Integrand size = 24, antiderivative size = 80 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {\sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)^3}+\frac {5 \sqrt {1-2 x} (1205+1867 x)}{9261 (2+3 x)^2}-\frac {78710 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \]
-78710/194481*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/63*(3+5*x)^2* (1-2*x)^(1/2)/(2+3*x)^3+5/9261*(1205+1867*x)*(1-2*x)^(1/2)/(2+3*x)^2
Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (13373+41155 x+31680 x^2\right )}{(2+3 x)^3}-78710 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{194481} \]
((21*Sqrt[1 - 2*x]*(13373 + 41155*x + 31680*x^2))/(2 + 3*x)^3 - 78710*Sqrt [21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/194481
Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {109, 27, 162, 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.
\(\displaystyle \int \frac {(5 x+3)^3}{\sqrt {1-2 x} (3 x+2)^4} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}-\frac {1}{63} \int -\frac {10 (5 x+3) (52 x+29)}{\sqrt {1-2 x} (3 x+2)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {10}{63} \int \frac {(5 x+3) (52 x+29)}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\) |
\(\Big \downarrow \) 162 |
\(\displaystyle \frac {10}{63} \left (\frac {7871}{294} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {\sqrt {1-2 x} (1867 x+1205)}{294 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {10}{63} \left (\frac {\sqrt {1-2 x} (1867 x+1205)}{294 (3 x+2)^2}-\frac {7871}{294} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {10}{63} \left (\frac {\sqrt {1-2 x} (1867 x+1205)}{294 (3 x+2)^2}-\frac {7871 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\) |
(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)^3) + (10*((Sqrt[1 - 2*x]*(1205 + 1867*x))/(294*(2 + 3*x)^2) - (7871*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147 *Sqrt[21])))/63
3.21.32.3.1 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[((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 = 1.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {63360 x^{3}+50630 x^{2}-14409 x -13373}{9261 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {78710 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(51\) |
pseudoelliptic | \(\frac {-78710 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}+21 \sqrt {1-2 x}\, \left (31680 x^{2}+41155 x +13373\right )}{194481 \left (2+3 x \right )^{3}}\) | \(55\) |
derivativedivides | \(\frac {-\frac {7040 \left (1-2 x \right )^{\frac {5}{2}}}{1029}+\frac {41620 \left (1-2 x \right )^{\frac {3}{2}}}{1323}-\frac {6836 \sqrt {1-2 x}}{189}}{\left (-4-6 x \right )^{3}}-\frac {78710 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(57\) |
default | \(\frac {-\frac {7040 \left (1-2 x \right )^{\frac {5}{2}}}{1029}+\frac {41620 \left (1-2 x \right )^{\frac {3}{2}}}{1323}-\frac {6836 \sqrt {1-2 x}}{189}}{\left (-4-6 x \right )^{3}}-\frac {78710 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(57\) |
trager | \(\frac {\left (31680 x^{2}+41155 x +13373\right ) \sqrt {1-2 x}}{9261 \left (2+3 x \right )^{3}}+\frac {39355 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{194481}\) | \(72\) |
-1/9261*(63360*x^3+50630*x^2-14409*x-13373)/(2+3*x)^3/(1-2*x)^(1/2)-78710/ 194481*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.05 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {39355 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (31680 \, x^{2} + 41155 \, x + 13373\right )} \sqrt {-2 \, x + 1}}{194481 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
1/194481*(39355*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)* sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(31680*x^2 + 41155*x + 13373)*sqrt(-2* x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
Timed out. \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.15 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {39355}{194481} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (15840 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 72835 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 83741 \, \sqrt {-2 \, x + 1}\right )}}{9261 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
39355/194481*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr t(-2*x + 1))) + 4/9261*(15840*(-2*x + 1)^(5/2) - 72835*(-2*x + 1)^(3/2) + 83741*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.05 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {39355}{194481} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {15840 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 72835 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 83741 \, \sqrt {-2 \, x + 1}}{18522 \, {\left (3 \, x + 2\right )}^{3}} \]
39355/194481*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) + 1/18522*(15840*(2*x - 1)^2*sqrt(-2*x + 1) - 72835 *(-2*x + 1)^(3/2) + 83741*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {\frac {6836\,\sqrt {1-2\,x}}{5103}-\frac {41620\,{\left (1-2\,x\right )}^{3/2}}{35721}+\frac {7040\,{\left (1-2\,x\right )}^{5/2}}{27783}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}-\frac {78710\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{194481} \]