Integrand size = 13, antiderivative size = 54 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {2}{3 a (a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {a+b x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \] Output:
2/3/a/(b*x+a)^(3/2)+2/a^2/(b*x+a)^(1/2)-2*arctanh((b*x+a)^(1/2)/a^(1/2))/a ^(5/2)
Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {2 (a+3 (a+b x))}{3 a^2 (a+b x)^{3/2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \] Input:
Integrate[1/(x*(a + b*x)^(5/2)),x]
Output:
(2*(a + 3*(a + b*x)))/(3*a^2*(a + b*x)^(3/2)) - (2*ArcTanh[Sqrt[a + b*x]/S qrt[a]])/a^(5/2)
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {61, 61, 73, 221}
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.
\(\displaystyle \int \frac {1}{x (a+b x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {\int \frac {1}{x (a+b x)^{3/2}}dx}{a}+\frac {2}{3 a (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {\frac {\int \frac {1}{x \sqrt {a+b x}}dx}{a}+\frac {2}{a \sqrt {a+b x}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {2 \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{a b}+\frac {2}{a \sqrt {a+b x}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {2}{a \sqrt {a+b x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\) |
Input:
Int[1/(x*(a + b*x)^(5/2)),x]
Output:
2/(3*a*(a + b*x)^(3/2)) + (2/(a*Sqrt[a + b*x]) - (2*ArcTanh[Sqrt[a + b*x]/ Sqrt[a]])/a^(3/2))/a
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_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {2}{3 a \left (b x +a \right )^{\frac {3}{2}}}+\frac {2}{a^{2} \sqrt {b x +a}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}\) | \(43\) |
default | \(\frac {2}{3 a \left (b x +a \right )^{\frac {3}{2}}}+\frac {2}{a^{2} \sqrt {b x +a}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}\) | \(43\) |
pseudoelliptic | \(-\frac {2 \left (\left (b x +a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-\sqrt {a}\, b x -\frac {4 a^{\frac {3}{2}}}{3}\right )}{\left (b x +a \right )^{\frac {3}{2}} a^{\frac {5}{2}}}\) | \(46\) |
Input:
int(1/x/(b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/3/a/(b*x+a)^(3/2)+2/a^2/(b*x+a)^(1/2)-2*arctanh((b*x+a)^(1/2)/a^(1/2))/a ^(5/2)
Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.22 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\left [\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3 \, a b x + 4 \, a^{2}\right )} \sqrt {b x + a}}{3 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}}, \frac {2 \, {\left (3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) + {\left (3 \, a b x + 4 \, a^{2}\right )} \sqrt {b x + a}\right )}}{3 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}}\right ] \] Input:
integrate(1/x/(b*x+a)^(5/2),x, algorithm="fricas")
Output:
[1/3*(3*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt( a) + 2*a)/x) + 2*(3*a*b*x + 4*a^2)*sqrt(b*x + a))/(a^3*b^2*x^2 + 2*a^4*b*x + a^5), 2/3*(3*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b* x + a)) + (3*a*b*x + 4*a^2)*sqrt(b*x + a))/(a^3*b^2*x^2 + 2*a^4*b*x + a^5) ]
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (48) = 96\).
Time = 1.26 (sec) , antiderivative size = 697, normalized size of antiderivative = 12.91 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {8 a^{7} \sqrt {1 + \frac {b x}{a}}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {3 a^{7} \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {6 a^{7} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {14 a^{6} b x \sqrt {1 + \frac {b x}{a}}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {9 a^{6} b x \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {18 a^{6} b x \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {6 a^{5} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {9 a^{5} b^{2} x^{2} \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {18 a^{5} b^{2} x^{2} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {3 a^{4} b^{3} x^{3} \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {6 a^{4} b^{3} x^{3} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} \] Input:
integrate(1/x/(b*x+a)**(5/2),x)
Output:
8*a**7*sqrt(1 + b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x **2 + 3*a**(13/2)*b**3*x**3) + 3*a**7*log(b*x/a)/(3*a**(19/2) + 9*a**(17/2 )*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*a**7*log(sqrt(1 + b*x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3* a**(13/2)*b**3*x**3) + 14*a**6*b*x*sqrt(1 + b*x/a)/(3*a**(19/2) + 9*a**(17 /2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*a**6*b*x*log( b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2 )*b**3*x**3) - 18*a**6*b*x*log(sqrt(1 + b*x/a) + 1)/(3*a**(19/2) + 9*a**(1 7/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*a**5*b**2*x* *2*sqrt(1 + b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*a**5*b**2*x**2*log(b*x/a)/(3*a**(19/2) + 9*a* *(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*a**5*b** 2*x**2*log(sqrt(1 + b*x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/ 2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 3*a**4*b**3*x**3*log(b*x/a)/(3*a** (19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*a**4*b**3*x**3*log(sqrt(1 + b*x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3)
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {\log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x + 4 \, a\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}} \] Input:
integrate(1/x/(b*x+a)^(5/2),x, algorithm="maxima")
Output:
log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/a^(5/2) + 2/3*(3* b*x + 4*a)/((b*x + a)^(3/2)*a^2)
Time = 0.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, {\left (3 \, b x + 4 \, a\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}} \] Input:
integrate(1/x/(b*x+a)^(5/2),x, algorithm="giac")
Output:
2*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2) + 2/3*(3*b*x + 4*a)/((b*x + a)^(3/2)*a^2)
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {\frac {2\,\left (a+b\,x\right )}{a^2}+\frac {2}{3\,a}}{{\left (a+b\,x\right )}^{3/2}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{a^{5/2}} \] Input:
int(1/(x*(a + b*x)^(5/2)),x)
Output:
((2*(a + b*x))/a^2 + 2/(3*a))/(a + b*x)^(3/2) - (2*atanh((a + b*x)^(1/2)/a ^(1/2)))/a^(5/2)
Time = 0.16 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.24 \[ \int \frac {1}{x (a+b x)^{5/2}} \, dx=\frac {3 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a +3 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b x -3 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a -3 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b x +8 a^{2}+6 a b x}{3 \sqrt {b x +a}\, a^{3} \left (b x +a \right )} \] Input:
int(1/x/(b*x+a)^(5/2),x)
Output:
(3*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*a + 3*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*b*x - 3*sqrt(a)*sqrt(a + b*x)*log(sqr t(a + b*x) + sqrt(a))*a - 3*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt (a))*b*x + 8*a**2 + 6*a*b*x)/(3*sqrt(a + b*x)*a**3*(a + b*x))