frac {5}{2}}}{9604}+\frac {691955 \left (1-2 x \right )^{\frac {3}{2}}}{4116}-\frac {74185 \sqrt {1-2 x}}{588}}{\left (-4-6 x \right )^{4}}-\frac {39185 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{605052}+\frac {5324}{16807 \sqrt {1-2 x}}\) | \(75\) |
| trager | \(-\frac {\left (2115990 x^{4}+4819755 x^{3}+4093057 x^{2}+1534434 x +213998\right ) \sqrt {1-2 x}}{57624 \left (2+3 x \right )^{4} \left (-1+2 x \right )}-\frac {39185 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1210104}\) | \(89\) |
1/57624*(2115990*x^4+4819755*x^3+4093057*x^2+1534434*x+213998)/(2+3*x)^4/( 1-2*x)^(1/2)-39185/605052*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {39185 \, \sqrt {21} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2115990 \, x^{4} + 4819755 \, x^{3} + 4093057 \, x^{2} + 1534434 \, x + 213998\right )} \sqrt {-2 \, x + 1}}{1210104 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
1/1210104*(39185*sqrt(21)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 1 6)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(2115990*x^4 + 4819755*x^3 + 4093057*x^2 + 1534434*x + 213998)*sqrt(-2*x + 1))/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)
Timed out. \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.99 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {39185}{1210104} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1057995 \, {\left (2 \, x - 1\right )}^{4} + 9051735 \, {\left (2 \, x - 1\right )}^{3} + 28993349 \, {\left (2 \, x - 1\right )}^{2} + 82402418 \, x - 19287625}{28812 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2401 \, \sqrt {-2 \, x + 1}\right )}} \]
39185/1210104*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq rt(-2*x + 1))) + 1/28812*(1057995*(2*x - 1)^4 + 9051735*(2*x - 1)^3 + 2899 3349*(2*x - 1)^2 + 82402418*x - 19287625)/(81*(-2*x + 1)^(9/2) - 756*(-2*x + 1)^(7/2) + 2646*(-2*x + 1)^(5/2) - 4116*(-2*x + 1)^(3/2) + 2401*sqrt(-2 *x + 1))
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.91 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {39185}{1210104} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {5324}{16807 \, \sqrt {-2 \, x + 1}} - \frac {2231037 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 15062817 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 33905795 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 25445455 \, \sqrt {-2 \, x + 1}}{3226944 \, {\left (3 \, x + 2\right )}^{4}} \]
39185/1210104*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) + 5324/16807/sqrt(-2*x + 1) - 1/3226944*(2231037*( 2*x - 1)^3*sqrt(-2*x + 1) + 15062817*(2*x - 1)^2*sqrt(-2*x + 1) - 33905795 *(-2*x + 1)^(3/2) + 25445455*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.82 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {\frac {840841\,x}{23814}+\frac {591701\,{\left (2\,x-1\right )}^2}{47628}+\frac {431035\,{\left (2\,x-1\right )}^3}{111132}+\frac {39185\,{\left (2\,x-1\right )}^4}{86436}-\frac {393625}{47628}}{\frac {2401\,\sqrt {1-2\,x}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{3/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{5/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{7/2}}{3}+{\left (1-2\,x\right )}^{9/2}}-\frac {39185\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{605052} \]
((840841*x)/23814 + (591701*(2*x - 1)^2)/47628 + (431035*(2*x - 1)^3)/1111 32 + (39185*(2*x - 1)^4)/86436 - 393625/47628)/((2401*(1 - 2*x)^(1/2))/81 - (1372*(1 - 2*x)^(3/2))/27 + (98*(1 - 2*x)^(5/2))/3 - (28*(1 - 2*x)^(7/2) )/3 + (1 - 2*x)^(9/2)) - (39185*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/ 7))/605052