Integrand size = 24, antiderivative size = 100 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=-\frac {4660 \sqrt {1-2 x}}{3087 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (297+470 x)}{441 (2+3 x)^3}-\frac {9320 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3087 \sqrt {21}} \]
-9320/64827*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+11/7*(3+5*x)^2/(2 +3*x)^3/(1-2*x)^(1/2)-4660/3087*(1-2*x)^(1/2)/(2+3*x)+2/441*(297+470*x)*(1 -2*x)^(1/2)/(2+3*x)^3
Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.65 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {2 \left (\frac {21 \left (29177+125154 x+178015 x^2+83880 x^3\right )}{2 \sqrt {1-2 x} (2+3 x)^3}-4660 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{64827} \]
(2*((21*(29177 + 125154*x + 178015*x^2 + 83880*x^3))/(2*Sqrt[1 - 2*x]*(2 + 3*x)^3) - 4660*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]))/64827
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {109, 27, 162, 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.
\(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^{3/2} (3 x+2)^4} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^3}-\frac {1}{7} \int -\frac {2 (5 x+3) (80 x+37)}{\sqrt {1-2 x} (3 x+2)^4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{7} \int \frac {(5 x+3) (80 x+37)}{\sqrt {1-2 x} (3 x+2)^4}dx+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
\(\Big \downarrow \) 162 |
\(\displaystyle \frac {2}{7} \left (\frac {2330}{63} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {\sqrt {1-2 x} (470 x+297)}{63 (3 x+2)^3}\right )+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {2}{7} \left (\frac {2330}{63} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (470 x+297)}{63 (3 x+2)^3}\right )+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2}{7} \left (\frac {2330}{63} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (470 x+297)}{63 (3 x+2)^3}\right )+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2}{7} \left (\frac {2330}{63} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (470 x+297)}{63 (3 x+2)^3}\right )+\frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) + (2*((Sqrt[1 - 2*x]*(297 + 470*x))/(63*(2 + 3*x)^3) + (2330*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTa nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/63))/7
3.22.2.3.1 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] :> 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 = 1.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.51
method | result | size |
risch | \(\frac {83880 x^{3}+178015 x^{2}+125154 x +29177}{3087 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {9320 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{64827}\) | \(51\) |
pseudoelliptic | \(-\frac {9320 \left (\sqrt {21}\, \left (\frac {2}{3}+x \right )^{3} \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )-7 x^{3}-\frac {249221 x^{2}}{16776}-\frac {48671 x}{4660}-\frac {204239}{83880}\right )}{2401 \sqrt {1-2 x}\, \left (2+3 x \right )^{3}}\) | \(61\) |
derivativedivides | \(\frac {-\frac {6634 \left (1-2 x \right )^{\frac {5}{2}}}{2401}+\frac {39196 \left (1-2 x \right )^{\frac {3}{2}}}{3087}-\frac {6434 \sqrt {1-2 x}}{441}}{\left (-4-6 x \right )^{3}}-\frac {9320 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{64827}+\frac {2662}{2401 \sqrt {1-2 x}}\) | \(66\) |
default | \(\frac {-\frac {6634 \left (1-2 x \right )^{\frac {5}{2}}}{2401}+\frac {39196 \left (1-2 x \right )^{\frac {3}{2}}}{3087}-\frac {6434 \sqrt {1-2 x}}{441}}{\left (-4-6 x \right )^{3}}-\frac {9320 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{64827}+\frac {2662}{2401 \sqrt {1-2 x}}\) | \(66\) |
trager | \(-\frac {\left (83880 x^{3}+178015 x^{2}+125154 x +29177\right ) \sqrt {1-2 x}}{3087 \left (2+3 x \right )^{3} \left (-1+2 x \right )}+\frac {4660 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{64827}\) | \(84\) |
1/3087*(83880*x^3+178015*x^2+125154*x+29177)/(2+3*x)^3/(1-2*x)^(1/2)-9320/ 64827*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {4660 \, \sqrt {21} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (83880 \, x^{3} + 178015 \, x^{2} + 125154 \, x + 29177\right )} \sqrt {-2 \, x + 1}}{64827 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
1/64827*(4660*sqrt(21)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log((3*x + sq rt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(83880*x^3 + 178015*x^2 + 12515 4*x + 29177)*sqrt(-2*x + 1))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)
Timed out. \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.01 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {4660}{64827} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (41940 \, {\left (2 \, x - 1\right )}^{3} + 303835 \, {\left (2 \, x - 1\right )}^{2} + 1464316 \, x - 145187\right )}}{3087 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}} \]
4660/64827*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt( -2*x + 1))) - 2/3087*(41940*(2*x - 1)^3 + 303835*(2*x - 1)^2 + 1464316*x - 145187)/(27*(-2*x + 1)^(7/2) - 189*(-2*x + 1)^(5/2) + 441*(-2*x + 1)^(3/2 ) - 343*sqrt(-2*x + 1))
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {4660}{64827} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2662}{2401 \, \sqrt {-2 \, x + 1}} + \frac {29853 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 137186 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 157633 \, \sqrt {-2 \, x + 1}}{86436 \, {\left (3 \, x + 2\right )}^{3}} \]
4660/64827*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2662/2401/sqrt(-2*x + 1) + 1/86436*(29853*(2*x - 1) ^2*sqrt(-2*x + 1) - 137186*(-2*x + 1)^(3/2) + 157633*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 1.35 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {\frac {59768\,x}{1701}+\frac {86810\,{\left (2\,x-1\right )}^2}{11907}+\frac {9320\,{\left (2\,x-1\right )}^3}{9261}-\frac {5926}{1701}}{\frac {343\,\sqrt {1-2\,x}}{27}-\frac {49\,{\left (1-2\,x\right )}^{3/2}}{3}+7\,{\left (1-2\,x\right )}^{5/2}-{\left (1-2\,x\right )}^{7/2}}-\frac {9320\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{64827} \]