Integrand size = 24, antiderivative size = 68 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {121}{14 \sqrt {1-2 x} (2+3 x)}-\frac {1091 \sqrt {1-2 x}}{294 (2+3 x)}+\frac {134 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \]
134/3087*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+121/14/(2+3*x)/(1-2* x)^(1/2)-1091/294*(1-2*x)^(1/2)/(2+3*x)
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {-2541+1091 (1-2 x)}{147 (-7+3 (1-2 x)) \sqrt {1-2 x}}+\frac {134 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \]
(-2541 + 1091*(1 - 2*x))/(147*(-7 + 3*(1 - 2*x))*Sqrt[1 - 2*x]) + (134*Arc Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 25, 87, 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.
\(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x)^{3/2} (3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {121}{14 \sqrt {1-2 x} (3 x+2)}-\frac {1}{14} \int -\frac {247-175 x}{\sqrt {1-2 x} (3 x+2)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{14} \int \frac {247-175 x}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {121}{14 \sqrt {1-2 x} (3 x+2)}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{14} \left (-\frac {134}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {1091 \sqrt {1-2 x}}{21 (3 x+2)}\right )+\frac {121}{14 \sqrt {1-2 x} (3 x+2)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{14} \left (\frac {134}{21} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {1091 \sqrt {1-2 x}}{21 (3 x+2)}\right )+\frac {121}{14 \sqrt {1-2 x} (3 x+2)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{14} \left (\frac {268 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-\frac {1091 \sqrt {1-2 x}}{21 (3 x+2)}\right )+\frac {121}{14 \sqrt {1-2 x} (3 x+2)}\) |
121/(14*Sqrt[1 - 2*x]*(2 + 3*x)) + ((-1091*Sqrt[1 - 2*x])/(21*(2 + 3*x)) + (268*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21]))/14
3.21.90.3.1 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*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 1.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.60
method | result | size |
risch | \(\frac {1091 x +725}{147 \left (2+3 x \right ) \sqrt {1-2 x}}+\frac {134 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}\) | \(41\) |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{441 \left (-\frac {4}{3}-2 x \right )}+\frac {134 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}+\frac {121}{49 \sqrt {1-2 x}}\) | \(45\) |
default | \(\frac {2 \sqrt {1-2 x}}{441 \left (-\frac {4}{3}-2 x \right )}+\frac {134 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3087}+\frac {121}{49 \sqrt {1-2 x}}\) | \(45\) |
pseudoelliptic | \(\frac {402 \sqrt {1-2 x}\, \sqrt {21}\, \left (\frac {2}{3}+x \right ) \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )+22911 x +15225}{\sqrt {1-2 x}\, \left (6174+9261 x \right )}\) | \(49\) |
trager | \(-\frac {\left (1091 x +725\right ) \sqrt {1-2 x}}{147 \left (6 x^{2}+x -2\right )}-\frac {67 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{3087}\) | \(70\) |
1/147*(1091*x+725)/(2+3*x)/(1-2*x)^(1/2)+134/3087*arctanh(1/7*21^(1/2)*(1- 2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {67 \, \sqrt {21} {\left (6 \, x^{2} + x - 2\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (1091 \, x + 725\right )} \sqrt {-2 \, x + 1}}{3087 \, {\left (6 \, x^{2} + x - 2\right )}} \]
1/3087*(67*sqrt(21)*(6*x^2 + x - 2)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5 )/(3*x + 2)) - 21*(1091*x + 725)*sqrt(-2*x + 1))/(6*x^2 + x - 2)
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (56) = 112\).
Time = 42.56 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.54 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=- \frac {68 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{3087} - \frac {4 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{21} + \frac {121}{49 \sqrt {1 - 2 x}} \]
-68*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(2 1)/3))/3087 - 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, (sqrt(1 - 2*x) > -sqrt(21) /3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/21 + 121/(49*sqrt(1 - 2*x))
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.96 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=-\frac {67}{3087} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (1091 \, x + 725\right )}}{147 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \]
-67/3087*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 *x + 1))) - 2/147*(1091*x + 725)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))
Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=-\frac {67}{3087} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2 \, {\left (1091 \, x + 725\right )}}{147 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \]
-67/3087*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/147*(1091*x + 725)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2* x + 1))
Time = 1.35 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {\frac {2182\,x}{441}+\frac {1450}{441}}{\frac {7\,\sqrt {1-2\,x}}{3}-{\left (1-2\,x\right )}^{3/2}}+\frac {134\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3087} \]