Integrand size = 24, antiderivative size = 108 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^3}-\frac {467 \sqrt {1-2 x}}{126 (2+3 x)^3}-\frac {905 \sqrt {1-2 x}}{882 (2+3 x)^2}-\frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}-\frac {905 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
-905/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+121/14/(2+3*x)^3/( 1-2*x)^(1/2)-467/126*(1-2*x)^(1/2)/(2+3*x)^3-905/882*(1-2*x)^(1/2)/(2+3*x) ^2-905/2058*(1-2*x)^(1/2)/(2+3*x)
Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {\frac {21 \left (2316+13747 x+26245 x^2+16290 x^3\right )}{2 \sqrt {1-2 x} (2+3 x)^3}-905 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609} \]
((21*(2316 + 13747*x + 26245*x^2 + 16290*x^3))/(2*Sqrt[1 - 2*x]*(2 + 3*x)^ 3) - 905*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/21609
Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {100, 27, 87, 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.
\(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x)^{3/2} (3 x+2)^4} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {121}{14 \sqrt {1-2 x} (3 x+2)^3}-\frac {1}{14} \int -\frac {7 (139-25 x)}{\sqrt {1-2 x} (3 x+2)^4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {139-25 x}{\sqrt {1-2 x} (3 x+2)^4}dx+\frac {121}{14 \sqrt {1-2 x} (3 x+2)^3}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{2} \left (\frac {1810}{63} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {467 \sqrt {1-2 x}}{63 (3 x+2)^3}\right )+\frac {121}{14 \sqrt {1-2 x} (3 x+2)^3}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (\frac {1810}{63} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {467 \sqrt {1-2 x}}{63 (3 x+2)^3}\right )+\frac {121}{14 \sqrt {1-2 x} (3 x+2)^3}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (\frac {1810}{63} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {467 \sqrt {1-2 x}}{63 (3 x+2)^3}\right )+\frac {121}{14 \sqrt {1-2 x} (3 x+2)^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {1810}{63} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {467 \sqrt {1-2 x}}{63 (3 x+2)^3}\right )+\frac {121}{14 \sqrt {1-2 x} (3 x+2)^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {1810}{63} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {467 \sqrt {1-2 x}}{63 (3 x+2)^3}\right )+\frac {121}{14 \sqrt {1-2 x} (3 x+2)^3}\) |
121/(14*Sqrt[1 - 2*x]*(2 + 3*x)^3) + ((-467*Sqrt[1 - 2*x])/(63*(2 + 3*x)^3 ) + (1810*(-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*Sqrt[21])))/14))/63)/2
3.21.92.3.1 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 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 = 1.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.47
method | result | size |
risch | \(\frac {16290 x^{3}+26245 x^{2}+13747 x +2316}{2058 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {905 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(51\) |
pseudoelliptic | \(-\frac {2715 \left (\sqrt {21}\, \left (\frac {2}{3}+x \right )^{3} \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )-7 x^{3}-\frac {203 x^{2}}{18}-\frac {96229 x}{16290}-\frac {2702}{2715}\right )}{2401 \sqrt {1-2 x}\, \left (2+3 x \right )^{3}}\) | \(61\) |
derivativedivides | \(\frac {\frac {5937 \left (1-2 x \right )^{\frac {5}{2}}}{2401}-\frac {11476 \left (1-2 x \right )^{\frac {3}{2}}}{1029}+\frac {1849 \sqrt {1-2 x}}{147}}{\left (-4-6 x \right )^{3}}-\frac {905 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}+\frac {484}{2401 \sqrt {1-2 x}}\) | \(66\) |
default | \(\frac {\frac {5937 \left (1-2 x \right )^{\frac {5}{2}}}{2401}-\frac {11476 \left (1-2 x \right )^{\frac {3}{2}}}{1029}+\frac {1849 \sqrt {1-2 x}}{147}}{\left (-4-6 x \right )^{3}}-\frac {905 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}+\frac {484}{2401 \sqrt {1-2 x}}\) | \(66\) |
trager | \(-\frac {\left (16290 x^{3}+26245 x^{2}+13747 x +2316\right ) \sqrt {1-2 x}}{2058 \left (2+3 x \right )^{3} \left (-1+2 x \right )}+\frac {905 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{43218}\) | \(84\) |
1/2058*(16290*x^3+26245*x^2+13747*x+2316)/(2+3*x)^3/(1-2*x)^(1/2)-905/2160 9*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 = 0.92 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {905 \, \sqrt {21} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (16290 \, x^{3} + 26245 \, x^{2} + 13747 \, x + 2316\right )} \sqrt {-2 \, x + 1}}{43218 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
1/43218*(905*sqrt(21)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log((3*x + sqr t(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(16290*x^3 + 26245*x^2 + 13747*x + 2316)*sqrt(-2*x + 1))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)
Timed out. \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {905}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8145 \, {\left (2 \, x - 1\right )}^{3} + 50680 \, {\left (2 \, x - 1\right )}^{2} + 208838 \, x - 33271}{1029 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}} \]
905/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(- 2*x + 1))) - 1/1029*(8145*(2*x - 1)^3 + 50680*(2*x - 1)^2 + 208838*x - 332 71)/(27*(-2*x + 1)^(7/2) - 189*(-2*x + 1)^(5/2) + 441*(-2*x + 1)^(3/2) - 3 43*sqrt(-2*x + 1))
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {905}{43218} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {484}{2401 \, \sqrt {-2 \, x + 1}} - \frac {17811 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 80332 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 90601 \, \sqrt {-2 \, x + 1}}{57624 \, {\left (3 \, x + 2\right )}^{3}} \]
905/43218*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 484/2401/sqrt(-2*x + 1) - 1/57624*(17811*(2*x - 1)^2 *sqrt(-2*x + 1) - 80332*(-2*x + 1)^(3/2) + 90601*sqrt(-2*x + 1))/(3*x + 2) ^3
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.76 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {\frac {4262\,x}{567}+\frac {7240\,{\left (2\,x-1\right )}^2}{3969}+\frac {905\,{\left (2\,x-1\right )}^3}{3087}-\frac {97}{81}}{\frac {343\,\sqrt {1-2\,x}}{27}-\frac {49\,{\left (1-2\,x\right )}^{3/2}}{3}+7\,{\left (1-2\,x\right )}^{5/2}-{\left (1-2\,x\right )}^{7/2}}-\frac {905\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609} \]