Integrand size = 15, antiderivative size = 68 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^{9/2}} \, dx=-\frac {2}{5 b x^{5/2}}+\frac {2 a}{3 b^2 x^{3/2}}-\frac {2 a^2}{b^3 \sqrt {x}}+\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{b^{7/2}} \] Output:
-2/5/b/x^(5/2)+2/3*a/b^2/x^(3/2)-2*a^2/b^3/x^(1/2)+2*a^(5/2)*arctan(1/a^(1 /2)/x^(1/2)*b^(1/2))/b^(7/2)
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^{9/2}} \, dx=-\frac {2 \left (3 b^2-5 a b x+15 a^2 x^2\right )}{15 b^3 x^{5/2}}-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \] Input:
Integrate[1/((a + b/x)*x^(9/2)),x]
Output:
(-2*(3*b^2 - 5*a*b*x + 15*a^2*x^2))/(15*b^3*x^(5/2)) - (2*a^(5/2)*ArcTan[( Sqrt[a]*Sqrt[x])/Sqrt[b]])/b^(7/2)
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {795, 61, 61, 61, 73, 218}
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.
\(\displaystyle \int \frac {1}{x^{9/2} \left (a+\frac {b}{x}\right )} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{x^{7/2} (a x+b)}dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {a \int \frac {1}{x^{5/2} (b+a x)}dx}{b}-\frac {2}{5 b x^{5/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {a \left (-\frac {a \int \frac {1}{x^{3/2} (b+a x)}dx}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{5 b x^{5/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {a \left (-\frac {a \left (-\frac {a \int \frac {1}{\sqrt {x} (b+a x)}dx}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{5 b x^{5/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {a \left (-\frac {a \left (-\frac {2 a \int \frac {1}{b+a x}d\sqrt {x}}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{5 b x^{5/2}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {a \left (-\frac {a \left (-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{5 b x^{5/2}}\) |
Input:
Int[1/((a + b/x)*x^(9/2)),x]
Output:
-2/(5*b*x^(5/2)) - (a*(-2/(3*b*x^(3/2)) - (a*(-2/(b*Sqrt[x]) - (2*Sqrt[a]* ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/b^(3/2)))/b))/b
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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {2 \left (15 a^{2} x^{2}-5 a b x +3 b^{2}\right )}{15 b^{3} x^{\frac {5}{2}}}-\frac {2 a^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}\) | \(53\) |
derivativedivides | \(-\frac {2}{5 b \,x^{\frac {5}{2}}}-\frac {2 a^{2}}{b^{3} \sqrt {x}}+\frac {2 a}{3 b^{2} x^{\frac {3}{2}}}-\frac {2 a^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}\) | \(54\) |
default | \(-\frac {2}{5 b \,x^{\frac {5}{2}}}-\frac {2 a^{2}}{b^{3} \sqrt {x}}+\frac {2 a}{3 b^{2} x^{\frac {3}{2}}}-\frac {2 a^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}\) | \(54\) |
Input:
int(1/(a+b/x)/x^(9/2),x,method=_RETURNVERBOSE)
Output:
-2/15*(15*a^2*x^2-5*a*b*x+3*b^2)/b^3/x^(5/2)-2*a^3/b^3/(a*b)^(1/2)*arctan( a*x^(1/2)/(a*b)^(1/2))
Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.04 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^{9/2}} \, dx=\left [\frac {15 \, a^{2} x^{3} \sqrt {-\frac {a}{b}} \log \left (\frac {a x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - b}{a x + b}\right ) - 2 \, {\left (15 \, a^{2} x^{2} - 5 \, a b x + 3 \, b^{2}\right )} \sqrt {x}}{15 \, b^{3} x^{3}}, -\frac {2 \, {\left (15 \, a^{2} x^{3} \sqrt {\frac {a}{b}} \arctan \left (\sqrt {x} \sqrt {\frac {a}{b}}\right ) + {\left (15 \, a^{2} x^{2} - 5 \, a b x + 3 \, b^{2}\right )} \sqrt {x}\right )}}{15 \, b^{3} x^{3}}\right ] \] Input:
integrate(1/(a+b/x)/x^(9/2),x, algorithm="fricas")
Output:
[1/15*(15*a^2*x^3*sqrt(-a/b)*log((a*x - 2*b*sqrt(x)*sqrt(-a/b) - b)/(a*x + b)) - 2*(15*a^2*x^2 - 5*a*b*x + 3*b^2)*sqrt(x))/(b^3*x^3), -2/15*(15*a^2* x^3*sqrt(a/b)*arctan(sqrt(x)*sqrt(a/b)) + (15*a^2*x^2 - 5*a*b*x + 3*b^2)*s qrt(x))/(b^3*x^3)]
Time = 16.88 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^{9/2}} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {2}{7 a x^{\frac {7}{2}}} & \text {for}\: b = 0 \\- \frac {a^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{b^{3} \sqrt {- \frac {b}{a}}} + \frac {a^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{b^{3} \sqrt {- \frac {b}{a}}} - \frac {2 a^{2}}{b^{3} \sqrt {x}} + \frac {2 a}{3 b^{2} x^{\frac {3}{2}}} - \frac {2}{5 b x^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x)/x**(9/2),x)
Output:
Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-2/(5*b*x**(5/2)), Eq(a, 0 )), (-2/(7*a*x**(7/2)), Eq(b, 0)), (-a**2*log(sqrt(x) - sqrt(-b/a))/(b**3* sqrt(-b/a)) + a**2*log(sqrt(x) + sqrt(-b/a))/(b**3*sqrt(-b/a)) - 2*a**2/(b **3*sqrt(x)) + 2*a/(3*b**2*x**(3/2)) - 2/(5*b*x**(5/2)), True))
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^{9/2}} \, dx=\frac {2 \, a^{3} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} b^{3}} - \frac {2 \, {\left (\frac {15 \, a^{2}}{\sqrt {x}} - \frac {5 \, a b}{x^{\frac {3}{2}}} + \frac {3 \, b^{2}}{x^{\frac {5}{2}}}\right )}}{15 \, b^{3}} \] Input:
integrate(1/(a+b/x)/x^(9/2),x, algorithm="maxima")
Output:
2*a^3*arctan(b/(sqrt(a*b)*sqrt(x)))/(sqrt(a*b)*b^3) - 2/15*(15*a^2/sqrt(x) - 5*a*b/x^(3/2) + 3*b^2/x^(5/2))/b^3
Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^{9/2}} \, dx=-\frac {2 \, a^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} - \frac {2 \, {\left (15 \, a^{2} x^{2} - 5 \, a b x + 3 \, b^{2}\right )}}{15 \, b^{3} x^{\frac {5}{2}}} \] Input:
integrate(1/(a+b/x)/x^(9/2),x, algorithm="giac")
Output:
-2*a^3*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^3) - 2/15*(15*a^2*x^2 - 5* a*b*x + 3*b^2)/(b^3*x^(5/2))
Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^{9/2}} \, dx=-\frac {\frac {2}{5\,b}+\frac {2\,a^2\,x^2}{b^3}-\frac {2\,a\,x}{3\,b^2}}{x^{5/2}}-\frac {2\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \] Input:
int(1/(x^(9/2)*(a + b/x)),x)
Output:
- (2/(5*b) + (2*a^2*x^2)/b^3 - (2*a*x)/(3*b^2))/x^(5/2) - (2*a^(5/2)*atan( (a^(1/2)*x^(1/2))/b^(1/2)))/b^(7/2)
Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^{9/2}} \, dx=\frac {-2 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} x^{2}-2 a^{2} b \,x^{2}+\frac {2 a \,b^{2} x}{3}-\frac {2 b^{3}}{5}}{\sqrt {x}\, b^{4} x^{2}} \] Input:
int(1/(a+b/x)/x^(9/2),x)
Output:
(2*( - 15*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*a**2 *x**2 - 15*a**2*b*x**2 + 5*a*b**2*x - 3*b**3))/(15*sqrt(x)*b**4*x**2)