Integrand size = 22, antiderivative size = 68 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {\sqrt {1-2 x}}{42 (2+3 x)^2}-\frac {67 \sqrt {1-2 x}}{294 (2+3 x)}-\frac {67 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \]
-67/3087*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/42*(1-2*x)^(1/2)/( 2+3*x)^2-67/294*(1-2*x)^(1/2)/(2+3*x)
Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {\sqrt {1-2 x} (127+201 x)}{294 (2+3 x)^2}-\frac {67 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \]
-1/294*(Sqrt[1 - 2*x]*(127 + 201*x))/(2 + 3*x)^2 - (67*ArcTanh[Sqrt[3/7]*S qrt[1 - 2*x]])/(147*Sqrt[21])
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {87, 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.
\(\displaystyle \int \frac {5 x+3}{\sqrt {1-2 x} (3 x+2)^3} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {67}{42} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {\sqrt {1-2 x}}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {67}{42} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {67}{42} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {67}{42} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{42 (3 x+2)^2}\) |
Sqrt[1 - 2*x]/(42*(2 + 3*x)^2) + (67*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*Ar cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/42
3.21.9.3.1 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 = 3.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {402 x^{2}+53 x -127}{294 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {67 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}\) | \(46\) |
derivativedivides | \(-\frac {36 \left (-\frac {67 \left (1-2 x \right )^{\frac {3}{2}}}{1764}+\frac {65 \sqrt {1-2 x}}{756}\right )}{\left (-4-6 x \right )^{2}}-\frac {67 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}\) | \(48\) |
default | \(-\frac {36 \left (-\frac {67 \left (1-2 x \right )^{\frac {3}{2}}}{1764}+\frac {65 \sqrt {1-2 x}}{756}\right )}{\left (-4-6 x \right )^{2}}-\frac {67 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}\) | \(48\) |
pseudoelliptic | \(\frac {-134 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}-21 \sqrt {1-2 x}\, \left (201 x +127\right )}{6174 \left (2+3 x \right )^{2}}\) | \(50\) |
trager | \(-\frac {\left (201 x +127\right ) \sqrt {1-2 x}}{294 \left (2+3 x \right )^{2}}-\frac {67 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{6174}\) | \(67\) |
1/294*(402*x^2+53*x-127)/(2+3*x)^2/(1-2*x)^(1/2)-67/3087*arctanh(1/7*21^(1 /2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {67 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (201 \, x + 127\right )} \sqrt {-2 \, x + 1}}{6174 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/6174*(67*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(201*x + 127)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (56) = 112\).
Time = 68.07 (sec) , antiderivative size = 289, normalized size of antiderivative = 4.25 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^3} \, dx=- \frac {20 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} - \frac {8 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} \]
-20*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(2 1)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(s qrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt( 1 - 2*x) < sqrt(21)/3)))/3 - 8*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(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) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/3
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {67}{6174} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {201 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 455 \, \sqrt {-2 \, x + 1}}{147 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
67/6174*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2* x + 1))) + 1/147*(201*(-2*x + 1)^(3/2) - 455*sqrt(-2*x + 1))/(9*(2*x - 1)^ 2 + 84*x + 7)
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {67}{6174} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {201 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 455 \, \sqrt {-2 \, x + 1}}{588 \, {\left (3 \, x + 2\right )}^{2}} \]
67/6174*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) + 1/588*(201*(-2*x + 1)^(3/2) - 455*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.79 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {67\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3087}-\frac {\frac {65\,\sqrt {1-2\,x}}{189}-\frac {67\,{\left (1-2\,x\right )}^{3/2}}{441}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \]