Integrand size = 24, antiderivative size = 120 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=-\frac {3755 \sqrt {1-2 x}}{3087 (2+3 x)^2}-\frac {3755 \sqrt {1-2 x}}{7203 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2 (1346+2027 x)}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {7510 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7203 \sqrt {21}} \]
11/21*(3+5*x)^2/(1-2*x)^(3/2)/(2+3*x)^3-7510/151263*arctanh(1/7*21^(1/2)*( 1-2*x)^(1/2))*21^(1/2)+2/441*(1346+2027*x)/(2+3*x)^3/(1-2*x)^(1/2)-3755/30 87*(1-2*x)^(1/2)/(2+3*x)^2-3755/7203*(1-2*x)^(1/2)/(2+3*x)
Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.57 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {-\frac {21 \left (-45383-150295 x-83306 x^2+150200 x^3+135180 x^4\right )}{(1-2 x)^{3/2} (2+3 x)^3}-7510 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{151263} \]
((-21*(-45383 - 150295*x - 83306*x^2 + 150200*x^3 + 135180*x^4))/((1 - 2*x )^(3/2)*(2 + 3*x)^3) - 7510*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/151 263
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, 27, 161, 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.
| \(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^{5/2} (3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac {1}{21} \int -\frac {2 (5 x+3) (75 x+34)}{(1-2 x)^{3/2} (3 x+2)^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{21} \int \frac {(5 x+3) (75 x+34)}{(1-2 x)^{3/2} (3 x+2)^4}dx+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 161 |
| \(\displaystyle \frac {2}{21} \left (\frac {3755}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {2027 x+1346}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2}{21} \left (\frac {3755}{21} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {2027 x+1346}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2}{21} \left (\frac {3755}{21} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {2027 x+1346}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2}{21} \left (\frac {3755}{21} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {2027 x+1346}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2}{21} \left (\frac {3755}{21} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {2027 x+1346}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
(11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (2*((1346 + 2027*x)/(2 1*Sqrt[1 - 2*x]*(2 + 3*x)^3) + (3755*(-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*S qrt[21])))/14))/21))/21
3.22.66.3.1 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^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1 ) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)/(b*d*(b*c - a*d)^2*( m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f* h*(2 + 3*n + 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*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b*c - a*d)^2*(m + 1)*( n + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c , d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -1]
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 = 1.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {135180 x^{4}+150200 x^{3}-83306 x^{2}-150295 x -45383}{7203 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {7510 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{151263}\) | \(63\) |
| pseudoelliptic | \(\frac {\frac {7510 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{3} \sqrt {21}}{151263}-\frac {45060 x^{4}}{2401}-\frac {150200 x^{3}}{7203}+\frac {83306 x^{2}}{7203}+\frac {150295 x}{7203}+\frac {45383}{7203}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{3}}\) | \(74\) |
| derivativedivides | \(\frac {-\frac {18708 \left (1-2 x \right )^{\frac {5}{2}}}{16807}+\frac {5260 \left (1-2 x \right )^{\frac {3}{2}}}{1029}-\frac {6040 \sqrt {1-2 x}}{1029}}{\left (-4-6 x \right )^{3}}-\frac {7510 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{151263}+\frac {2662}{7203 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {6534}{16807 \sqrt {1-2 x}}\) | \(75\) |
| default | \(\frac {-\frac {18708 \left (1-2 x \right )^{\frac {5}{2}}}{16807}+\frac {5260 \left (1-2 x \right )^{\frac {3}{2}}}{1029}-\frac {6040 \sqrt {1-2 x}}{1029}}{\left (-4-6 x \right )^{3}}-\frac {7510 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{151263}+\frac {2662}{7203 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {6534}{16807 \sqrt {1-2 x}}\) | \(75\) |
| trager | \(-\frac {\left (135180 x^{4}+150200 x^{3}-83306 x^{2}-150295 x -45383\right ) \sqrt {1-2 x}}{7203 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2}}+\frac {3755 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{151263}\) | \(89\) |
1/7203*(135180*x^4+150200*x^3-83306*x^2-150295*x-45383)/(2+3*x)^3/(1-2*x)^ (1/2)/(-1+2*x)-7510/151263*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {3755 \, \sqrt {21} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (135180 \, x^{4} + 150200 \, x^{3} - 83306 \, x^{2} - 150295 \, x - 45383\right )} \sqrt {-2 \, x + 1}}{151263 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
1/151263*(3755*sqrt(21)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*lo g((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(135180*x^4 + 150200 *x^3 - 83306*x^2 - 150295*x - 45383)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)
Timed out. \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {3755}{151263} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (33795 \, {\left (2 \, x - 1\right )}^{4} + 210280 \, {\left (2 \, x - 1\right )}^{3} + 344764 \, {\left (2 \, x - 1\right )}^{2} - 213444 \, x - 349811\right )}}{7203 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 343 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
3755/151263*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt (-2*x + 1))) + 2/7203*(33795*(2*x - 1)^4 + 210280*(2*x - 1)^3 + 344764*(2* x - 1)^2 - 213444*x - 349811)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2) + 441*(-2*x + 1)^(5/2) - 343*(-2*x + 1)^(3/2))
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {3755}{151263} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (33795 \, {\left (2 \, x - 1\right )}^{4} + 210280 \, {\left (2 \, x - 1\right )}^{3} + 344764 \, {\left (2 \, x - 1\right )}^{2} - 213444 \, x - 349811\right )}}{7203 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}^{3}} \]
3755/151263*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/7203*(33795*(2*x - 1)^4 + 210280*(2*x - 1)^3 + 3 44764*(2*x - 1)^2 - 213444*x - 349811)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))^3
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.77 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=-\frac {\frac {14072\,{\left (2\,x-1\right )}^2}{3969}-\frac {968\,x}{441}+\frac {60080\,{\left (2\,x-1\right )}^3}{27783}+\frac {7510\,{\left (2\,x-1\right )}^4}{21609}-\frac {14278}{3969}}{\frac {343\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {49\,{\left (1-2\,x\right )}^{5/2}}{3}+7\,{\left (1-2\,x\right )}^{7/2}-{\left (1-2\,x\right )}^{9/2}}-\frac {7510\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{151263} \]