Integrand size = 22, antiderivative size = 56 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)} \, dx=\frac {11}{21 (1-2 x)^{3/2}}-\frac {2}{49 \sqrt {1-2 x}}+\frac {2}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \] Output:
11/21/(1-2*x)^(3/2)-2/49/(1-2*x)^(1/2)+2/343*21^(1/2)*arctanh(1/7*21^(1/2) *(1-2*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)} \, dx=\frac {\frac {7 (71+12 x)}{(1-2 x)^{3/2}}+6 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029} \] Input:
Integrate[(3 + 5*x)/((1 - 2*x)^(5/2)*(2 + 3*x)),x]
Output:
((7*(71 + 12*x))/(1 - 2*x)^(3/2) + 6*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2 *x]])/1029
Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {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.
\(\displaystyle \int \frac {5 x+3}{(1-2 x)^{5/2} (3 x+2)} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {11}{21 (1-2 x)^{3/2}}-\frac {1}{7} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{7} \left (-\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {2}{7 \sqrt {1-2 x}}\right )+\frac {11}{21 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{7} \left (\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {2}{7 \sqrt {1-2 x}}\right )+\frac {11}{21 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2}{7 \sqrt {1-2 x}}\right )+\frac {11}{21 (1-2 x)^{3/2}}\) |
Input:
Int[(3 + 5*x)/((1 - 2*x)^(5/2)*(2 + 3*x)),x]
Output:
11/(21*(1 - 2*x)^(3/2)) + (-2/(7*Sqrt[1 - 2*x]) + (2*Sqrt[3/7]*ArcTanh[Sqr t[3/7]*Sqrt[1 - 2*x]])/7)/7
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)^(-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 = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(\frac {11}{21 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2}{49 \sqrt {1-2 x}}+\frac {2 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}\) | \(38\) |
default | \(\frac {11}{21 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2}{49 \sqrt {1-2 x}}+\frac {2 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}\) | \(38\) |
pseudoelliptic | \(\frac {\frac {2 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (1-2 x \right )^{\frac {3}{2}}}{343}+\frac {4 x}{49}+\frac {71}{147}}{\left (1-2 x \right )^{\frac {3}{2}}}\) | \(39\) |
trager | \(\frac {\left (12 x +71\right ) \sqrt {1-2 x}}{147 \left (-1+2 x \right )^{2}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{343}\) | \(67\) |
Input:
int((3+5*x)/(1-2*x)^(5/2)/(2+3*x),x,method=_RETURNVERBOSE)
Output:
11/21/(1-2*x)^(3/2)-2/49/(1-2*x)^(1/2)+2/343*21^(1/2)*arctanh(1/7*21^(1/2) *(1-2*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.23 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)} \, dx=\frac {3 \, \sqrt {\frac {3}{7}} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {3 \, x - 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + {\left (12 \, x + 71\right )} \sqrt {-2 \, x + 1}}{147 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:
integrate((3+5*x)/(1-2*x)^(5/2)/(2+3*x),x, algorithm="fricas")
Output:
1/147*(3*sqrt(3/7)*(4*x^2 - 4*x + 1)*log((3*x - 7*sqrt(3/7)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + (12*x + 71)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)
Time = 2.41 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)} \, dx=- \frac {\sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{343} - \frac {2}{49 \sqrt {1 - 2 x}} + \frac {11}{21 \left (1 - 2 x\right )^{\frac {3}{2}}} \] Input:
integrate((3+5*x)/(1-2*x)**(5/2)/(2+3*x),x)
Output:
-sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/ 3))/343 - 2/(49*sqrt(1 - 2*x)) + 11/(21*(1 - 2*x)**(3/2))
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)} \, dx=-\frac {1}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {12 \, x + 71}{147 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \] Input:
integrate((3+5*x)/(1-2*x)^(5/2)/(2+3*x),x, algorithm="maxima")
Output:
-1/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/147*(12*x + 71)/(-2*x + 1)^(3/2)
Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)} \, dx=-\frac {1}{343} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {12 \, x + 71}{147 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \] Input:
integrate((3+5*x)/(1-2*x)^(5/2)/(2+3*x),x, algorithm="giac")
Output:
-1/343*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3* sqrt(-2*x + 1))) - 1/147*(12*x + 71)/((2*x - 1)*sqrt(-2*x + 1))
Time = 1.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.57 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)} \, dx=\frac {\frac {4\,x}{49}+\frac {71}{147}}{{\left (1-2\,x\right )}^{3/2}}+\frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343} \] Input:
int((5*x + 3)/((1 - 2*x)^(5/2)*(3*x + 2)),x)
Output:
((4*x)/49 + 71/147)/(1 - 2*x)^(3/2) + (2*21^(1/2)*atanh((21^(1/2)*(1 - 2*x )^(1/2))/7))/343
Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.07 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)} \, dx=\frac {-6 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +3 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+6 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -3 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )-84 x -497}{1029 \sqrt {-2 x +1}\, \left (2 x -1\right )} \] Input:
int((3+5*x)/(1-2*x)^(5/2)/(2+3*x),x)
Output:
( - 6*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 3*s qrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 6*sqrt( - 2* x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 3*sqrt( - 2*x + 1)* sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)) - 84*x - 497)/(1029*sqrt( - 2* x + 1)*(2*x - 1))