Integrand size = 24, antiderivative size = 140 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}-\frac {177635 \sqrt {1-2 x}}{172872 (2+3 x)^2}-\frac {177635 \sqrt {1-2 x}}{403368 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}-\frac {177635 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{201684 \sqrt {21}} \]
11/21*(3+5*x)^2/(1-2*x)^(3/2)/(2+3*x)^4-177635/4235364*arctanh(1/7*21^(1/2 )*(1-2*x)^(1/2))*21^(1/2)+1/4116*(57069+85754*x)/(2+3*x)^4/(1-2*x)^(1/2)-3 5527/12348*(1-2*x)^(1/2)/(2+3*x)^3-177635/172872*(1-2*x)^(1/2)/(2+3*x)^2-1 77635/403368*(1-2*x)^(1/2)/(2+3*x)
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.54 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {-\frac {21 \left (-2094250-10307138 x-12952519 x^2+10906789 x^3+34105920 x^4+19184580 x^5\right )}{2 (1-2 x)^{3/2} (2+3 x)^4}-177635 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4235364} \]
((-21*(-2094250 - 10307138*x - 12952519*x^2 + 10906789*x^3 + 34105920*x^4 + 19184580*x^5))/(2*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - 177635*Sqrt[21]*ArcTanh [Sqrt[3/7]*Sqrt[1 - 2*x]])/4235364
Time = 0.22 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {109, 25, 161, 52, 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.
| \(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^{5/2} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}-\frac {1}{21} \int -\frac {(5 x+3) (315 x+167)}{(1-2 x)^{3/2} (3 x+2)^5}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{21} \int \frac {(5 x+3) (315 x+167)}{(1-2 x)^{3/2} (3 x+2)^5}dx+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 161 |
| \(\displaystyle \frac {1}{21} \left (\frac {35527}{28} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4}dx+\frac {85754 x+57069}{196 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{21} \left (\frac {35527}{28} \left (\frac {5}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {85754 x+57069}{196 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{21} \left (\frac {35527}{28} \left (\frac {5}{21} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {85754 x+57069}{196 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{21} \left (\frac {35527}{28} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {85754 x+57069}{196 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{21} \left (\frac {35527}{28} \left (\frac {5}{21} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {85754 x+57069}{196 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{21} \left (\frac {35527}{28} \left (\frac {5}{21} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {85754 x+57069}{196 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}\) |
(11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + ((57069 + 85754*x)/(19 6*Sqrt[1 - 2*x]*(2 + 3*x)^4) + (35527*(-1/21*Sqrt[1 - 2*x]/(2 + 3*x)^3 + ( 5*(-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))/21))/28)/21
3.22.67.3.1 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 = 3.54 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(\frac {19184580 x^{5}+34105920 x^{4}+10906789 x^{3}-12952519 x^{2}-10307138 x -2094250}{403368 \left (2+3 x \right )^{4} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {177635 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4235364}\) | \(68\) |
| pseudoelliptic | \(\frac {\frac {177635 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{4} \sqrt {21}}{4235364}-\frac {1598715 x^{5}}{33614}-\frac {1421080 x^{4}}{16807}-\frac {10906789 x^{3}}{403368}+\frac {12952519 x^{2}}{403368}+\frac {5153569 x}{201684}+\frac {1047125}{201684}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{4}}\) | \(79\) |
| derivativedivides | \(\frac {\frac {1782045 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {1707607 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {1636345 \left (1-2 x \right )^{\frac {3}{2}}}{28812}-\frac {174235 \sqrt {1-2 x}}{4116}}{\left (-4-6 x \right )^{4}}-\frac {177635 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4235364}+\frac {5324}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {29040}{117649 \sqrt {1-2 x}}\) | \(84\) |
| default | \(\frac {\frac {1782045 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {1707607 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {1636345 \left (1-2 x \right )^{\frac {3}{2}}}{28812}-\frac {174235 \sqrt {1-2 x}}{4116}}{\left (-4-6 x \right )^{4}}-\frac {177635 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4235364}+\frac {5324}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {29040}{117649 \sqrt {1-2 x}}\) | \(84\) |
| trager | \(-\frac {\left (19184580 x^{5}+34105920 x^{4}+10906789 x^{3}-12952519 x^{2}-10307138 x -2094250\right ) \sqrt {1-2 x}}{403368 \left (2+3 x \right )^{4} \left (-1+2 x \right )^{2}}+\frac {177635 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{8470728}\) | \(94\) |
1/403368*(19184580*x^5+34105920*x^4+10906789*x^3-12952519*x^2-10307138*x-2 094250)/(2+3*x)^4/(1-2*x)^(1/2)/(-1+2*x)-177635/4235364*arctanh(1/7*21^(1/ 2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.92 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {177635 \, \sqrt {21} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (19184580 \, x^{5} + 34105920 \, x^{4} + 10906789 \, x^{3} - 12952519 \, x^{2} - 10307138 \, x - 2094250\right )} \sqrt {-2 \, x + 1}}{8470728 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]
1/8470728*(177635*sqrt(21)*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(191 84580*x^5 + 34105920*x^4 + 10906789*x^3 - 12952519*x^2 - 10307138*x - 2094 250)*sqrt(-2*x + 1))/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32* x + 16)
Timed out. \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {177635}{8470728} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4796145 \, {\left (2 \, x - 1\right )}^{5} + 41033685 \, {\left (2 \, x - 1\right )}^{4} + 127080079 \, {\left (2 \, x - 1\right )}^{3} + 157094539 \, {\left (2 \, x - 1\right )}^{2} + 63748608 \, x - 83006000}{201684 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 2401 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
177635/8470728*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*s qrt(-2*x + 1))) - 1/201684*(4796145*(2*x - 1)^5 + 41033685*(2*x - 1)^4 + 1 27080079*(2*x - 1)^3 + 157094539*(2*x - 1)^2 + 63748608*x - 83006000)/(81* (-2*x + 1)^(11/2) - 756*(-2*x + 1)^(9/2) + 2646*(-2*x + 1)^(7/2) - 4116*(- 2*x + 1)^(5/2) + 2401*(-2*x + 1)^(3/2))
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {177635}{8470728} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {484 \, {\left (360 \, x - 257\right )}}{352947 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {5346135 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 35859747 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 80180905 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 59762605 \, \sqrt {-2 \, x + 1}}{22588608 \, {\left (3 \, x + 2\right )}^{4}} \]
177635/8470728*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt( 21) + 3*sqrt(-2*x + 1))) + 484/352947*(360*x - 257)/((2*x - 1)*sqrt(-2*x + 1)) - 1/22588608*(5346135*(2*x - 1)^3*sqrt(-2*x + 1) + 35859747*(2*x - 1) ^2*sqrt(-2*x + 1) - 80180905*(-2*x + 1)^(3/2) + 59762605*sqrt(-2*x + 1))/( 3*x + 2)^4
Time = 1.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.77 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=-\frac {177635\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4235364}-\frac {\frac {15488\,x}{3969}+\frac {3206011\,{\left (2\,x-1\right )}^2}{333396}+\frac {2593471\,{\left (2\,x-1\right )}^3}{333396}+\frac {1953985\,{\left (2\,x-1\right )}^4}{777924}+\frac {177635\,{\left (2\,x-1\right )}^5}{605052}-\frac {60500}{11907}}{\frac {2401\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{7/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{9/2}}{3}+{\left (1-2\,x\right )}^{11/2}} \]
- (177635*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/4235364 - ((15488* x)/3969 + (3206011*(2*x - 1)^2)/333396 + (2593471*(2*x - 1)^3)/333396 + (1 953985*(2*x - 1)^4)/777924 + (177635*(2*x - 1)^5)/605052 - 60500/11907)/(( 2401*(1 - 2*x)^(3/2))/81 - (1372*(1 - 2*x)^(5/2))/27 + (98*(1 - 2*x)^(7/2) )/3 - (28*(1 - 2*x)^(9/2))/3 + (1 - 2*x)^(11/2))