Integrand size = 22, antiderivative size = 88 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac {52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac {52 \sqrt {1-2 x}}{189 (2+3 x)}-\frac {104 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}} \]
1/63*(1-2*x)^(5/2)/(2+3*x)^3-52/189*(1-2*x)^(3/2)/(2+3*x)^2-104/3969*arcta nh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+52/189*(1-2*x)^(1/2)/(2+3*x)
Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {8 \left (\frac {21 \sqrt {1-2 x} \left (107+664 x+792 x^2\right )}{8 (2+3 x)^3}-13 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{3969} \]
(8*((21*Sqrt[1 - 2*x]*(107 + 664*x + 792*x^2))/(8*(2 + 3*x)^3) - 13*Sqrt[2 1]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]))/3969
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, 51, 51, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)}{(3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {104}{63} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^3}dx+\frac {(1-2 x)^{5/2}}{63 (3 x+2)^3}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {104}{63} \left (-\frac {1}{2} \int \frac {\sqrt {1-2 x}}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )+\frac {(1-2 x)^{5/2}}{63 (3 x+2)^3}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {104}{63} \left (\frac {1}{2} \left (\frac {1}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {\sqrt {1-2 x}}{3 (3 x+2)}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )+\frac {(1-2 x)^{5/2}}{63 (3 x+2)^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {104}{63} \left (\frac {1}{2} \left (\frac {\sqrt {1-2 x}}{3 (3 x+2)}-\frac {1}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )+\frac {(1-2 x)^{5/2}}{63 (3 x+2)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {104}{63} \left (\frac {1}{2} \left (\frac {\sqrt {1-2 x}}{3 (3 x+2)}-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )+\frac {(1-2 x)^{5/2}}{63 (3 x+2)^3}\) |
(1 - 2*x)^(5/2)/(63*(2 + 3*x)^3) + (104*(-1/6*(1 - 2*x)^(3/2)/(2 + 3*x)^2 + (Sqrt[1 - 2*x]/(3*(2 + 3*x)) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*S qrt[21]))/2))/63
3.19.67.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[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 = 0.99 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(-\frac {1584 x^{3}+536 x^{2}-450 x -107}{189 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {104 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\) | \(51\) |
| pseudoelliptic | \(\frac {-104 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}+21 \sqrt {1-2 x}\, \left (792 x^{2}+664 x +107\right )}{3969 \left (2+3 x \right )^{3}}\) | \(55\) |
| derivativedivides | \(\frac {-\frac {176 \left (1-2 x \right )^{\frac {5}{2}}}{21}+\frac {832 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {728 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{3}}-\frac {104 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\) | \(57\) |
| default | \(\frac {-\frac {176 \left (1-2 x \right )^{\frac {5}{2}}}{21}+\frac {832 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {728 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{3}}-\frac {104 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\) | \(57\) |
| trager | \(\frac {\left (792 x^{2}+664 x +107\right ) \sqrt {1-2 x}}{189 \left (2+3 x \right )^{3}}+\frac {52 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{3969}\) | \(72\) |
-1/189*(1584*x^3+536*x^2-450*x-107)/(2+3*x)^3/(1-2*x)^(1/2)-104/3969*arcta nh(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 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {52 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (792 \, x^{2} + 664 \, x + 107\right )} \sqrt {-2 \, x + 1}}{3969 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
1/3969*(52*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*(792*x^2 + 664*x + 107)*sqrt(-2*x + 1))/(27 *x^3 + 54*x^2 + 36*x + 8)
Time = 151.43 (sec) , antiderivative size = 542, normalized size of antiderivative = 6.16 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {20 \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 {64 \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 {728 \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 )}{9} + \frac {784 \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 )}{27} \]
20*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21 )/3))/567 + 64*Piecewise((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) < sqrt(21)/3)))/3 + 728*Piecewise((sqrt(21)*(3*log(sqr t(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) < sqr t(21)/3)))/9 + 784*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)))/2 7
Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {52}{3969} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {8 \, {\left (198 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 728 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 637 \, \sqrt {-2 \, x + 1}\right )}}{189 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
52/3969*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2* x + 1))) + 8/189*(198*(-2*x + 1)^(5/2) - 728*(-2*x + 1)^(3/2) + 637*sqrt(- 2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {52}{3969} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {198 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 728 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 637 \, \sqrt {-2 \, x + 1}}{189 \, {\left (3 \, x + 2\right )}^{3}} \]
52/3969*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) + 1/189*(198*(2*x - 1)^2*sqrt(-2*x + 1) - 728*(-2*x + 1) ^(3/2) + 637*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {\frac {728\,\sqrt {1-2\,x}}{729}-\frac {832\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {176\,{\left (1-2\,x\right )}^{5/2}}{567}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}-\frac {104\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3969} \]