Integrand size = 24, antiderivative size = 81 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=-\frac {130}{1029 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac {365}{294 \sqrt {1-2 x} (2+3 x)}+\frac {130 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{343 \sqrt {21}} \]
121/42/(1-2*x)^(3/2)/(2+3*x)+130/7203*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))* 21^(1/2)-130/1029/(1-2*x)^(1/2)-365/294/(2+3*x)/(1-2*x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {-5929+3465 (1-2 x)-390 (1-2 x)^2}{1029 (-7+3 (1-2 x)) (1-2 x)^{3/2}}+\frac {130 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{343 \sqrt {21}} \]
(-5929 + 3465*(1 - 2*x) - 390*(1 - 2*x)^2)/(1029*(-7 + 3*(1 - 2*x))*(1 - 2 *x)^(3/2)) + (130*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(343*Sqrt[21])
Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 27, 87, 61, 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.
\(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x)^{5/2} (3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {121}{42 (1-2 x)^{3/2} (3 x+2)}-\frac {1}{42} \int -\frac {15 (1-35 x)}{(1-2 x)^{3/2} (3 x+2)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {5}{14} \int \frac {1-35 x}{(1-2 x)^{3/2} (3 x+2)^2}dx+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {5}{14} \left (-\frac {26}{21} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx-\frac {73}{21 \sqrt {1-2 x} (3 x+2)}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {5}{14} \left (-\frac {26}{21} \left (\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{7 \sqrt {1-2 x}}\right )-\frac {73}{21 \sqrt {1-2 x} (3 x+2)}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {5}{14} \left (-\frac {26}{21} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {73}{21 \sqrt {1-2 x} (3 x+2)}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {5}{14} \left (-\frac {26}{21} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {73}{21 \sqrt {1-2 x} (3 x+2)}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)}\) |
121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)) + (5*(-73/(21*Sqrt[1 - 2*x]*(2 + 3*x)) - (26*(2/(7*Sqrt[1 - 2*x]) - (2*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] )/7))/21))/14
3.22.53.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((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 = 1.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.65
method | result | size |
risch | \(-\frac {780 x^{2}+2685 x +1427}{1029 \left (-1+2 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )}+\frac {130 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7203}\) | \(53\) |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{1029 \left (-\frac {4}{3}-2 x \right )}+\frac {130 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7203}+\frac {121}{147 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {44}{343 \sqrt {1-2 x}}\) | \(54\) |
default | \(\frac {2 \sqrt {1-2 x}}{1029 \left (-\frac {4}{3}-2 x \right )}+\frac {130 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7203}+\frac {121}{147 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {44}{343 \sqrt {1-2 x}}\) | \(54\) |
pseudoelliptic | \(-\frac {780 \left (\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (6 x^{2}+x -2\right ) \sqrt {21}}{6}-7 x^{2}-\frac {1253 x}{52}-\frac {9989}{780}\right )}{\left (1-2 x \right )^{\frac {3}{2}} \left (14406+21609 x \right )}\) | \(60\) |
trager | \(\frac {\left (780 x^{2}+2685 x +1427\right ) \sqrt {1-2 x}}{1029 \left (-1+2 x \right )^{2} \left (2+3 x \right )}+\frac {65 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{7203}\) | \(79\) |
-1/1029*(780*x^2+2685*x+1427)/(-1+2*x)/(1-2*x)^(1/2)/(2+3*x)+130/7203*arct anh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {65 \, \sqrt {21} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \, {\left (780 \, x^{2} + 2685 \, x + 1427\right )} \sqrt {-2 \, x + 1}}{7203 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
1/7203*(65*sqrt(21)*(12*x^3 - 4*x^2 - 5*x + 2)*log((3*x - sqrt(21)*sqrt(-2 *x + 1) - 5)/(3*x + 2)) + 7*(780*x^2 + 2685*x + 1427)*sqrt(-2*x + 1))/(12* x^3 - 4*x^2 - 5*x + 2)
Time = 40.91 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.28 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=- \frac {22 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{2401} - \frac {4 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{49} - \frac {44}{343 \sqrt {1 - 2 x}} + \frac {121}{147 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
-22*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(2 1)/3))/2401 - 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)))/49 - 44/(343*sqrt(1 - 2*x)) + 121/(14 7*(1 - 2*x)**(3/2))
Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=-\frac {65}{7203} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (195 \, {\left (2 \, x - 1\right )}^{2} + 3465 \, x + 1232\right )}}{1029 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 7 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
-65/7203*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 *x + 1))) - 2/1029*(195*(2*x - 1)^2 + 3465*x + 1232)/(3*(-2*x + 1)^(5/2) - 7*(-2*x + 1)^(3/2))
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=-\frac {65}{7203} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {11 \, {\left (24 \, x + 65\right )}}{1029 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {\sqrt {-2 \, x + 1}}{343 \, {\left (3 \, x + 2\right )}} \]
-65/7203*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 11/1029*(24*x + 65)/((2*x - 1)*sqrt(-2*x + 1)) - 1/34 3*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.68 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {\frac {110\,x}{49}+\frac {130\,{\left (2\,x-1\right )}^2}{1029}+\frac {352}{441}}{\frac {7\,{\left (1-2\,x\right )}^{3/2}}{3}-{\left (1-2\,x\right )}^{5/2}}+\frac {130\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7203} \]