Integrand size = 24, antiderivative size = 61 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx=-\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (2+3 x)}+\frac {46 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]
46/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-25/9*(1-2*x)^(1/2)-1/6 3*(1-2*x)^(1/2)/(2+3*x)
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx=-\frac {\sqrt {1-2 x} (117+175 x)}{42+63 x}+\frac {46 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]
-((Sqrt[1 - 2*x]*(117 + 175*x))/(42 + 63*x)) + (46*ArcTanh[Sqrt[3/7]*Sqrt[ 1 - 2*x]])/(21*Sqrt[21])
Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {100, 90, 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.
\(\displaystyle \int \frac {(5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {1}{63} \int \frac {525 x+281}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{63} \left (-69 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-175 \sqrt {1-2 x}\right )-\frac {\sqrt {1-2 x}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{63} \left (69 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-175 \sqrt {1-2 x}\right )-\frac {\sqrt {1-2 x}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{63} \left (46 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-175 \sqrt {1-2 x}\right )-\frac {\sqrt {1-2 x}}{63 (3 x+2)}\) |
-1/63*Sqrt[1 - 2*x]/(2 + 3*x) + (-175*Sqrt[1 - 2*x] + 46*Sqrt[3/7]*ArcTanh [Sqrt[3/7]*Sqrt[1 - 2*x]])/63
3.21.19.3.1 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_))*((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 = 3.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(-\frac {25 \sqrt {1-2 x}}{9}+\frac {2 \sqrt {1-2 x}}{189 \left (-\frac {4}{3}-2 x \right )}+\frac {46 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(45\) |
default | \(-\frac {25 \sqrt {1-2 x}}{9}+\frac {2 \sqrt {1-2 x}}{189 \left (-\frac {4}{3}-2 x \right )}+\frac {46 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(45\) |
risch | \(\frac {350 x^{2}+59 x -117}{21 \left (2+3 x \right ) \sqrt {1-2 x}}+\frac {46 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(46\) |
pseudoelliptic | \(\frac {46 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}-21 \sqrt {1-2 x}\, \left (117+175 x \right )}{882+1323 x}\) | \(47\) |
trager | \(-\frac {\left (117+175 x \right ) \sqrt {1-2 x}}{21 \left (2+3 x \right )}+\frac {23 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{441}\) | \(67\) |
-25/9*(1-2*x)^(1/2)+2/189*(1-2*x)^(1/2)/(-4/3-2*x)+46/441*arctanh(1/7*21^( 1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx=\frac {23 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (175 \, x + 117\right )} \sqrt {-2 \, x + 1}}{441 \, {\left (3 \, x + 2\right )}} \]
1/441*(23*sqrt(21)*(3*x + 2)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(175*x + 117)*sqrt(-2*x + 1))/(3*x + 2)
Time = 32.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.87 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx=- \frac {25 \sqrt {1 - 2 x}}{9} - \frac {10 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{189} - \frac {4 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} \]
-25*sqrt(1 - 2*x)/9 - 10*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(s qrt(1 - 2*x) + sqrt(21)/3))/189 - 4*Piecewise((sqrt(21)*(-log(sqrt(21)*sqr t(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)))/9
Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.02 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx=-\frac {23}{441} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {25}{9} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{63 \, {\left (3 \, x + 2\right )}} \]
-23/441*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2* x + 1))) - 25/9*sqrt(-2*x + 1) - 1/63*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx=-\frac {23}{441} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {25}{9} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{63 \, {\left (3 \, x + 2\right )}} \]
-23/441*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) - 25/9*sqrt(-2*x + 1) - 1/63*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx=\frac {46\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441}-\frac {2\,\sqrt {1-2\,x}}{189\,\left (2\,x+\frac {4}{3}\right )}-\frac {25\,\sqrt {1-2\,x}}{9} \]