Integrand size = 24, antiderivative size = 120 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=-\frac {7588 (2+3 x)^2}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^3}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {6 \sqrt {1-2 x} (114092+38025 x)}{33275}-\frac {68 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \]
7/33*(2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)-68/1830125*arctanh(1/11*55^(1/2)*(1-2 *x)^(1/2))*55^(1/2)-7588/6655*(2+3*x)^2/(1-2*x)^(1/2)-38/1815*(2+3*x)^3/(3 +5*x)/(1-2*x)^(1/2)-6/33275*(114092+38025*x)*(1-2*x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=-\frac {55 \left (7204728-10671002 x-28677318 x^2+16171650 x^3+1617165 x^4\right )-204 \sqrt {55} \sqrt {1-2 x} \left (-3+x+10 x^2\right ) \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5490375 (1-2 x)^{3/2} (3+5 x)} \]
-1/5490375*(55*(7204728 - 10671002*x - 28677318*x^2 + 16171650*x^3 + 16171 65*x^4) - 204*Sqrt[55]*Sqrt[1 - 2*x]*(-3 + x + 10*x^2)*ArcTanh[Sqrt[5/11]* Sqrt[1 - 2*x]])/((1 - 2*x)^(3/2)*(3 + 5*x))
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, 167, 25, 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.
| \(\displaystyle \int \frac {(3 x+2)^5}{(1-2 x)^{5/2} (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {1}{33} \int \frac {2 (3 x+2)^3 (153 x+88)}{(1-2 x)^{3/2} (5 x+3)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {2}{33} \int \frac {(3 x+2)^3 (153 x+88)}{(1-2 x)^{3/2} (5 x+3)^2}dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {2}{33} \left (\frac {1}{55} \int \frac {3 (3 x+2)^2 (1740 x+1027)}{(1-2 x)^{3/2} (5 x+3)}dx+\frac {19 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {2}{33} \left (\frac {3}{55} \int \frac {(3 x+2)^2 (1740 x+1027)}{(1-2 x)^{3/2} (5 x+3)}dx+\frac {19 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {2}{33} \left (\frac {3}{55} \left (\frac {1}{11} \int -\frac {(3 x+2) (114075 x+68462)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {3794 (3 x+2)^2}{11 \sqrt {1-2 x}}\right )+\frac {19 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {2}{33} \left (\frac {3}{55} \left (\frac {3794 (3 x+2)^2}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(3 x+2) (114075 x+68462)}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {19 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {2}{33} \left (\frac {3}{55} \left (\frac {1}{11} \left (\frac {3}{5} \sqrt {1-2 x} (38025 x+114092)-\frac {17}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {3794 (3 x+2)^2}{11 \sqrt {1-2 x}}\right )+\frac {19 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {2}{33} \left (\frac {3}{55} \left (\frac {1}{11} \left (\frac {17}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {3}{5} \sqrt {1-2 x} (38025 x+114092)\right )+\frac {3794 (3 x+2)^2}{11 \sqrt {1-2 x}}\right )+\frac {19 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {2}{33} \left (\frac {3}{55} \left (\frac {1}{11} \left (\frac {34 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}+\frac {3}{5} \sqrt {1-2 x} (38025 x+114092)\right )+\frac {3794 (3 x+2)^2}{11 \sqrt {1-2 x}}\right )+\frac {19 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )\) |
(7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)) - (2*((19*(2 + 3*x)^3)/(55* Sqrt[1 - 2*x]*(3 + 5*x)) + (3*((3794*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]) + ((3 *Sqrt[1 - 2*x]*(114092 + 38025*x))/5 + (34*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x ]])/(5*Sqrt[55]))/11))/55))/33
3.22.81.3.1 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[(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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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 = 3.55 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {1617165 x^{4}+16171650 x^{3}-28677318 x^{2}-10671002 x +7204728}{99825 \sqrt {1-2 x}\, \left (3+5 x \right ) \left (-1+2 x \right )}-\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1830125}\) | \(63\) |
| pseudoelliptic | \(\frac {204 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (10 x^{2}+x -3\right ) \sqrt {55}-88944075 x^{4}-889440750 x^{3}+1577252490 x^{2}+586905110 x -396260040}{\left (1-2 x \right )^{\frac {3}{2}} \left (16471125+27451875 x \right )}\) | \(70\) |
| derivativedivides | \(\frac {81 \left (1-2 x \right )^{\frac {3}{2}}}{200}-\frac {8829 \sqrt {1-2 x}}{1000}+\frac {2 \sqrt {1-2 x}}{831875 \left (-\frac {6}{5}-2 x \right )}-\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1830125}+\frac {16807}{2904 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {228095}{10648 \sqrt {1-2 x}}\) | \(72\) |
| default | \(\frac {81 \left (1-2 x \right )^{\frac {3}{2}}}{200}-\frac {8829 \sqrt {1-2 x}}{1000}+\frac {2 \sqrt {1-2 x}}{831875 \left (-\frac {6}{5}-2 x \right )}-\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1830125}+\frac {16807}{2904 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {228095}{10648 \sqrt {1-2 x}}\) | \(72\) |
| trager | \(-\frac {\left (1617165 x^{4}+16171650 x^{3}-28677318 x^{2}-10671002 x +7204728\right ) \sqrt {1-2 x}}{99825 \left (-1+2 x \right )^{2} \left (3+5 x \right )}+\frac {34 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1830125}\) | \(89\) |
1/99825*(1617165*x^4+16171650*x^3-28677318*x^2-10671002*x+7204728)/(1-2*x) ^(1/2)/(3+5*x)/(-1+2*x)-68/1830125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55 ^(1/2)
Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.78 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {102 \, \sqrt {55} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (1617165 \, x^{4} + 16171650 \, x^{3} - 28677318 \, x^{2} - 10671002 \, x + 7204728\right )} \sqrt {-2 \, x + 1}}{5490375 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
1/5490375*(102*sqrt(55)*(20*x^3 - 8*x^2 - 7*x + 3)*log((5*x + sqrt(55)*sqr t(-2*x + 1) - 8)/(5*x + 3)) - 55*(1617165*x^4 + 16171650*x^3 - 28677318*x^ 2 - 10671002*x + 7204728)*sqrt(-2*x + 1))/(20*x^3 - 8*x^2 - 7*x + 3)
Time = 81.18 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.74 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {81 \left (1 - 2 x\right )^{\frac {3}{2}}}{200} - \frac {8829 \sqrt {1 - 2 x}}{1000} + \frac {169 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{9150625} - \frac {4 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15125} - \frac {228095}{10648 \sqrt {1 - 2 x}} + \frac {16807}{2904 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
81*(1 - 2*x)**(3/2)/200 - 8829*sqrt(1 - 2*x)/1000 + 169*sqrt(55)*(log(sqrt (1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/9150625 - 4*Pie cewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqr t(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(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)))/15125 - 228095/(10648*sqrt(1 - 2*x)) + 16807/(2904*( 1 - 2*x)**(3/2))
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.77 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {81}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {34}{1830125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8829}{1000} \, \sqrt {-2 \, x + 1} - \frac {427678077 \, {\left (2 \, x - 1\right )}^{2} + 2112880000 \, x - 802234125}{3993000 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
81/200*(-2*x + 1)^(3/2) + 34/1830125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 8829/1000*sqrt(-2*x + 1) - 1/39930 00*(427678077*(2*x - 1)^2 + 2112880000*x - 802234125)/(5*(-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))
Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {81}{200} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {34}{1830125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {8829}{1000} \, \sqrt {-2 \, x + 1} - \frac {2401 \, {\left (285 \, x - 104\right )}}{15972 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {\sqrt {-2 \, x + 1}}{166375 \, {\left (5 \, x + 3\right )}} \]
81/200*(-2*x + 1)^(3/2) + 34/1830125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10 *sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 8829/1000*sqrt(-2*x + 1) - 2401/15972*(285*x - 104)/((2*x - 1)*sqrt(-2*x + 1)) - 1/166375*sqrt(-2* x + 1)/(5*x + 3)
Time = 1.40 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {\frac {38416\,x}{363}+\frac {142559359\,{\left (2\,x-1\right )}^2}{6655000}-\frac {194481}{4840}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {8829\,\sqrt {1-2\,x}}{1000}+\frac {81\,{\left (1-2\,x\right )}^{3/2}}{200}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,68{}\mathrm {i}}{1830125} \]