Integrand size = 24, antiderivative size = 100 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {487+720 x}{294 \sqrt {1-2 x} (2+3 x)^2}+\frac {905 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
11/21*(3+5*x)^2/(1-2*x)^(3/2)/(2+3*x)^2+905/21609*arctanh(1/7*21^(1/2)*(1- 2*x)^(1/2))*21^(1/2)+1/294*(-487-720*x)/(2+3*x)^2/(1-2*x)^(1/2)+905/2058*( 1-2*x)^(1/2)/(2+3*x)
Time = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.65 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {\frac {21 \left (8103+29593 x+33410 x^2+10860 x^3\right )}{2 (1-2 x)^{3/2} (2+3 x)^2}+905 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609} \]
((21*(8103 + 29593*x + 33410*x^2 + 10860*x^3))/(2*(1 - 2*x)^(3/2)*(2 + 3*x )^2) + 905*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/21609
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {109, 161, 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.
| \(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^{5/2} (3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {1}{21} \int \frac {(5 x+3) (15 x+31)}{(1-2 x)^{3/2} (3 x+2)^3}dx\) |
| \(\Big \downarrow \) 161 |
| \(\displaystyle \frac {1}{21} \left (-\frac {905}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {720 x+487}{14 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{21} \left (-\frac {905}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {720 x+487}{14 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{21} \left (-\frac {905}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {720 x+487}{14 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{21} \left (-\frac {905}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {720 x+487}{14 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
(11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (-1/14*(487 + 720*x)/( Sqrt[1 - 2*x]*(2 + 3*x)^2) - (905*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTa nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/14)/21
3.22.65.3.1 Defintions of rubi rules used
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 = 3.49 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(-\frac {10860 x^{3}+33410 x^{2}+29593 x +8103}{2058 \left (-1+2 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{2}}+\frac {905 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(58\) |
| derivativedivides | \(-\frac {18 \left (-\frac {199 \left (1-2 x \right )^{\frac {3}{2}}}{18}+\frac {1379 \sqrt {1-2 x}}{54}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {905 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}+\frac {1331}{1029 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {726}{2401 \sqrt {1-2 x}}\) | \(66\) |
| default | \(-\frac {18 \left (-\frac {199 \left (1-2 x \right )^{\frac {3}{2}}}{18}+\frac {1379 \sqrt {1-2 x}}{54}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {905 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}+\frac {1331}{1029 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {726}{2401 \sqrt {1-2 x}}\) | \(66\) |
| pseudoelliptic | \(-\frac {1810 \left (\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{2} \sqrt {21}}{18}-7 x^{3}-\frac {23387 x^{2}}{1086}-\frac {207151 x}{10860}-\frac {18907}{3620}\right )}{2401 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{2}}\) | \(69\) |
| trager | \(\frac {\left (10860 x^{3}+33410 x^{2}+29593 x +8103\right ) \sqrt {1-2 x}}{2058 \left (6 x^{2}+x -2\right )^{2}}-\frac {905 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{43218}\) | \(80\) |
-1/2058*(10860*x^3+33410*x^2+29593*x+8103)/(-1+2*x)/(1-2*x)^(1/2)/(2+3*x)^ 2+905/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {905 \, \sqrt {21} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (10860 \, x^{3} + 33410 \, x^{2} + 29593 \, x + 8103\right )} \sqrt {-2 \, x + 1}}{43218 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
1/43218*(905*sqrt(21)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log((3*x - sqrt (21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(10860*x^3 + 33410*x^2 + 29593*x + 8103)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)
Time = 146.01 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.54 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=- \frac {363 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{16807} - \frac {404 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{1029} - \frac {8 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{147} - \frac {726}{2401 \sqrt {1 - 2 x}} + \frac {1331}{1029 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
-363*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt( 21)/3))/16807 - 404*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1 )/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt (21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/1029 - 8*Piecewise((sqrt(21)*(3*l og(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/ 16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x) /7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt (1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/147 - 726/(2401*sqrt(1 - 2*x)) + 1331/(1029*(1 - 2*x)**(3 /2))
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.92 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {905}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2715 \, {\left (2 \, x - 1\right )}^{3} + 24850 \, {\left (2 \, x - 1\right )}^{2} + 142296 \, x - 5929}{1029 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 49 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
-905/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt( -2*x + 1))) + 1/1029*(2715*(2*x - 1)^3 + 24850*(2*x - 1)^2 + 142296*x - 59 29)/(9*(-2*x + 1)^(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))
Time = 0.31 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.89 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {905}{43218} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {121 \, {\left (36 \, x + 59\right )}}{7203 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {597 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1379 \, \sqrt {-2 \, x + 1}}{28812 \, {\left (3 \, x + 2\right )}^{2}} \]
-905/43218*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 121/7203*(36*x + 59)/((2*x - 1)*sqrt(-2*x + 1)) + 1 /28812*(597*(-2*x + 1)^(3/2) - 1379*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.71 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {905\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}+\frac {\frac {968\,x}{63}+\frac {3550\,{\left (2\,x-1\right )}^2}{1323}+\frac {905\,{\left (2\,x-1\right )}^3}{3087}-\frac {121}{189}}{\frac {49\,{\left (1-2\,x\right )}^{3/2}}{9}-\frac {14\,{\left (1-2\,x\right )}^{5/2}}{3}+{\left (1-2\,x\right )}^{7/2}} \]