Integrand size = 24, antiderivative size = 107 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (18016+26075 x)}{3969 (2+3 x)}-\frac {92996 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}} \]
-92996/83349*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-53/189*(3+5*x)^2 *(1-2*x)^(1/2)/(2+3*x)^2-1/9*(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^3+2/3969*(180 16+26075*x)*(1-2*x)^(1/2)/(2+3*x)
Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (112187+484618 x+695043 x^2+330750 x^3\right )}{(2+3 x)^3}-92996 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{83349} \]
((21*Sqrt[1 - 2*x]*(112187 + 484618*x + 695043*x^2 + 330750*x^3))/(2 + 3*x )^3 - 92996*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/83349
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {108, 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.
\(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^4} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {1}{9} \int \frac {(12-35 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\) |
\(\Big \downarrow \) 166 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{42} \int \frac {4 (136-745 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {2}{21} \int \frac {(136-745 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\) |
\(\Big \downarrow \) 163 |
\(\displaystyle \frac {1}{9} \left (\frac {2}{21} \left (\frac {23249}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {\sqrt {1-2 x} (26075 x+18016)}{21 (3 x+2)}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{9} \left (\frac {2}{21} \left (\frac {\sqrt {1-2 x} (26075 x+18016)}{21 (3 x+2)}-\frac {23249}{21} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{9} \left (\frac {2}{21} \left (\frac {\sqrt {1-2 x} (26075 x+18016)}{21 (3 x+2)}-\frac {46498 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\) |
-1/9*(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^3 + ((-53*Sqrt[1 - 2*x]*(3 + 5* x)^2)/(21*(2 + 3*x)^2) + (2*((Sqrt[1 - 2*x]*(18016 + 26075*x))/(21*(2 + 3* x)) - (46498*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21])))/21)/9
3.19.23.3.1 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 = 0.98 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {661500 x^{4}+1059336 x^{3}+274193 x^{2}-260244 x -112187}{3969 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {92996 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\) | \(56\) |
pseudoelliptic | \(\frac {-92996 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}+21 \sqrt {1-2 x}\, \left (330750 x^{3}+695043 x^{2}+484618 x +112187\right )}{83349 \left (2+3 x \right )^{3}}\) | \(60\) |
derivativedivides | \(\frac {250 \sqrt {1-2 x}}{81}+\frac {-\frac {7454 \left (1-2 x \right )^{\frac {5}{2}}}{441}+\frac {44092 \left (1-2 x \right )^{\frac {3}{2}}}{567}-\frac {7246 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{3}}-\frac {92996 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\) | \(66\) |
default | \(\frac {250 \sqrt {1-2 x}}{81}+\frac {-\frac {7454 \left (1-2 x \right )^{\frac {5}{2}}}{441}+\frac {44092 \left (1-2 x \right )^{\frac {3}{2}}}{567}-\frac {7246 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{3}}-\frac {92996 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\) | \(66\) |
trager | \(\frac {\left (330750 x^{3}+695043 x^{2}+484618 x +112187\right ) \sqrt {1-2 x}}{3969 \left (2+3 x \right )^{3}}+\frac {46498 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{83349}\) | \(77\) |
-1/3969*(661500*x^4+1059336*x^3+274193*x^2-260244*x-112187)/(2+3*x)^3/(1-2 *x)^(1/2)-92996/83349*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {46498 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (330750 \, x^{3} + 695043 \, x^{2} + 484618 \, x + 112187\right )} \sqrt {-2 \, x + 1}}{83349 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
1/83349*(46498*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*s qrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(330750*x^3 + 695043*x^2 + 484618*x + 1 12187)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\text {Timed out} \]
Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {46498}{83349} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {250}{81} \, \sqrt {-2 \, x + 1} + \frac {2 \, {\left (33543 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 154322 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 177527 \, \sqrt {-2 \, x + 1}\right )}}{3969 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
46498/83349*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt (-2*x + 1))) + 250/81*sqrt(-2*x + 1) + 2/3969*(33543*(-2*x + 1)^(5/2) - 15 4322*(-2*x + 1)^(3/2) + 177527*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {46498}{83349} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {250}{81} \, \sqrt {-2 \, x + 1} + \frac {33543 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 154322 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 177527 \, \sqrt {-2 \, x + 1}}{15876 \, {\left (3 \, x + 2\right )}^{3}} \]
46498/83349*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 250/81*sqrt(-2*x + 1) + 1/15876*(33543*(2*x - 1)^2 *sqrt(-2*x + 1) - 154322*(-2*x + 1)^(3/2) + 177527*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {250\,\sqrt {1-2\,x}}{81}-\frac {92996\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{83349}+\frac {\frac {7246\,\sqrt {1-2\,x}}{2187}-\frac {44092\,{\left (1-2\,x\right )}^{3/2}}{15309}+\frac {7454\,{\left (1-2\,x\right )}^{5/2}}{11907}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]