Integrand size = 22, antiderivative size = 88 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {\sqrt {1-2 x}}{63 (2+3 x)^3}-\frac {50 \sqrt {1-2 x}}{441 (2+3 x)^2}-\frac {50 \sqrt {1-2 x}}{1029 (2+3 x)}-\frac {100 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
-100/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/63*(1-2*x)^(1/2) /(2+3*x)^3-50/441*(1-2*x)^(1/2)/(2+3*x)^2-50/1029*(1-2*x)^(1/2)/(2+3*x)
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {4 \left (-\frac {21 \sqrt {1-2 x} \left (417+950 x+450 x^2\right )}{4 (2+3 x)^3}-25 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{21609} \]
(4*((-21*Sqrt[1 - 2*x]*(417 + 950*x + 450*x^2))/(4*(2 + 3*x)^3) - 25*Sqrt[ 21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]))/21609
Time = 0.18 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {87, 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.
\(\displaystyle \int \frac {5 x+3}{\sqrt {1-2 x} (3 x+2)^4} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {100}{63} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x}}{63 (3 x+2)^3}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {100}{63} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{63 (3 x+2)^3}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {100}{63} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{63 (3 x+2)^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {100}{63} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{63 (3 x+2)^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {100}{63} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{63 (3 x+2)^3}\) |
Sqrt[1 - 2*x]/(63*(2 + 3*x)^3) + (100*(-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))/63
3.21.10.3.1 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_))*((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 = 1.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.58
method | result | size |
risch | \(\frac {900 x^{3}+1450 x^{2}-116 x -417}{1029 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {100 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(51\) |
pseudoelliptic | \(\frac {-100 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}-21 \sqrt {1-2 x}\, \left (450 x^{2}+950 x +417\right )}{21609 \left (2+3 x \right )^{3}}\) | \(55\) |
derivativedivides | \(\frac {\frac {300 \left (1-2 x \right )^{\frac {5}{2}}}{343}-\frac {800 \left (1-2 x \right )^{\frac {3}{2}}}{147}+\frac {164 \sqrt {1-2 x}}{21}}{\left (-4-6 x \right )^{3}}-\frac {100 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(57\) |
default | \(\frac {\frac {300 \left (1-2 x \right )^{\frac {5}{2}}}{343}-\frac {800 \left (1-2 x \right )^{\frac {3}{2}}}{147}+\frac {164 \sqrt {1-2 x}}{21}}{\left (-4-6 x \right )^{3}}-\frac {100 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(57\) |
trager | \(-\frac {\left (450 x^{2}+950 x +417\right ) \sqrt {1-2 x}}{1029 \left (2+3 x \right )^{3}}-\frac {50 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{21609}\) | \(72\) |
1/1029*(900*x^3+1450*x^2-116*x-417)/(2+3*x)^3/(1-2*x)^(1/2)-100/21609*arct anh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {50 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (450 \, x^{2} + 950 \, x + 417\right )} \sqrt {-2 \, x + 1}}{21609 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
1/21609*(50*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*sqrt (-2*x + 1) - 5)/(3*x + 2)) - 21*(450*x^2 + 950*x + 417)*sqrt(-2*x + 1))/(2 7*x^3 + 54*x^2 + 36*x + 8)
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (75) = 150\).
Time = 107.25 (sec) , antiderivative size = 379, normalized size of antiderivative = 4.31 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {40 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} + \frac {16 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{32} + \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{32} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}}\right )}{7203} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} \]
40*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqr t(21)*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/ (16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -s qrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/3 + 16*Piecewise((sqrt(21)*(-5 *log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1 )/32 - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2* x)/7 + 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21) *sqrt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48* (sqrt(21)*sqrt(1 - 2*x)/7 - 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/3
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {50}{21609} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4 \, {\left (225 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 1400 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2009 \, \sqrt {-2 \, x + 1}\right )}}{1029 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
50/21609*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 *x + 1))) - 4/1029*(225*(-2*x + 1)^(5/2) - 1400*(-2*x + 1)^(3/2) + 2009*sq rt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {50}{21609} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {225 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1400 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2009 \, \sqrt {-2 \, x + 1}}{2058 \, {\left (3 \, x + 2\right )}^{3}} \]
50/21609*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/2058*(225*(2*x - 1)^2*sqrt(-2*x + 1) - 1400*(-2*x + 1)^(3/2) + 2009*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 1.55 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.82 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^4} \, dx=-\frac {100\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}-\frac {\frac {164\,\sqrt {1-2\,x}}{567}-\frac {800\,{\left (1-2\,x\right )}^{3/2}}{3969}+\frac {100\,{\left (1-2\,x\right )}^{5/2}}{3087}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]