Integrand size = 24, antiderivative size = 120 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {10836 \sqrt {1-2 x} (2+3 x)^2}{15125}-\frac {36 \sqrt {1-2 x} (2+3 x)^3}{605 (3+5 x)}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} (3+5 x)}+\frac {504 \sqrt {1-2 x} (4499+1500 x)}{75625}-\frac {336 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{75625 \sqrt {55}} \]
-336/4159375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+7/11*(2+3*x)^4/ (3+5*x)/(1-2*x)^(1/2)+10836/15125*(2+3*x)^2*(1-2*x)^(1/2)-36/605*(2+3*x)^3 *(1-2*x)^(1/2)/(3+5*x)+504/75625*(4499+1500*x)*(1-2*x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.57 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {-\frac {55 \left (-8186648-6264264 x+14309460 x^2+3789720 x^3+735075 x^4\right )}{\sqrt {1-2 x} (3+5 x)}-336 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375} \]
((-55*(-8186648 - 6264264*x + 14309460*x^2 + 3789720*x^3 + 735075*x^4))/(S qrt[1 - 2*x]*(3 + 5*x)) - 336*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/ 4159375
Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {109, 27, 166, 27, 170, 27, 164, 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.
| \(\displaystyle \int \frac {(3 x+2)^5}{(1-2 x)^{3/2} (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)}-\frac {1}{11} \int \frac {12 (3 x+2)^3 (26 x+15)}{\sqrt {1-2 x} (5 x+3)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)}-\frac {12}{11} \int \frac {(3 x+2)^3 (26 x+15)}{\sqrt {1-2 x} (5 x+3)^2}dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)}-\frac {12}{11} \left (\frac {1}{55} \int \frac {7 (3 x+2)^2 (129 x+77)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {3 \sqrt {1-2 x} (3 x+2)^3}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)}-\frac {12}{11} \left (\frac {7}{55} \int \frac {(3 x+2)^2 (129 x+77)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {3 \sqrt {1-2 x} (3 x+2)^3}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)}-\frac {12}{11} \left (\frac {7}{55} \left (-\frac {1}{25} \int -\frac {2 (3 x+2) (4500 x+2699)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {129}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {3 \sqrt {1-2 x} (3 x+2)^3}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)}-\frac {12}{11} \left (\frac {7}{55} \left (\frac {2}{25} \int \frac {(3 x+2) (4500 x+2699)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {129}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {3 \sqrt {1-2 x} (3 x+2)^3}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)}-\frac {12}{11} \left (\frac {7}{55} \left (\frac {2}{25} \left (-\frac {1}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-\frac {3}{5} \sqrt {1-2 x} (1500 x+4499)\right )-\frac {129}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {3 \sqrt {1-2 x} (3 x+2)^3}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)}-\frac {12}{11} \left (\frac {7}{55} \left (\frac {2}{25} \left (\frac {1}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {3}{5} \sqrt {1-2 x} (1500 x+4499)\right )-\frac {129}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {3 \sqrt {1-2 x} (3 x+2)^3}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)}-\frac {12}{11} \left (\frac {7}{55} \left (\frac {2}{25} \left (\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}-\frac {3}{5} \sqrt {1-2 x} (1500 x+4499)\right )-\frac {129}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {3 \sqrt {1-2 x} (3 x+2)^3}{55 (5 x+3)}\right )\) |
(7*(2 + 3*x)^4)/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (12*((3*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(55*(3 + 5*x)) + (7*((-129*Sqrt[1 - 2*x]*(2 + 3*x)^2)/25 + (2*((-3 *Sqrt[1 - 2*x]*(4499 + 1500*x))/5 + (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/ (5*Sqrt[55])))/25))/55))/11
3.22.16.3.1 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[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(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*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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 = 3.45 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.47
| method | result | size |
| risch | \(-\frac {735075 x^{4}+3789720 x^{3}+14309460 x^{2}-6264264 x -8186648}{75625 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {336 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{4159375}\) | \(56\) |
| pseudoelliptic | \(-\frac {1680 \left (\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}}{5}+\frac {2695275 x^{4}}{112}+\frac {1736955 x^{3}}{14}+\frac {13117005 x^{2}}{28}-\frac {2871121 x}{14}-\frac {11256641}{42}\right )}{\sqrt {1-2 x}\, \left (12478125+20796875 x \right )}\) | \(67\) |
| derivativedivides | \(\frac {243 \left (1-2 x \right )^{\frac {5}{2}}}{1000}-\frac {2943 \left (1-2 x \right )^{\frac {3}{2}}}{1000}+\frac {107109 \sqrt {1-2 x}}{5000}+\frac {2 \sqrt {1-2 x}}{378125 \left (-\frac {6}{5}-2 x \right )}-\frac {336 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{4159375}+\frac {16807}{968 \sqrt {1-2 x}}\) | \(72\) |
| default | \(\frac {243 \left (1-2 x \right )^{\frac {5}{2}}}{1000}-\frac {2943 \left (1-2 x \right )^{\frac {3}{2}}}{1000}+\frac {107109 \sqrt {1-2 x}}{5000}+\frac {2 \sqrt {1-2 x}}{378125 \left (-\frac {6}{5}-2 x \right )}-\frac {336 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{4159375}+\frac {16807}{968 \sqrt {1-2 x}}\) | \(72\) |
| trager | \(\frac {\left (735075 x^{4}+3789720 x^{3}+14309460 x^{2}-6264264 x -8186648\right ) \sqrt {1-2 x}}{756250 x^{2}+75625 x -226875}-\frac {168 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{4159375}\) | \(85\) |
-1/75625*(735075*x^4+3789720*x^3+14309460*x^2-6264264*x-8186648)/(3+5*x)/( 1-2*x)^(1/2)-336/4159375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {168 \, \sqrt {55} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (735075 \, x^{4} + 3789720 \, x^{3} + 14309460 \, x^{2} - 6264264 \, x - 8186648\right )} \sqrt {-2 \, x + 1}}{4159375 \, {\left (10 \, x^{2} + x - 3\right )}} \]
1/4159375*(168*sqrt(55)*(10*x^2 + x - 3)*log((5*x + sqrt(55)*sqrt(-2*x + 1 ) - 8)/(5*x + 3)) + 55*(735075*x^4 + 3789720*x^3 + 14309460*x^2 - 6264264* x - 8186648)*sqrt(-2*x + 1))/(10*x^2 + x - 3)
Time = 81.86 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.74 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {243 \left (1 - 2 x\right )^{\frac {5}{2}}}{1000} - \frac {2943 \left (1 - 2 x\right )^{\frac {3}{2}}}{1000} + \frac {107109 \sqrt {1 - 2 x}}{5000} + \frac {167 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{4159375} - \frac {4 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{6875} + \frac {16807}{968 \sqrt {1 - 2 x}} \]
243*(1 - 2*x)**(5/2)/1000 - 2943*(1 - 2*x)**(3/2)/1000 + 107109*sqrt(1 - 2 *x)/5000 + 167*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2* x) + sqrt(55)/5))/4159375 - 4*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqr t(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/6875 + 16807/(968*s qrt(1 - 2*x))
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.77 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {243}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {2943}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {168}{4159375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {107109}{5000} \, \sqrt {-2 \, x + 1} - \frac {52521891 \, x + 31513117}{302500 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
243/1000*(-2*x + 1)^(5/2) - 2943/1000*(-2*x + 1)^(3/2) + 168/4159375*sqrt( 55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 10 7109/5000*sqrt(-2*x + 1) - 1/302500*(52521891*x + 31513117)/(5*(-2*x + 1)^ (3/2) - 11*sqrt(-2*x + 1))
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.85 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {243}{1000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {2943}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {168}{4159375} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {107109}{5000} \, \sqrt {-2 \, x + 1} - \frac {52521891 \, x + 31513117}{302500 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
243/1000*(2*x - 1)^2*sqrt(-2*x + 1) - 2943/1000*(-2*x + 1)^(3/2) + 168/415 9375*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s qrt(-2*x + 1))) + 107109/5000*sqrt(-2*x + 1) - 1/302500*(52521891*x + 3151 3117)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))
Time = 1.47 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {\frac {52521891\,x}{1512500}+\frac {31513117}{1512500}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}+\frac {107109\,\sqrt {1-2\,x}}{5000}-\frac {2943\,{\left (1-2\,x\right )}^{3/2}}{1000}+\frac {243\,{\left (1-2\,x\right )}^{5/2}}{1000}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,336{}\mathrm {i}}{4159375} \]