Integrand size = 24, antiderivative size = 80 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)}+\frac {2 \sqrt {1-2 x} (1978+2975 x)}{147 (2+3 x)}-\frac {68 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \]
-68/3087*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+11/7*(3+5*x)^2/(2+3* x)/(1-2*x)^(1/2)+2/147*(1978+2975*x)*(1-2*x)^(1/2)/(2+3*x)
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {-21 \left (-6035-4968 x+6125 x^2\right )-68 \sqrt {21-42 x} (2+3 x) \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3087 \sqrt {1-2 x} (2+3 x)} \]
(-21*(-6035 - 4968*x + 6125*x^2) - 68*Sqrt[21 - 42*x]*(2 + 3*x)*ArcTanh[Sq rt[3/7]*Sqrt[1 - 2*x]])/(3087*Sqrt[1 - 2*x]*(2 + 3*x))
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, 27, 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.
\(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^{3/2} (3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)}-\frac {1}{7} \int \frac {2 (5 x+3) (85 x+62)}{\sqrt {1-2 x} (3 x+2)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)}-\frac {2}{7} \int \frac {(5 x+3) (85 x+62)}{\sqrt {1-2 x} (3 x+2)^2}dx\) |
\(\Big \downarrow \) 163 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)}-\frac {2}{7} \left (-\frac {17}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x} (2975 x+1978)}{21 (3 x+2)}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)}-\frac {2}{7} \left (\frac {17}{21} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x} (2975 x+1978)}{21 (3 x+2)}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)}-\frac {2}{7} \left (\frac {34 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-\frac {\sqrt {1-2 x} (2975 x+1978)}{21 (3 x+2)}\right )\) |
(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)) - (2*(-1/21*(Sqrt[1 - 2*x]*(1 978 + 2975*x))/(2 + 3*x) + (34*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[ 21])))/7
3.21.100.3.1 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] :> 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 = 1.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.58
method | result | size |
risch | \(-\frac {6125 x^{2}-4968 x -6035}{147 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}\) | \(46\) |
derivativedivides | \(\frac {125 \sqrt {1-2 x}}{18}-\frac {2 \sqrt {1-2 x}}{1323 \left (-\frac {4}{3}-2 x \right )}-\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}+\frac {1331}{98 \sqrt {1-2 x}}\) | \(54\) |
default | \(\frac {125 \sqrt {1-2 x}}{18}-\frac {2 \sqrt {1-2 x}}{1323 \left (-\frac {4}{3}-2 x \right )}-\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}+\frac {1331}{98 \sqrt {1-2 x}}\) | \(54\) |
pseudoelliptic | \(-\frac {204 \left (\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}}{3}+\frac {42875 x^{2}}{68}-\frac {8694 x}{17}-\frac {2485}{4}\right )}{\sqrt {1-2 x}\, \left (6174+9261 x \right )}\) | \(57\) |
trager | \(\frac {\left (6125 x^{2}-4968 x -6035\right ) \sqrt {1-2 x}}{882 x^{2}+147 x -294}+\frac {34 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{3087}\) | \(75\) |
-1/147*(6125*x^2-4968*x-6035)/(2+3*x)/(1-2*x)^(1/2)-68/3087*arctanh(1/7*21 ^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {34 \, \sqrt {21} {\left (6 \, x^{2} + x - 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (6125 \, x^{2} - 4968 \, x - 6035\right )} \sqrt {-2 \, x + 1}}{3087 \, {\left (6 \, x^{2} + x - 2\right )}} \]
1/3087*(34*sqrt(21)*(6*x^2 + x - 2)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5 )/(3*x + 2)) + 21*(6125*x^2 - 4968*x - 6035)*sqrt(-2*x + 1))/(6*x^2 + x - 2)
Time = 54.16 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.31 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {125 \sqrt {1 - 2 x}}{18} + \frac {103 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{9261} + \frac {4 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{63} + \frac {1331}{98 \sqrt {1 - 2 x}} \]
125*sqrt(1 - 2*x)/18 + 103*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log (sqrt(1 - 2*x) + sqrt(21)/3))/9261 + 4*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)))/63 + 1331/(98* sqrt(1 - 2*x))
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {34}{3087} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {125}{18} \, \sqrt {-2 \, x + 1} - \frac {35933 \, x + 23960}{441 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \]
34/3087*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2* x + 1))) + 125/18*sqrt(-2*x + 1) - 1/441*(35933*x + 23960)/(3*(-2*x + 1)^( 3/2) - 7*sqrt(-2*x + 1))
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {34}{3087} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {125}{18} \, \sqrt {-2 \, x + 1} - \frac {35933 \, x + 23960}{441 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \]
34/3087*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) + 125/18*sqrt(-2*x + 1) - 1/441*(35933*x + 23960)/(3*(-2 *x + 1)^(3/2) - 7*sqrt(-2*x + 1))
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.69 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {\frac {35933\,x}{1323}+\frac {23960}{1323}}{\frac {7\,\sqrt {1-2\,x}}{3}-{\left (1-2\,x\right )}^{3/2}}-\frac {68\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3087}+\frac {125\,\sqrt {1-2\,x}}{18} \]