Integrand size = 22, antiderivative size = 121 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {655}{28812 \sqrt {1-2 x}}+\frac {1}{84 \sqrt {1-2 x} (2+3 x)^4}-\frac {131}{1764 \sqrt {1-2 x} (2+3 x)^3}-\frac {131}{3528 \sqrt {1-2 x} (2+3 x)^2}-\frac {655}{24696 \sqrt {1-2 x} (2+3 x)}-\frac {655 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604 \sqrt {21}} \]
-655/201684*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+655/28812/(1-2*x) ^(1/2)+1/84/(2+3*x)^4/(1-2*x)^(1/2)-131/1764/(2+3*x)^3/(1-2*x)^(1/2)-131/3 528/(2+3*x)^2/(1-2*x)^(1/2)-655/24696/(2+3*x)/(1-2*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {\frac {21 \left (-2566+10742 x+60391 x^2+80565 x^3+35370 x^4\right )}{2 \sqrt {1-2 x} (2+3 x)^4}-655 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{201684} \]
((21*(-2566 + 10742*x + 60391*x^2 + 80565*x^3 + 35370*x^4))/(2*Sqrt[1 - 2* x]*(2 + 3*x)^4) - 655*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/201684
Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {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.
| \(\displaystyle \int \frac {5 x+3}{(1-2 x)^{3/2} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {131}{84} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^4}dx+\frac {1}{84 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {131}{84} \left (\frac {1}{3} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^3}dx-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {1}{84 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {131}{84} \left (\frac {1}{3} \left (\frac {5}{14} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^2}dx-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {1}{84 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {131}{84} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {1}{84 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {131}{84} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{7 \sqrt {1-2 x}}\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {1}{84 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {131}{84} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {1}{84 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {131}{84} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {1}{84 \sqrt {1-2 x} (3 x+2)^4}\) |
1/(84*Sqrt[1 - 2*x]*(2 + 3*x)^4) + (131*(-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))/84
3.21.81.3.1 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 = 1.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(\frac {35370 x^{4}+80565 x^{3}+60391 x^{2}+10742 x -2566}{19208 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {655 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{201684}\) | \(56\) |
| pseudoelliptic | \(-\frac {17685 \left (\sqrt {21}\, \left (\frac {2}{3}+x \right )^{4} \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )-7 x^{4}-\frac {287 x^{3}}{18}-\frac {3227 x^{2}}{270}-\frac {287 x}{135}+\frac {8981}{17685}\right )}{67228 \sqrt {1-2 x}\, \left (2+3 x \right )^{4}}\) | \(66\) |
| derivativedivides | \(\frac {\frac {66771 \left (1-2 x \right )^{\frac {7}{2}}}{67228}-\frac {75273 \left (1-2 x \right )^{\frac {5}{2}}}{9604}+\frac {28925 \left (1-2 x \right )^{\frac {3}{2}}}{1372}-\frac {3735 \sqrt {1-2 x}}{196}}{\left (-4-6 x \right )^{4}}-\frac {655 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{201684}+\frac {176}{16807 \sqrt {1-2 x}}\) | \(75\) |
| default | \(\frac {\frac {66771 \left (1-2 x \right )^{\frac {7}{2}}}{67228}-\frac {75273 \left (1-2 x \right )^{\frac {5}{2}}}{9604}+\frac {28925 \left (1-2 x \right )^{\frac {3}{2}}}{1372}-\frac {3735 \sqrt {1-2 x}}{196}}{\left (-4-6 x \right )^{4}}-\frac {655 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{201684}+\frac {176}{16807 \sqrt {1-2 x}}\) | \(75\) |
| trager | \(-\frac {\left (35370 x^{4}+80565 x^{3}+60391 x^{2}+10742 x -2566\right ) \sqrt {1-2 x}}{19208 \left (2+3 x \right )^{4} \left (-1+2 x \right )}+\frac {655 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{403368}\) | \(89\) |
1/19208*(35370*x^4+80565*x^3+60391*x^2+10742*x-2566)/(2+3*x)^4/(1-2*x)^(1/ 2)-655/201684*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.94 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {655 \, \sqrt {21} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (35370 \, x^{4} + 80565 \, x^{3} + 60391 \, x^{2} + 10742 \, x - 2566\right )} \sqrt {-2 \, x + 1}}{403368 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
1/403368*(655*sqrt(21)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)* log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(35370*x^4 + 80565 *x^3 + 60391*x^2 + 10742*x - 2566)*sqrt(-2*x + 1))/(162*x^5 + 351*x^4 + 21 6*x^3 - 24*x^2 - 64*x - 16)
Timed out. \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {655}{403368} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {17685 \, {\left (2 \, x - 1\right )}^{4} + 151305 \, {\left (2 \, x - 1\right )}^{3} + 468587 \, {\left (2 \, x - 1\right )}^{2} + 1193934 \, x - 355495}{9604 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2401 \, \sqrt {-2 \, x + 1}\right )}} \]
655/403368*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt( -2*x + 1))) + 1/9604*(17685*(2*x - 1)^4 + 151305*(2*x - 1)^3 + 468587*(2*x - 1)^2 + 1193934*x - 355495)/(81*(-2*x + 1)^(9/2) - 756*(-2*x + 1)^(7/2) + 2646*(-2*x + 1)^(5/2) - 4116*(-2*x + 1)^(3/2) + 2401*sqrt(-2*x + 1))
Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.90 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {655}{403368} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {176}{16807 \, \sqrt {-2 \, x + 1}} - \frac {66771 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 526911 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1417325 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1281105 \, \sqrt {-2 \, x + 1}}{1075648 \, {\left (3 \, x + 2\right )}^{4}} \]
655/403368*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 176/16807/sqrt(-2*x + 1) - 1/1075648*(66771*(2*x - 1)^3*sqrt(-2*x + 1) + 526911*(2*x - 1)^2*sqrt(-2*x + 1) - 1417325*(-2*x + 1)^(3/2) + 1281105*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 1.52 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.81 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {\frac {4061\,x}{2646}+\frac {9563\,{\left (2\,x-1\right )}^2}{15876}+\frac {7205\,{\left (2\,x-1\right )}^3}{37044}+\frac {655\,{\left (2\,x-1\right )}^4}{28812}-\frac {7255}{15876}}{\frac {2401\,\sqrt {1-2\,x}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{3/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{5/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{7/2}}{3}+{\left (1-2\,x\right )}^{9/2}}-\frac {655\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{201684} \]
((4061*x)/2646 + (9563*(2*x - 1)^2)/15876 + (7205*(2*x - 1)^3)/37044 + (65 5*(2*x - 1)^4)/28812 - 7255/15876)/((2401*(1 - 2*x)^(1/2))/81 - (1372*(1 - 2*x)^(3/2))/27 + (98*(1 - 2*x)^(5/2))/3 - (28*(1 - 2*x)^(7/2))/3 + (1 - 2 *x)^(9/2)) - (655*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/201684