Integrand size = 22, antiderivative size = 136 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {215}{9604 (1-2 x)^{3/2}}+\frac {1935}{67228 \sqrt {1-2 x}}+\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}-\frac {43}{588 (1-2 x)^{3/2} (2+3 x)^3}-\frac {129}{2744 (1-2 x)^{3/2} (2+3 x)^2}-\frac {129}{2744 (1-2 x)^{3/2} (2+3 x)}-\frac {1935 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228} \]
215/9604/(1-2*x)^(3/2)+1/84/(1-2*x)^(3/2)/(2+3*x)^4-43/588/(1-2*x)^(3/2)/( 2+3*x)^3-129/2744/(1-2*x)^(3/2)/(2+3*x)^2-129/2744/(1-2*x)^(3/2)/(2+3*x)-1 935/470596*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1935/67228/(1-2*x) ^(1/2)
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.55 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {-\frac {7 \left (-48490-611202 x-1034451 x^2+1069281 x^3+3343680 x^4+1880820 x^5\right )}{2 (1-2 x)^{3/2} (2+3 x)^4}-5805 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1411788} \]
((-7*(-48490 - 611202*x - 1034451*x^2 + 1069281*x^3 + 3343680*x^4 + 188082 0*x^5))/(2*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - 5805*Sqrt[21]*ArcTanh[Sqrt[3/7]* Sqrt[1 - 2*x]])/1411788
Time = 0.21 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {87, 52, 52, 52, 61, 61, 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.
| \(\displaystyle \int \frac {5 x+3}{(1-2 x)^{5/2} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {43}{28} \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^4}dx+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {43}{28} \left (\frac {3}{7} \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^3}dx-\frac {1}{21 (1-2 x)^{3/2} (3 x+2)^3}\right )+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {43}{28} \left (\frac {3}{7} \left (\frac {1}{2} \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^2}dx-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )-\frac {1}{21 (1-2 x)^{3/2} (3 x+2)^3}\right )+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {43}{28} \left (\frac {3}{7} \left (\frac {1}{2} \left (\frac {5}{7} \int \frac {1}{(1-2 x)^{5/2} (3 x+2)}dx-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )-\frac {1}{21 (1-2 x)^{3/2} (3 x+2)^3}\right )+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {43}{28} \left (\frac {3}{7} \left (\frac {1}{2} \left (\frac {5}{7} \left (\frac {3}{7} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )-\frac {1}{21 (1-2 x)^{3/2} (3 x+2)^3}\right )+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {43}{28} \left (\frac {3}{7} \left (\frac {1}{2} \left (\frac {5}{7} \left (\frac {3}{7} \left (\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{7 \sqrt {1-2 x}}\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )-\frac {1}{21 (1-2 x)^{3/2} (3 x+2)^3}\right )+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {43}{28} \left (\frac {3}{7} \left (\frac {1}{2} \left (\frac {5}{7} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )-\frac {1}{21 (1-2 x)^{3/2} (3 x+2)^3}\right )+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {43}{28} \left (\frac {3}{7} \left (\frac {1}{2} \left (\frac {5}{7} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )-\frac {1}{21 (1-2 x)^{3/2} (3 x+2)^3}\right )+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}\) |
1/(84*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + (43*(-1/21*1/((1 - 2*x)^(3/2)*(2 + 3* x)^3) + (3*(-1/14*1/((1 - 2*x)^(3/2)*(2 + 3*x)^2) + (-1/7*1/((1 - 2*x)^(3/ 2)*(2 + 3*x)) + (5*(2/(21*(1 - 2*x)^(3/2)) + (3*(2/(7*Sqrt[1 - 2*x]) - (2* Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7))/7))/7)/2))/7))/28
3.22.45.3.1 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] :> 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((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)^(-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 = 3.50 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.50
| method | result | size |
| risch | \(\frac {1880820 x^{5}+3343680 x^{4}+1069281 x^{3}-1034451 x^{2}-611202 x -48490}{403368 \left (2+3 x \right )^{4} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {1935 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{470596}\) | \(68\) |
| pseudoelliptic | \(\frac {\frac {1935 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{4} \sqrt {21}}{470596}-\frac {156735 x^{5}}{33614}-\frac {139320 x^{4}}{16807}-\frac {356427 x^{3}}{134456}+\frac {344817 x^{2}}{134456}+\frac {101867 x}{67228}+\frac {24245}{201684}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{4}}\) | \(79\) |
| derivativedivides | \(\frac {\frac {423225 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {461403 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {169335 \left (1-2 x \right )^{\frac {3}{2}}}{9604}-\frac {20805 \sqrt {1-2 x}}{1372}}{\left (-4-6 x \right )^{4}}-\frac {1935 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{470596}+\frac {176}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2080}{117649 \sqrt {1-2 x}}\) | \(84\) |
| default | \(\frac {\frac {423225 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {461403 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {169335 \left (1-2 x \right )^{\frac {3}{2}}}{9604}-\frac {20805 \sqrt {1-2 x}}{1372}}{\left (-4-6 x \right )^{4}}-\frac {1935 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{470596}+\frac {176}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2080}{117649 \sqrt {1-2 x}}\) | \(84\) |
| trager | \(-\frac {\left (1880820 x^{5}+3343680 x^{4}+1069281 x^{3}-1034451 x^{2}-611202 x -48490\right ) \sqrt {1-2 x}}{403368 \left (2+3 x \right )^{4} \left (-1+2 x \right )^{2}}-\frac {1935 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{941192}\) | \(94\) |
1/403368*(1880820*x^5+3343680*x^4+1069281*x^3-1034451*x^2-611202*x-48490)/ (2+3*x)^4/(1-2*x)^(1/2)/(-1+2*x)-1935/470596*arctanh(1/7*21^(1/2)*(1-2*x)^ (1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {5805 \, \sqrt {7} \sqrt {3} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (1880820 \, x^{5} + 3343680 \, x^{4} + 1069281 \, x^{3} - 1034451 \, x^{2} - 611202 \, x - 48490\right )} \sqrt {-2 \, x + 1}}{2823576 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]
1/2823576*(5805*sqrt(7)*sqrt(3)*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 10 4*x^2 + 32*x + 16)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2 )) - 7*(1880820*x^5 + 3343680*x^4 + 1069281*x^3 - 1034451*x^2 - 611202*x - 48490)*sqrt(-2*x + 1))/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)
Timed out. \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.94 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {1935}{941192} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {470205 \, {\left (2 \, x - 1\right )}^{5} + 4022865 \, {\left (2 \, x - 1\right )}^{4} + 12458691 \, {\left (2 \, x - 1\right )}^{3} + 15872031 \, {\left (2 \, x - 1\right )}^{2} + 11327232 \, x - 7353920}{201684 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 2401 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
1935/941192*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt (-2*x + 1))) - 1/201684*(470205*(2*x - 1)^5 + 4022865*(2*x - 1)^4 + 124586 91*(2*x - 1)^3 + 15872031*(2*x - 1)^2 + 11327232*x - 7353920)/(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 = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.89 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {1935}{941192} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16 \, {\left (780 \, x - 467\right )}}{352947 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {3 \, {\left (141075 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 1076607 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 2765805 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2378705 \, \sqrt {-2 \, x + 1}\right )}}{7529536 \, {\left (3 \, x + 2\right )}^{4}} \]
1935/941192*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/352947*(780*x - 467)/((2*x - 1)*sqrt(-2*x + 1)) - 3/7529536*(141075*(2*x - 1)^3*sqrt(-2*x + 1) + 1076607*(2*x - 1)^2*sqrt (-2*x + 1) - 2765805*(-2*x + 1)^(3/2) + 2378705*sqrt(-2*x + 1))/(3*x + 2)^ 4
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=-\frac {1935\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{470596}-\frac {\frac {2752\,x}{3969}+\frac {1333\,{\left (2\,x-1\right )}^2}{1372}+\frac {3139\,{\left (2\,x-1\right )}^3}{4116}+\frac {2365\,{\left (2\,x-1\right )}^4}{9604}+\frac {1935\,{\left (2\,x-1\right )}^5}{67228}-\frac {5360}{11907}}{\frac {2401\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{7/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{9/2}}{3}+{\left (1-2\,x\right )}^{11/2}} \]
- (1935*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/470596 - ((2752*x)/3 969 + (1333*(2*x - 1)^2)/1372 + (3139*(2*x - 1)^3)/4116 + (2365*(2*x - 1)^ 4)/9604 + (1935*(2*x - 1)^5)/67228 - 5360/11907)/((2401*(1 - 2*x)^(3/2))/8 1 - (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))