Integrand size = 24, antiderivative size = 80 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {\sqrt {1-2 x} (3+5 x)^2}{42 (2+3 x)^2}-\frac {\sqrt {1-2 x} (8329+12425 x)}{882 (2+3 x)}+\frac {2381 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \]
2381/9261*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/42*(3+5*x)^2*(1-2 *x)^(1/2)/(2+3*x)^2-1/882*(8329+12425*x)*(1-2*x)^(1/2)/(2+3*x)
Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {\sqrt {1-2 x} \left (16469+49207 x+36750 x^2\right )}{882 (2+3 x)^2}+\frac {2381 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \]
-1/882*(Sqrt[1 - 2*x]*(16469 + 49207*x + 36750*x^2))/(2 + 3*x)^2 + (2381*A rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(441*Sqrt[21])
Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {109, 25, 163, 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.
\(\displaystyle \int \frac {(5 x+3)^3}{\sqrt {1-2 x} (3 x+2)^3} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {\sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}-\frac {1}{42} \int -\frac {(5 x+3) (355 x+191)}{\sqrt {1-2 x} (3 x+2)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{42} \int \frac {(5 x+3) (355 x+191)}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {\sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 163 |
\(\displaystyle \frac {1}{42} \left (-\frac {2381}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x} (12425 x+8329)}{21 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{42} \left (\frac {2381}{21} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x} (12425 x+8329)}{21 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{42} \left (\frac {4762 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-\frac {\sqrt {1-2 x} (12425 x+8329)}{21 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^2}\) |
(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(42*(2 + 3*x)^2) + (-1/21*(Sqrt[1 - 2*x]*(8329 + 12425*x))/(2 + 3*x) + (4762*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[ 21]))/42
3.21.31.3.1 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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* (m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f *h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* d*(b*c - a*d)*(m + 1)*(m + n + 3)) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]
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 = 1.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.64
method | result | size |
risch | \(\frac {73500 x^{3}+61664 x^{2}-16269 x -16469}{882 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {2381 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\) | \(51\) |
pseudoelliptic | \(\frac {4762 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}-21 \sqrt {1-2 x}\, \left (36750 x^{2}+49207 x +16469\right )}{18522 \left (2+3 x \right )^{2}}\) | \(55\) |
derivativedivides | \(-\frac {125 \sqrt {1-2 x}}{27}-\frac {2 \left (-\frac {69 \left (1-2 x \right )^{\frac {3}{2}}}{98}+\frac {205 \sqrt {1-2 x}}{126}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {2381 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\) | \(57\) |
default | \(-\frac {125 \sqrt {1-2 x}}{27}-\frac {2 \left (-\frac {69 \left (1-2 x \right )^{\frac {3}{2}}}{98}+\frac {205 \sqrt {1-2 x}}{126}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {2381 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\) | \(57\) |
trager | \(-\frac {\left (36750 x^{2}+49207 x +16469\right ) \sqrt {1-2 x}}{882 \left (2+3 x \right )^{2}}-\frac {2381 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{18522}\) | \(72\) |
1/882*(73500*x^3+61664*x^2-16269*x-16469)/(2+3*x)^2/(1-2*x)^(1/2)+2381/926 1*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {2381 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (36750 \, x^{2} + 49207 \, x + 16469\right )} \sqrt {-2 \, x + 1}}{18522 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/18522*(2381*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(36750*x^2 + 49207*x + 16469)*sqrt(-2*x + 1))/(9*x ^2 + 12*x + 4)
Time = 124.91 (sec) , antiderivative size = 343, normalized size of antiderivative = 4.29 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^3} \, dx=- \frac {125 \sqrt {1 - 2 x}}{27} - \frac {25 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{189} - \frac {20 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} - \frac {8 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{27} \]
-125*sqrt(1 - 2*x)/27 - 25*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log (sqrt(1 - 2*x) + sqrt(21)/3))/189 - 20*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, (sq rt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/9 - 8*Piecewis e((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(2 1)*sqrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(1 6*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/27
Time = 0.35 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {2381}{18522} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {125}{27} \, \sqrt {-2 \, x + 1} + \frac {621 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1435 \, \sqrt {-2 \, x + 1}}{1323 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
-2381/18522*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt (-2*x + 1))) - 125/27*sqrt(-2*x + 1) + 1/1323*(621*(-2*x + 1)^(3/2) - 1435 *sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)
Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {2381}{18522} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {125}{27} \, \sqrt {-2 \, x + 1} + \frac {621 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1435 \, \sqrt {-2 \, x + 1}}{5292 \, {\left (3 \, x + 2\right )}^{2}} \]
-2381/18522*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/27*sqrt(-2*x + 1) + 1/5292*(621*(-2*x + 1)^(3/ 2) - 1435*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 1.47 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {2381\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9261}-\frac {125\,\sqrt {1-2\,x}}{27}-\frac {\frac {205\,\sqrt {1-2\,x}}{1701}-\frac {23\,{\left (1-2\,x\right )}^{3/2}}{441}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \]