Integrand size = 24, antiderivative size = 128 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (2+3 x)}+\frac {8051 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \]
-1/315*(1-2*x)^(7/2)/(2+3*x)^5+347/8820*(1-2*x)^(7/2)/(2+3*x)^4-8051/26460 *(1-2*x)^(5/2)/(2+3*x)^3+8051/31752*(1-2*x)^(3/2)/(2+3*x)^2+8051/333396*ar ctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-8051/31752*(1-2*x)^(1/2)/(2+3*x )
Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (503276+2919346 x+8277204 x^2+12406455 x^3+7323345 x^4\right )}{2 (2+3 x)^5}+40255 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1666980} \]
((-21*Sqrt[1 - 2*x]*(503276 + 2919346*x + 8277204*x^2 + 12406455*x^3 + 732 3345*x^4))/(2*(2 + 3*x)^5) + 40255*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x ]])/1666980
Time = 0.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {100, 87, 51, 51, 51, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^2}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {1}{315} \int \frac {(1-2 x)^{5/2} (2625 x+1403)}{(3 x+2)^5}dx-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{315} \left (\frac {24153}{28} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^4}dx+\frac {347 (1-2 x)^{7/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{315} \left (\frac {24153}{28} \left (-\frac {5}{9} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^3}dx-\frac {(1-2 x)^{5/2}}{9 (3 x+2)^3}\right )+\frac {347 (1-2 x)^{7/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{315} \left (\frac {24153}{28} \left (-\frac {5}{9} \left (-\frac {1}{2} \int \frac {\sqrt {1-2 x}}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{9 (3 x+2)^3}\right )+\frac {347 (1-2 x)^{7/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{315} \left (\frac {24153}{28} \left (-\frac {5}{9} \left (\frac {1}{2} \left (\frac {1}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {\sqrt {1-2 x}}{3 (3 x+2)}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{9 (3 x+2)^3}\right )+\frac {347 (1-2 x)^{7/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{315} \left (\frac {24153}{28} \left (-\frac {5}{9} \left (\frac {1}{2} \left (\frac {\sqrt {1-2 x}}{3 (3 x+2)}-\frac {1}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{9 (3 x+2)^3}\right )+\frac {347 (1-2 x)^{7/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{315} \left (\frac {24153}{28} \left (-\frac {5}{9} \left (\frac {1}{2} \left (\frac {\sqrt {1-2 x}}{3 (3 x+2)}-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{9 (3 x+2)^3}\right )+\frac {347 (1-2 x)^{7/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}\) |
-1/315*(1 - 2*x)^(7/2)/(2 + 3*x)^5 + ((347*(1 - 2*x)^(7/2))/(28*(2 + 3*x)^ 4) + (24153*(-1/9*(1 - 2*x)^(5/2)/(2 + 3*x)^3 - (5*(-1/6*(1 - 2*x)^(3/2)/( 2 + 3*x)^2 + (Sqrt[1 - 2*x]/(3*(2 + 3*x)) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21]))/2))/9))/28)/315
3.20.52.3.1 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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[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_))*((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*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 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 = 1.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {14646690 x^{5}+17489565 x^{4}+4147953 x^{3}-2438512 x^{2}-1912794 x -503276}{158760 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {8051 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\) | \(61\) |
| pseudoelliptic | \(\frac {80510 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}-21 \sqrt {1-2 x}\, \left (7323345 x^{4}+12406455 x^{3}+8277204 x^{2}+2919346 x +503276\right )}{3333960 \left (2+3 x \right )^{5}}\) | \(65\) |
| derivativedivides | \(-\frac {3888 \left (-\frac {54247 \left (1-2 x \right )^{\frac {9}{2}}}{2286144}+\frac {12269 \left (1-2 x \right )^{\frac {7}{2}}}{69984}-\frac {16102 \left (1-2 x \right )^{\frac {5}{2}}}{32805}+\frac {394499 \left (1-2 x \right )^{\frac {3}{2}}}{629856}-\frac {394499 \sqrt {1-2 x}}{1259712}\right )}{\left (-4-6 x \right )^{5}}+\frac {8051 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\) | \(75\) |
| default | \(-\frac {3888 \left (-\frac {54247 \left (1-2 x \right )^{\frac {9}{2}}}{2286144}+\frac {12269 \left (1-2 x \right )^{\frac {7}{2}}}{69984}-\frac {16102 \left (1-2 x \right )^{\frac {5}{2}}}{32805}+\frac {394499 \left (1-2 x \right )^{\frac {3}{2}}}{629856}-\frac {394499 \sqrt {1-2 x}}{1259712}\right )}{\left (-4-6 x \right )^{5}}+\frac {8051 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\) | \(75\) |
| trager | \(-\frac {\left (7323345 x^{4}+12406455 x^{3}+8277204 x^{2}+2919346 x +503276\right ) \sqrt {1-2 x}}{158760 \left (2+3 x \right )^{5}}-\frac {8051 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{666792}\) | \(82\) |
1/158760*(14646690*x^5+17489565*x^4+4147953*x^3-2438512*x^2-1912794*x-5032 76)/(2+3*x)^5/(1-2*x)^(1/2)+8051/333396*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2) )*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {40255 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (7323345 \, x^{4} + 12406455 \, x^{3} + 8277204 \, x^{2} + 2919346 \, x + 503276\right )} \sqrt {-2 \, x + 1}}{3333960 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/3333960*(40255*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(7323345*x^4 + 12406455*x^3 + 8277204*x^2 + 2919346*x + 503276)*sqrt(-2*x + 1))/(243*x ^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {8051}{666792} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {7323345 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 54106290 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 151487616 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 193304510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 96652255 \, \sqrt {-2 \, x + 1}}{79380 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
-8051/666792*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr t(-2*x + 1))) - 1/79380*(7323345*(-2*x + 1)^(9/2) - 54106290*(-2*x + 1)^(7 /2) + 151487616*(-2*x + 1)^(5/2) - 193304510*(-2*x + 1)^(3/2) + 96652255*s qrt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 3 0870*(2*x - 1)^2 + 72030*x - 19208)
Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {8051}{666792} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {7323345 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 54106290 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 151487616 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 193304510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 96652255 \, \sqrt {-2 \, x + 1}}{2540160 \, {\left (3 \, x + 2\right )}^{5}} \]
-8051/666792*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) - 1/2540160*(7323345*(2*x - 1)^4*sqrt(-2*x + 1) + 5 4106290*(2*x - 1)^3*sqrt(-2*x + 1) + 151487616*(2*x - 1)^2*sqrt(-2*x + 1) - 193304510*(-2*x + 1)^(3/2) + 96652255*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {8051\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{333396}-\frac {\frac {394499\,\sqrt {1-2\,x}}{78732}-\frac {394499\,{\left (1-2\,x\right )}^{3/2}}{39366}+\frac {257632\,{\left (1-2\,x\right )}^{5/2}}{32805}-\frac {12269\,{\left (1-2\,x\right )}^{7/2}}{4374}+\frac {54247\,{\left (1-2\,x\right )}^{9/2}}{142884}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
(8051*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/333396 - ((394499*(1 - 2*x)^(1/2))/78732 - (394499*(1 - 2*x)^(3/2))/39366 + (257632*(1 - 2*x)^(5 /2))/32805 - (12269*(1 - 2*x)^(7/2))/4374 + (54247*(1 - 2*x)^(9/2))/142884 )/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243)