Integrand size = 24, antiderivative size = 94 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {2873}{567} \sqrt {1-2 x}+\frac {2873 (1-2 x)^{3/2}}{3969}-\frac {(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac {47 (1-2 x)^{5/2}}{294 (2+3 x)}-\frac {2873 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \]
2873/3969*(1-2*x)^(3/2)-1/126*(1-2*x)^(5/2)/(2+3*x)^2+47/294*(1-2*x)^(5/2) /(2+3*x)-2873/1701*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2873/567*( 1-2*x)^(1/2)
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (3803+10195 x+5520 x^2-1800 x^3\right )}{162 (2+3 x)^2}-\frac {2873 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \]
(Sqrt[1 - 2*x]*(3803 + 10195*x + 5520*x^2 - 1800*x^3))/(162*(2 + 3*x)^2) - (2873*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])
Time = 0.19 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 87, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^2}{(3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {1}{126} \int \frac {(1-2 x)^{3/2} (1050 x+559)}{(3 x+2)^2}dx-\frac {(1-2 x)^{5/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{126} \left (\frac {2873}{7} \int \frac {(1-2 x)^{3/2}}{3 x+2}dx+\frac {141 (1-2 x)^{5/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{5/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{126} \left (\frac {2873}{7} \left (\frac {7}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {141 (1-2 x)^{5/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{5/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{126} \left (\frac {2873}{7} \left (\frac {7}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {141 (1-2 x)^{5/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{5/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{126} \left (\frac {2873}{7} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {141 (1-2 x)^{5/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{5/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{126} \left (\frac {2873}{7} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {141 (1-2 x)^{5/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{5/2}}{126 (3 x+2)^2}\) |
-1/126*(1 - 2*x)^(5/2)/(2 + 3*x)^2 + ((141*(1 - 2*x)^(5/2))/(7*(2 + 3*x)) + (2873*((2*(1 - 2*x)^(3/2))/9 + (7*((2*Sqrt[1 - 2*x])/3 - (2*Sqrt[7/3]*Ar cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3))/3))/7)/126
3.19.77.3.1 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*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 = 0.99 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.60
| method | result | size |
| risch | \(\frac {3600 x^{4}-12840 x^{3}-14870 x^{2}+2589 x +3803}{162 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {2873 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1701}\) | \(56\) |
| pseudoelliptic | \(\frac {-5746 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}-21 \sqrt {1-2 x}\, \left (1800 x^{3}-5520 x^{2}-10195 x -3803\right )}{3402 \left (2+3 x \right )^{2}}\) | \(60\) |
| derivativedivides | \(\frac {50 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {130 \sqrt {1-2 x}}{27}+\frac {-\frac {145 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {1001 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{2}}-\frac {2873 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1701}\) | \(66\) |
| default | \(\frac {50 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {130 \sqrt {1-2 x}}{27}+\frac {-\frac {145 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {1001 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{2}}-\frac {2873 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1701}\) | \(66\) |
| trager | \(-\frac {\left (1800 x^{3}-5520 x^{2}-10195 x -3803\right ) \sqrt {1-2 x}}{162 \left (2+3 x \right )^{2}}+\frac {2873 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{3402}\) | \(77\) |
1/162*(3600*x^4-12840*x^3-14870*x^2+2589*x+3803)/(2+3*x)^2/(1-2*x)^(1/2)-2 873/1701*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {2873 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (1800 \, x^{3} - 5520 \, x^{2} - 10195 \, x - 3803\right )} \sqrt {-2 \, x + 1}}{3402 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/3402*(2873*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x + sqrt(21)*sqrt(-2*x + 1 ) - 5)/(3*x + 2)) - 21*(1800*x^3 - 5520*x^2 - 10195*x - 3803)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)
Time = 104.35 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.77 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {50 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} + \frac {130 \sqrt {1 - 2 x}}{27} + \frac {503 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{567} + \frac {2072 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{81} + \frac {392 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{81} \]
50*(1 - 2*x)**(3/2)/81 + 130*sqrt(1 - 2*x)/27 + 503*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/567 + 2072*Piecewis e((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt (1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < s qrt(21)/3)))/81 + 392*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)))/81
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {2873}{3402} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {130}{27} \, \sqrt {-2 \, x + 1} - \frac {435 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1001 \, \sqrt {-2 \, x + 1}}{81 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
50/81*(-2*x + 1)^(3/2) + 2873/3402*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 130/27*sqrt(-2*x + 1) - 1/81*(435*(- 2*x + 1)^(3/2) - 1001*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)
Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {2873}{3402} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {130}{27} \, \sqrt {-2 \, x + 1} - \frac {435 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1001 \, \sqrt {-2 \, x + 1}}{324 \, {\left (3 \, x + 2\right )}^{2}} \]
50/81*(-2*x + 1)^(3/2) + 2873/3402*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sq rt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 130/27*sqrt(-2*x + 1) - 1/3 24*(435*(-2*x + 1)^(3/2) - 1001*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {130\,\sqrt {1-2\,x}}{27}+\frac {50\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {\frac {1001\,\sqrt {1-2\,x}}{729}-\frac {145\,{\left (1-2\,x\right )}^{3/2}}{243}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,2873{}\mathrm {i}}{1701} \]