Integrand size = 22, antiderivative size = 69 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{2+3 x} \, dx=-\frac {14}{27} \sqrt {1-2 x}-\frac {2}{27} (1-2 x)^{3/2}-\frac {1}{3} (1-2 x)^{5/2}+\frac {14}{27} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
-2/27*(1-2*x)^(3/2)-1/3*(1-2*x)^(5/2)+14/81*arctanh(1/7*21^(1/2)*(1-2*x)^( 1/2))*21^(1/2)-14/27*(1-2*x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{2+3 x} \, dx=\frac {1}{81} \left (-3 \sqrt {1-2 x} \left (25-40 x+36 x^2\right )+14 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \]
Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {90, 60, 60, 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.
\(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)}{3 x+2} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -\frac {1}{3} \int \frac {(1-2 x)^{3/2}}{3 x+2}dx-\frac {1}{3} (1-2 x)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} \left (-\frac {7}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx-\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {1}{3} (1-2 x)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} \left (-\frac {7}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )-\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {1}{3} (1-2 x)^{5/2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (-\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {1}{3} (1-2 x)^{5/2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (-\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {1}{3} (1-2 x)^{5/2}\) |
-1/3*(1 - 2*x)^(5/2) + ((-2*(1 - 2*x)^(3/2))/9 - (7*((2*Sqrt[1 - 2*x])/3 - (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3))/3)/3
3.19.64.3.1 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 + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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 = 0.96 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
pseudoelliptic | \(\frac {14 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{81}-\frac {\sqrt {1-2 x}\, \left (36 x^{2}-40 x +25\right )}{27}\) | \(39\) |
risch | \(\frac {\left (36 x^{2}-40 x +25\right ) \left (-1+2 x \right )}{27 \sqrt {1-2 x}}+\frac {14 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{81}\) | \(44\) |
derivativedivides | \(-\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {\left (1-2 x \right )^{\frac {5}{2}}}{3}+\frac {14 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{81}-\frac {14 \sqrt {1-2 x}}{27}\) | \(47\) |
default | \(-\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {\left (1-2 x \right )^{\frac {5}{2}}}{3}+\frac {14 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{81}-\frac {14 \sqrt {1-2 x}}{27}\) | \(47\) |
trager | \(\left (-\frac {4}{3} x^{2}+\frac {40}{27} x -\frac {25}{27}\right ) \sqrt {1-2 x}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{81}\) | \(64\) |
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{2+3 x} \, dx=\frac {7}{81} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) - \frac {1}{27} \, {\left (36 \, x^{2} - 40 \, x + 25\right )} \sqrt {-2 \, x + 1} \]
7/81*sqrt(7)*sqrt(3)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) - 1/27*(36*x^2 - 40*x + 25)*sqrt(-2*x + 1)
Time = 1.36 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{2+3 x} \, dx=- \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{3} - \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{27} - \frac {14 \sqrt {1 - 2 x}}{27} - \frac {7 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{81} \]
-(1 - 2*x)**(5/2)/3 - 2*(1 - 2*x)**(3/2)/27 - 14*sqrt(1 - 2*x)/27 - 7*sqrt (21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/8 1
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{2+3 x} \, dx=-\frac {1}{3} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {2}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7}{81} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {14}{27} \, \sqrt {-2 \, x + 1} \]
-1/3*(-2*x + 1)^(5/2) - 2/27*(-2*x + 1)^(3/2) - 7/81*sqrt(21)*log(-(sqrt(2 1) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 14/27*sqrt(-2*x + 1)
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{2+3 x} \, dx=-\frac {1}{3} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {2}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7}{81} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {14}{27} \, \sqrt {-2 \, x + 1} \]
-1/3*(2*x - 1)^2*sqrt(-2*x + 1) - 2/27*(-2*x + 1)^(3/2) - 7/81*sqrt(21)*lo g(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 14/27*sqrt(-2*x + 1)
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{2+3 x} \, dx=-\frac {14\,\sqrt {1-2\,x}}{27}-\frac {2\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {{\left (1-2\,x\right )}^{5/2}}{3}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,14{}\mathrm {i}}{81} \]