Integrand size = 24, antiderivative size = 74 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {142}{189} \sqrt {1-2 x}-\frac {25}{27} (1-2 x)^{3/2}-\frac {(1-2 x)^{3/2}}{63 (2+3 x)}+\frac {142 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}} \]
-25/27*(1-2*x)^(3/2)-1/63*(1-2*x)^(3/2)/(2+3*x)+142/567*arctanh(1/7*21^(1/ 2)*(1-2*x)^(1/2))*21^(1/2)-142/189*(1-2*x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx=\frac {\sqrt {1-2 x} \left (-91-35 x+150 x^2\right )}{54+81 x}+\frac {142 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}} \]
(Sqrt[1 - 2*x]*(-91 - 35*x + 150*x^2))/(54 + 81*x) + (142*ArcTanh[Sqrt[3/7 ]*Sqrt[1 - 2*x]])/(27*Sqrt[21])
Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 27, 90, 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.
\(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {1}{63} \int \frac {3 \sqrt {1-2 x} (175 x+93)}{3 x+2}dx-\frac {(1-2 x)^{3/2}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{21} \int \frac {\sqrt {1-2 x} (175 x+93)}{3 x+2}dx-\frac {(1-2 x)^{3/2}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{21} \left (-\frac {71}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx-\frac {175}{9} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{3/2}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{21} \left (-\frac {71}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )-\frac {175}{9} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{3/2}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{21} \left (-\frac {71}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {175}{9} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{3/2}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{21} \left (-\frac {71}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {175}{9} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{3/2}}{63 (3 x+2)}\) |
-1/63*(1 - 2*x)^(3/2)/(2 + 3*x) + ((-175*(1 - 2*x)^(3/2))/9 - (71*((2*Sqrt [1 - 2*x])/3 - (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3))/3)/21
3.19.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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*((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 = 0.96 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {300 x^{3}-220 x^{2}-147 x +91}{27 \left (2+3 x \right ) \sqrt {1-2 x}}+\frac {142 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{567}\) | \(51\) |
pseudoelliptic | \(\frac {142 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}+21 \sqrt {1-2 x}\, \left (150 x^{2}-35 x -91\right )}{1134+1701 x}\) | \(52\) |
derivativedivides | \(-\frac {25 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {20 \sqrt {1-2 x}}{27}+\frac {2 \sqrt {1-2 x}}{81 \left (-\frac {4}{3}-2 x \right )}+\frac {142 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{567}\) | \(54\) |
default | \(-\frac {25 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {20 \sqrt {1-2 x}}{27}+\frac {2 \sqrt {1-2 x}}{81 \left (-\frac {4}{3}-2 x \right )}+\frac {142 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{567}\) | \(54\) |
trager | \(\frac {\sqrt {1-2 x}\, \left (150 x^{2}-35 x -91\right )}{54+81 x}-\frac {71 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{567}\) | \(72\) |
-1/27*(300*x^3-220*x^2-147*x+91)/(2+3*x)/(1-2*x)^(1/2)+142/567*arctanh(1/7 *21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx=\frac {71 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (150 \, x^{2} - 35 \, x - 91\right )} \sqrt {-2 \, x + 1}}{567 \, {\left (3 \, x + 2\right )}} \]
1/567*(71*sqrt(21)*(3*x + 2)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(150*x^2 - 35*x - 91)*sqrt(-2*x + 1))/(3*x + 2)
Time = 32.62 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.53 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx=- \frac {25 \left (1 - 2 x\right )^{\frac {3}{2}}}{27} - \frac {20 \sqrt {1 - 2 x}}{27} - \frac {8 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{63} - \frac {28 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{27} \]
-25*(1 - 2*x)**(3/2)/27 - 20*sqrt(1 - 2*x)/27 - 8*sqrt(21)*(log(sqrt(1 - 2 *x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/63 - 28*Piecewise((sq rt(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(2 1)/3)))/27
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {25}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {71}{567} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20}{27} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{27 \, {\left (3 \, x + 2\right )}} \]
-25/27*(-2*x + 1)^(3/2) - 71/567*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1 ))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 20/27*sqrt(-2*x + 1) - 1/27*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {25}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {71}{567} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {20}{27} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{27 \, {\left (3 \, x + 2\right )}} \]
-25/27*(-2*x + 1)^(3/2) - 71/567*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt (-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 20/27*sqrt(-2*x + 1) - 1/27*s qrt(-2*x + 1)/(3*x + 2)
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {2\,\sqrt {1-2\,x}}{81\,\left (2\,x+\frac {4}{3}\right )}-\frac {20\,\sqrt {1-2\,x}}{27}-\frac {25\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,142{}\mathrm {i}}{567} \]