Integrand size = 24, antiderivative size = 161 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx=-\frac {209 \sqrt {1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11 \sqrt {1-2 x} (3911+6475 x)}{15876 (2+3 x)}-\frac {146971 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7938 \sqrt {21}} \]
-1/15*(1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^5+11/18*(1-2*x)^(3/2)*(3+5*x)^3/(2+3 *x)^4-146971/166698*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-209/756*( 3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^2+11/9*(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^3-11 /15876*(3911+6475*x)*(1-2*x)^(1/2)/(2+3*x)
Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (7933096+56745266 x+157178184 x^2+207486855 x^3+126578745 x^4+26460000 x^5\right )}{2 (2+3 x)^5}-734855 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{833490} \]
((21*Sqrt[1 - 2*x]*(7933096 + 56745266*x + 157178184*x^2 + 207486855*x^3 + 126578745*x^4 + 26460000*x^5))/(2*(2 + 3*x)^5) - 734855*Sqrt[21]*ArcTanh[ Sqrt[3/7]*Sqrt[1 - 2*x]])/833490
Time = 0.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {108, 27, 166, 27, 166, 27, 166, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^3}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{15} \int -\frac {55 (1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^5}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {11}{3} \int \frac {(1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^5}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {11}{3} \left (-\frac {1}{12} \int \frac {6 \sqrt {1-2 x} (3 x+4) (5 x+3)^2}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {11}{3} \left (-\frac {1}{2} \int \frac {\sqrt {1-2 x} (3 x+4) (5 x+3)^2}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {11}{3} \left (\frac {1}{2} \left (\frac {1}{9} \int -\frac {3 (4 x+9) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {2 \sqrt {1-2 x} (5 x+3)^3}{3 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {11}{3} \left (\frac {1}{2} \left (-\frac {1}{3} \int \frac {(4 x+9) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {2 \sqrt {1-2 x} (5 x+3)^3}{3 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {11}{3} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {19 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}-\frac {1}{42} \int \frac {(5 x+3) (185 x+529)}{\sqrt {1-2 x} (3 x+2)^2}dx\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^3}{3 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 163 |
| \(\displaystyle -\frac {11}{3} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{42} \left (\frac {\sqrt {1-2 x} (6475 x+3911)}{21 (3 x+2)}-\frac {13361}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {19 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^3}{3 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {11}{3} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{42} \left (\frac {13361}{21} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {\sqrt {1-2 x} (6475 x+3911)}{21 (3 x+2)}\right )+\frac {19 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^3}{3 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {11}{3} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{42} \left (\frac {26722 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}+\frac {\sqrt {1-2 x} (6475 x+3911)}{21 (3 x+2)}\right )+\frac {19 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^3}{3 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
-1/15*((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^5 - (11*(-1/6*((1 - 2*x)^(3/ 2)*(3 + 5*x)^3)/(2 + 3*x)^4 + ((-2*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(3*(2 + 3*x) ^3) + ((19*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(42*(2 + 3*x)^2) + ((Sqrt[1 - 2*x]*( 3911 + 6475*x))/(21*(2 + 3*x)) + (26722*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/ (21*Sqrt[21]))/42)/3)/2))/3
3.20.65.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 = 1.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.41
| method | result | size |
| risch | \(-\frac {52920000 x^{6}+226697490 x^{5}+288394965 x^{4}+106869513 x^{3}-43687652 x^{2}-40879074 x -7933096}{79380 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {146971 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{166698}\) | \(66\) |
| pseudoelliptic | \(\frac {-1469710 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}+21 \sqrt {1-2 x}\, \left (26460000 x^{5}+126578745 x^{4}+207486855 x^{3}+157178184 x^{2}+56745266 x +7933096\right )}{1666980 \left (2+3 x \right )^{5}}\) | \(70\) |
| derivativedivides | \(\frac {1000 \sqrt {1-2 x}}{729}+\frac {-\frac {284287 \left (1-2 x \right )^{\frac {9}{2}}}{294}+\frac {226727 \left (1-2 x \right )^{\frac {7}{2}}}{27}-\frac {11068432 \left (1-2 x \right )^{\frac {5}{2}}}{405}+\frac {9599737 \left (1-2 x \right )^{\frac {3}{2}}}{243}-\frac {31200211 \sqrt {1-2 x}}{1458}}{\left (-4-6 x \right )^{5}}-\frac {146971 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{166698}\) | \(84\) |
| default | \(\frac {1000 \sqrt {1-2 x}}{729}+\frac {-\frac {284287 \left (1-2 x \right )^{\frac {9}{2}}}{294}+\frac {226727 \left (1-2 x \right )^{\frac {7}{2}}}{27}-\frac {11068432 \left (1-2 x \right )^{\frac {5}{2}}}{405}+\frac {9599737 \left (1-2 x \right )^{\frac {3}{2}}}{243}-\frac {31200211 \sqrt {1-2 x}}{1458}}{\left (-4-6 x \right )^{5}}-\frac {146971 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{166698}\) | \(84\) |
| trager | \(\frac {\left (26460000 x^{5}+126578745 x^{4}+207486855 x^{3}+157178184 x^{2}+56745266 x +7933096\right ) \sqrt {1-2 x}}{79380 \left (2+3 x \right )^{5}}+\frac {146971 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{333396}\) | \(87\) |
-1/79380*(52920000*x^6+226697490*x^5+288394965*x^4+106869513*x^3-43687652* x^2-40879074*x-7933096)/(2+3*x)^5/(1-2*x)^(1/2)-146971/166698*arctanh(1/7* 21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {734855 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (26460000 \, x^{5} + 126578745 \, x^{4} + 207486855 \, x^{3} + 157178184 \, x^{2} + 56745266 \, x + 7933096\right )} \sqrt {-2 \, x + 1}}{1666980 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/1666980*(734855*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(26460000*x ^5 + 126578745*x^4 + 207486855*x^3 + 157178184*x^2 + 56745266*x + 7933096) *sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {146971}{333396} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1000}{729} \, \sqrt {-2 \, x + 1} + \frac {345408705 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 2999598210 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 9762357024 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 14111613390 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 7644051695 \, \sqrt {-2 \, x + 1}}{357210 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
146971/333396*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq rt(-2*x + 1))) + 1000/729*sqrt(-2*x + 1) + 1/357210*(345408705*(-2*x + 1)^ (9/2) - 2999598210*(-2*x + 1)^(7/2) + 9762357024*(-2*x + 1)^(5/2) - 141116 13390*(-2*x + 1)^(3/2) + 7644051695*sqrt(-2*x + 1))/(243*(2*x - 1)^5 + 283 5*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {146971}{333396} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1000}{729} \, \sqrt {-2 \, x + 1} + \frac {345408705 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 2999598210 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 9762357024 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 14111613390 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 7644051695 \, \sqrt {-2 \, x + 1}}{11430720 \, {\left (3 \, x + 2\right )}^{5}} \]
146971/333396*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) + 1000/729*sqrt(-2*x + 1) + 1/11430720*(345408705* (2*x - 1)^4*sqrt(-2*x + 1) + 2999598210*(2*x - 1)^3*sqrt(-2*x + 1) + 97623 57024*(2*x - 1)^2*sqrt(-2*x + 1) - 14111613390*(-2*x + 1)^(3/2) + 76440516 95*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 1.38 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {1000\,\sqrt {1-2\,x}}{729}-\frac {146971\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{166698}+\frac {\frac {31200211\,\sqrt {1-2\,x}}{354294}-\frac {9599737\,{\left (1-2\,x\right )}^{3/2}}{59049}+\frac {11068432\,{\left (1-2\,x\right )}^{5/2}}{98415}-\frac {226727\,{\left (1-2\,x\right )}^{7/2}}{6561}+\frac {284287\,{\left (1-2\,x\right )}^{9/2}}{71442}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
(1000*(1 - 2*x)^(1/2))/729 - (146971*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1 /2))/7))/166698 + ((31200211*(1 - 2*x)^(1/2))/354294 - (9599737*(1 - 2*x)^ (3/2))/59049 + (11068432*(1 - 2*x)^(5/2))/98415 - (226727*(1 - 2*x)^(7/2)) /6561 + (284287*(1 - 2*x)^(9/2))/71442)/((24010*x)/81 + (3430*(2*x - 1)^2) /27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243)