Integrand size = 24, antiderivative size = 95 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)} \, dx=\frac {\sqrt {1-2 x}}{2 (2+3 x)^2}+\frac {69 \sqrt {1-2 x}}{14 (2+3 x)}+\frac {793}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-10 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
793/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-10*arctanh(1/11*55^(1/ 2)*(1-2*x)^(1/2))*55^(1/2)+1/2*(1-2*x)^(1/2)/(2+3*x)^2+69/14*(1-2*x)^(1/2) /(2+3*x)
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} (145+207 x)}{14 (2+3 x)^2}+\frac {793}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-10 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(145 + 207*x))/(14*(2 + 3*x)^2) + (793*Sqrt[3/7]*ArcTanh[Sq rt[3/7]*Sqrt[1 - 2*x]])/7 - 10*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {110, 25, 168, 174, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^3 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\sqrt {1-2 x}}{2 (3 x+2)^2}-\frac {1}{2} \int -\frac {13-15 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \int \frac {13-15 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \int \frac {563-345 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {69 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (3850 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-2379 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {69 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (2379 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-3850 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {69 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (1586 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-140 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {69 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2}\) |
Sqrt[1 - 2*x]/(2*(2 + 3*x)^2) + ((69*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + (1586* Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 140*Sqrt[55]*ArcTanh[Sqrt[5/1 1]*Sqrt[1 - 2*x]])/7)/2
3.19.35.3.1 Defintions of rubi rules used
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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 = 1.00 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {414 x^{2}+83 x -145}{14 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {793 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}-10 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(64\) |
| derivativedivides | \(-10 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {18 \left (\frac {23 \left (1-2 x \right )^{\frac {3}{2}}}{14}-\frac {71 \sqrt {1-2 x}}{18}\right )}{\left (-4-6 x \right )^{2}}+\frac {793 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(66\) |
| default | \(-10 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {18 \left (\frac {23 \left (1-2 x \right )^{\frac {3}{2}}}{14}-\frac {71 \sqrt {1-2 x}}{18}\right )}{\left (-4-6 x \right )^{2}}+\frac {793 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(66\) |
| pseudoelliptic | \(\frac {1586 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}-980 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{2} \sqrt {55}+7 \sqrt {1-2 x}\, \left (207 x +145\right )}{98 \left (2+3 x \right )^{2}}\) | \(75\) |
| trager | \(\frac {\left (207 x +145\right ) \sqrt {1-2 x}}{14 \left (2+3 x \right )^{2}}+\frac {793 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{98}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )\) | \(111\) |
-1/14*(414*x^2+83*x-145)/(2+3*x)^2/(1-2*x)^(1/2)+793/49*arctanh(1/7*21^(1/ 2)*(1-2*x)^(1/2))*21^(1/2)-10*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2 )
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)} \, dx=\frac {793 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 490 \, \sqrt {55} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 7 \, {\left (207 \, x + 145\right )} \sqrt {-2 \, x + 1}}{98 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/98*(793*sqrt(7)*sqrt(3)*(9*x^2 + 12*x + 4)*log(-(sqrt(7)*sqrt(3)*sqrt(-2 *x + 1) - 3*x + 5)/(3*x + 2)) + 490*sqrt(55)*(9*x^2 + 12*x + 4)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 7*(207*x + 145)*sqrt(-2*x + 1)) /(9*x^2 + 12*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (80) = 160\).
Time = 39.64 (sec) , antiderivative size = 367, normalized size of antiderivative = 3.86 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)} \, dx=- \frac {55 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{7} + 5 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right ) + 132 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 56 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) \]
-55*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(2 1)/3))/7 + 5*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5)) + 132*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) > -sq rt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3))) - 56*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)))
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)} \, dx=5 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {793}{98} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {207 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 497 \, \sqrt {-2 \, x + 1}}{7 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
5*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1) )) - 793/98*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt (-2*x + 1))) - 1/7*(207*(-2*x + 1)^(3/2) - 497*sqrt(-2*x + 1))/(9*(2*x - 1 )^2 + 84*x + 7)
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)} \, dx=5 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {793}{98} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {207 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 497 \, \sqrt {-2 \, x + 1}}{28 \, {\left (3 \, x + 2\right )}^{2}} \]
5*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt (-2*x + 1))) - 793/98*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1)) /(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/28*(207*(-2*x + 1)^(3/2) - 497*sqrt(-2 *x + 1))/(3*x + 2)^2
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)} \, dx=\frac {793\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-10\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {71\,\sqrt {1-2\,x}}{9}-\frac {23\,{\left (1-2\,x\right )}^{3/2}}{7}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \]