Integrand size = 24, antiderivative size = 141 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {20465}{201684 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac {727}{588 \sqrt {1-2 x} (2+3 x)^4}-\frac {4093}{12348 \sqrt {1-2 x} (2+3 x)^3}-\frac {4093}{24696 \sqrt {1-2 x} (2+3 x)^2}-\frac {20465}{172872 \sqrt {1-2 x} (2+3 x)}-\frac {20465 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228 \sqrt {21}} \]
121/42/(1-2*x)^(3/2)/(2+3*x)^4-20465/1411788*arctanh(1/7*21^(1/2)*(1-2*x)^ (1/2))*21^(1/2)+20465/201684/(1-2*x)^(1/2)-727/588/(2+3*x)^4/(1-2*x)^(1/2) -4093/12348/(2+3*x)^3/(1-2*x)^(1/2)-4093/24696/(2+3*x)^2/(1-2*x)^(1/2)-204 65/172872/(2+3*x)/(1-2*x)^(1/2)
Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.53 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {-\frac {7 \left (-401410-2528226 x-3646863 x^2+3769653 x^3+11787840 x^4+6630660 x^5\right )}{2 (1-2 x)^{3/2} (2+3 x)^4}-20465 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1411788} \]
((-7*(-401410 - 2528226*x - 3646863*x^2 + 3769653*x^3 + 11787840*x^4 + 663 0660*x^5))/(2*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - 20465*Sqrt[21]*ArcTanh[Sqrt[3 /7]*Sqrt[1 - 2*x]])/1411788
Time = 0.22 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.19, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {100, 27, 87, 52, 52, 52, 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.
| \(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x)^{5/2} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {121}{42 (1-2 x)^{3/2} (3 x+2)^4}-\frac {1}{42} \int -\frac {3 (368-175 x)}{(1-2 x)^{3/2} (3 x+2)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {368-175 x}{(1-2 x)^{3/2} (3 x+2)^5}dx+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{14} \left (\frac {4093}{42} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^4}dx-\frac {727}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{14} \left (\frac {4093}{42} \left (\frac {1}{3} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^3}dx-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {727}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{14} \left (\frac {4093}{42} \left (\frac {1}{3} \left (\frac {5}{14} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^2}dx-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {727}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{14} \left (\frac {4093}{42} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {727}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{14} \left (\frac {4093}{42} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{7 \sqrt {1-2 x}}\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {727}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{14} \left (\frac {4093}{42} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {727}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{14} \left (\frac {4093}{42} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {727}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^4}\) |
121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + (-727/(42*Sqrt[1 - 2*x]*(2 + 3*x)^4 ) + (4093*(-1/21*1/(Sqrt[1 - 2*x]*(2 + 3*x)^3) + (-1/14*1/(Sqrt[1 - 2*x]*( 2 + 3*x)^2) + (5*(-1/7*1/(Sqrt[1 - 2*x]*(2 + 3*x)) + (3*(2/(7*Sqrt[1 - 2*x ]) - (2*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7))/7))/14)/3))/42)/14
3.22.56.3.1 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] :> 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*((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 = 1.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {6630660 x^{5}+11787840 x^{4}+3769653 x^{3}-3646863 x^{2}-2528226 x -401410}{403368 \left (2+3 x \right )^{4} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {20465 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1411788}\) | \(68\) |
| pseudoelliptic | \(\frac {\frac {20465 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{4} \sqrt {21}}{1411788}-\frac {552555 x^{5}}{33614}-\frac {491160 x^{4}}{16807}-\frac {1256551 x^{3}}{134456}+\frac {1215621 x^{2}}{134456}+\frac {421371 x}{67228}+\frac {200705}{201684}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{4}}\) | \(79\) |
| derivativedivides | \(\frac {\frac {1159245 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {1220439 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {425155 \left (1-2 x \right )^{\frac {3}{2}}}{9604}-\frac {49065 \sqrt {1-2 x}}{1372}}{\left (-4-6 x \right )^{4}}-\frac {20465 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1411788}+\frac {968}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {8360}{117649 \sqrt {1-2 x}}\) | \(84\) |
| default | \(\frac {\frac {1159245 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {1220439 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {425155 \left (1-2 x \right )^{\frac {3}{2}}}{9604}-\frac {49065 \sqrt {1-2 x}}{1372}}{\left (-4-6 x \right )^{4}}-\frac {20465 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1411788}+\frac {968}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {8360}{117649 \sqrt {1-2 x}}\) | \(84\) |
| trager | \(-\frac {\left (6630660 x^{5}+11787840 x^{4}+3769653 x^{3}-3646863 x^{2}-2528226 x -401410\right ) \sqrt {1-2 x}}{403368 \left (2+3 x \right )^{4} \left (-1+2 x \right )^{2}}-\frac {20465 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2823576}\) | \(94\) |
1/403368*(6630660*x^5+11787840*x^4+3769653*x^3-3646863*x^2-2528226*x-40141 0)/(2+3*x)^4/(1-2*x)^(1/2)/(-1+2*x)-20465/1411788*arctanh(1/7*21^(1/2)*(1- 2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.91 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {20465 \, \sqrt {21} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 7 \, {\left (6630660 \, x^{5} + 11787840 \, x^{4} + 3769653 \, x^{3} - 3646863 \, x^{2} - 2528226 \, x - 401410\right )} \sqrt {-2 \, x + 1}}{2823576 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]
1/2823576*(20465*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)) - 7*(66306 60*x^5 + 11787840*x^4 + 3769653*x^3 - 3646863*x^2 - 2528226*x - 401410)*sq rt(-2*x + 1))/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)
Timed out. \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {20465}{2823576} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1657665 \, {\left (2 \, x - 1\right )}^{5} + 14182245 \, {\left (2 \, x - 1\right )}^{4} + 43921983 \, {\left (2 \, x - 1\right )}^{3} + 55955403 \, {\left (2 \, x - 1\right )}^{2} + 36945216 \, x - 27769280}{201684 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 2401 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
20465/2823576*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq rt(-2*x + 1))) - 1/201684*(1657665*(2*x - 1)^5 + 14182245*(2*x - 1)^4 + 43 921983*(2*x - 1)^3 + 55955403*(2*x - 1)^2 + 36945216*x - 27769280)/(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 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {20465}{2823576} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {176 \, {\left (285 \, x - 181\right )}}{352947 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {1159245 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 8543073 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 20832595 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 16829295 \, \sqrt {-2 \, x + 1}}{7529536 \, {\left (3 \, x + 2\right )}^{4}} \]
20465/2823576*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) + 176/352947*(285*x - 181)/((2*x - 1)*sqrt(-2*x + 1)) - 1/7529536*(1159245*(2*x - 1)^3*sqrt(-2*x + 1) + 8543073*(2*x - 1)^2* sqrt(-2*x + 1) - 20832595*(-2*x + 1)^(3/2) + 16829295*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 1.49 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.77 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=-\frac {20465\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1411788}-\frac {\frac {2992\,x}{1323}+\frac {126883\,{\left (2\,x-1\right )}^2}{37044}+\frac {298789\,{\left (2\,x-1\right )}^3}{111132}+\frac {225115\,{\left (2\,x-1\right )}^4}{259308}+\frac {20465\,{\left (2\,x-1\right )}^5}{201684}-\frac {20240}{11907}}{\frac {2401\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{7/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{9/2}}{3}+{\left (1-2\,x\right )}^{11/2}} \]
- (20465*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1411788 - ((2992*x) /1323 + (126883*(2*x - 1)^2)/37044 + (298789*(2*x - 1)^3)/111132 + (225115 *(2*x - 1)^4)/259308 + (20465*(2*x - 1)^5)/201684 - 20240/11907)/((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))