Integrand size = 15, antiderivative size = 60 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{7/2}} \, dx=-\frac {2}{b^2 \sqrt {x}}-\frac {a}{b^2 \left (a+\frac {b}{x}\right ) \sqrt {x}}+\frac {3 \sqrt {a} \arctan \left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{b^{5/2}} \] Output:
-2/b^2/x^(1/2)-a/b^2/(a+b/x)/x^(1/2)+3*a^(1/2)*arctan(1/a^(1/2)/x^(1/2)*b^ (1/2))/b^(5/2)
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{7/2}} \, dx=\frac {-2 b-3 a x}{b^2 \sqrt {x} (b+a x)}-\frac {3 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2}} \] Input:
Integrate[1/((a + b/x)^2*x^(7/2)),x]
Output:
(-2*b - 3*a*x)/(b^2*Sqrt[x]*(b + a*x)) - (3*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[x ])/Sqrt[b]])/b^(5/2)
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {795, 52, 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.
\(\displaystyle \int \frac {1}{x^{7/2} \left (a+\frac {b}{x}\right )^2} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{x^{3/2} (a x+b)^2}dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {3 \int \frac {1}{x^{3/2} (b+a x)}dx}{2 b}+\frac {1}{b \sqrt {x} (a x+b)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {3 \left (-\frac {a \int \frac {1}{\sqrt {x} (b+a x)}dx}{b}-\frac {2}{b \sqrt {x}}\right )}{2 b}+\frac {1}{b \sqrt {x} (a x+b)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3 \left (-\frac {2 a \int \frac {1}{b+a x}d\sqrt {x}}{b}-\frac {2}{b \sqrt {x}}\right )}{2 b}+\frac {1}{b \sqrt {x} (a x+b)}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {3 \left (-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2}{b \sqrt {x}}\right )}{2 b}+\frac {1}{b \sqrt {x} (a x+b)}\) |
Input:
Int[1/((a + b/x)^2*x^(7/2)),x]
Output:
1/(b*Sqrt[x]*(b + a*x)) + (3*(-2/(b*Sqrt[x]) - (2*Sqrt[a]*ArcTan[(Sqrt[a]* Sqrt[x])/Sqrt[b]])/b^(3/2)))/(2*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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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 = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(-\frac {2}{\sqrt {x}\, b^{2}}-\frac {2 a \left (\frac {\sqrt {x}}{2 a x +2 b}+\frac {3 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2}}\) | \(47\) |
default | \(-\frac {2}{\sqrt {x}\, b^{2}}-\frac {2 a \left (\frac {\sqrt {x}}{2 a x +2 b}+\frac {3 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2}}\) | \(47\) |
risch | \(-\frac {2}{\sqrt {x}\, b^{2}}-\frac {a \sqrt {x}}{b^{2} \left (a x +b \right )}-\frac {3 a \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(48\) |
Input:
int(1/(a+b/x)^2/x^(7/2),x,method=_RETURNVERBOSE)
Output:
-2/x^(1/2)/b^2-2*a/b^2*(1/2*x^(1/2)/(a*x+b)+3/2/(a*b)^(1/2)*arctan(a*x^(1/ 2)/(a*b)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.38 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{7/2}} \, dx=\left [\frac {3 \, {\left (a x^{2} + b x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {a x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - b}{a x + b}\right ) - 2 \, {\left (3 \, a x + 2 \, b\right )} \sqrt {x}}{2 \, {\left (a b^{2} x^{2} + b^{3} x\right )}}, -\frac {3 \, {\left (a x^{2} + b x\right )} \sqrt {\frac {a}{b}} \arctan \left (\sqrt {x} \sqrt {\frac {a}{b}}\right ) + {\left (3 \, a x + 2 \, b\right )} \sqrt {x}}{a b^{2} x^{2} + b^{3} x}\right ] \] Input:
integrate(1/(a+b/x)^2/x^(7/2),x, algorithm="fricas")
Output:
[1/2*(3*(a*x^2 + b*x)*sqrt(-a/b)*log((a*x - 2*b*sqrt(x)*sqrt(-a/b) - b)/(a *x + b)) - 2*(3*a*x + 2*b)*sqrt(x))/(a*b^2*x^2 + b^3*x), -(3*(a*x^2 + b*x) *sqrt(a/b)*arctan(sqrt(x)*sqrt(a/b)) + (3*a*x + 2*b)*sqrt(x))/(a*b^2*x^2 + b^3*x)]
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (53) = 106\).
Time = 48.44 (sec) , antiderivative size = 384, normalized size of antiderivative = 6.40 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{7/2}} \, dx=\begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{b^{2} \sqrt {x}} & \text {for}\: a = 0 \\- \frac {2}{5 a^{2} x^{\frac {5}{2}}} & \text {for}\: b = 0 \\- \frac {3 a x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{2 a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}} + 2 b^{3} \sqrt {x} \sqrt {- \frac {b}{a}}} + \frac {3 a x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{2 a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}} + 2 b^{3} \sqrt {x} \sqrt {- \frac {b}{a}}} - \frac {6 a x \sqrt {- \frac {b}{a}}}{2 a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}} + 2 b^{3} \sqrt {x} \sqrt {- \frac {b}{a}}} - \frac {3 b \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{2 a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}} + 2 b^{3} \sqrt {x} \sqrt {- \frac {b}{a}}} + \frac {3 b \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{2 a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}} + 2 b^{3} \sqrt {x} \sqrt {- \frac {b}{a}}} - \frac {4 b \sqrt {- \frac {b}{a}}}{2 a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}} + 2 b^{3} \sqrt {x} \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x)**2/x**(7/2),x)
Output:
Piecewise((zoo/sqrt(x), Eq(a, 0) & Eq(b, 0)), (-2/(b**2*sqrt(x)), Eq(a, 0) ), (-2/(5*a**2*x**(5/2)), Eq(b, 0)), (-3*a*x**(3/2)*log(sqrt(x) - sqrt(-b/ a))/(2*a*b**2*x**(3/2)*sqrt(-b/a) + 2*b**3*sqrt(x)*sqrt(-b/a)) + 3*a*x**(3 /2)*log(sqrt(x) + sqrt(-b/a))/(2*a*b**2*x**(3/2)*sqrt(-b/a) + 2*b**3*sqrt( x)*sqrt(-b/a)) - 6*a*x*sqrt(-b/a)/(2*a*b**2*x**(3/2)*sqrt(-b/a) + 2*b**3*s qrt(x)*sqrt(-b/a)) - 3*b*sqrt(x)*log(sqrt(x) - sqrt(-b/a))/(2*a*b**2*x**(3 /2)*sqrt(-b/a) + 2*b**3*sqrt(x)*sqrt(-b/a)) + 3*b*sqrt(x)*log(sqrt(x) + sq rt(-b/a))/(2*a*b**2*x**(3/2)*sqrt(-b/a) + 2*b**3*sqrt(x)*sqrt(-b/a)) - 4*b *sqrt(-b/a)/(2*a*b**2*x**(3/2)*sqrt(-b/a) + 2*b**3*sqrt(x)*sqrt(-b/a)), Tr ue))
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{7/2}} \, dx=-\frac {a}{{\left (a b^{2} + \frac {b^{3}}{x}\right )} \sqrt {x}} + \frac {3 \, a \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} b^{2}} - \frac {2}{b^{2} \sqrt {x}} \] Input:
integrate(1/(a+b/x)^2/x^(7/2),x, algorithm="maxima")
Output:
-a/((a*b^2 + b^3/x)*sqrt(x)) + 3*a*arctan(b/(sqrt(a*b)*sqrt(x)))/(sqrt(a*b )*b^2) - 2/(b^2*sqrt(x))
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{7/2}} \, dx=-\frac {3 \, a \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} - \frac {3 \, a x + 2 \, b}{{\left (a x^{\frac {3}{2}} + b \sqrt {x}\right )} b^{2}} \] Input:
integrate(1/(a+b/x)^2/x^(7/2),x, algorithm="giac")
Output:
-3*a*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^2) - (3*a*x + 2*b)/((a*x^(3/ 2) + b*sqrt(x))*b^2)
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{7/2}} \, dx=-\frac {\frac {2}{b}+\frac {3\,a\,x}{b^2}}{a\,x^{3/2}+b\,\sqrt {x}}-\frac {3\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{b^{5/2}} \] Input:
int(1/(x^(7/2)*(a + b/x)^2),x)
Output:
- (2/b + (3*a*x)/b^2)/(a*x^(3/2) + b*x^(1/2)) - (3*a^(1/2)*atan((a^(1/2)*x ^(1/2))/b^(1/2)))/b^(5/2)
Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{7/2}} \, dx=\frac {-3 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) a x -3 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) b -3 a b x -2 b^{2}}{\sqrt {x}\, b^{3} \left (a x +b \right )} \] Input:
int(1/(a+b/x)^2/x^(7/2),x)
Output:
( - 3*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*a*x - 3* sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*b - 3*a*b*x - 2*b**2)/(sqrt(x)*b**3*(a*x + b))