Optimal. Leaf size=83 \[ -\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{4 b^{5/2}}+\frac {3 a \sqrt {a+\frac {b}{x}}}{4 b^2 \sqrt {x}}-\frac {\sqrt {a+\frac {b}{x}}}{2 b x^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {337, 321, 217, 206} \[ -\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{4 b^{5/2}}+\frac {3 a \sqrt {a+\frac {b}{x}}}{4 b^2 \sqrt {x}}-\frac {\sqrt {a+\frac {b}{x}}}{2 b x^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 321
Rule 337
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x}} x^{7/2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {\sqrt {a+\frac {b}{x}}}{2 b x^{3/2}}+\frac {(3 a) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{2 b}\\ &=-\frac {\sqrt {a+\frac {b}{x}}}{2 b x^{3/2}}+\frac {3 a \sqrt {a+\frac {b}{x}}}{4 b^2 \sqrt {x}}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{4 b^2}\\ &=-\frac {\sqrt {a+\frac {b}{x}}}{2 b x^{3/2}}+\frac {3 a \sqrt {a+\frac {b}{x}}}{4 b^2 \sqrt {x}}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{4 b^2}\\ &=-\frac {\sqrt {a+\frac {b}{x}}}{2 b x^{3/2}}+\frac {3 a \sqrt {a+\frac {b}{x}}}{4 b^2 \sqrt {x}}-\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{4 b^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 93, normalized size = 1.12 \[ \frac {\sqrt {b} \left (3 a^2 x^2+a b x-2 b^2\right )-3 a^{5/2} x^{5/2} \sqrt {\frac {b}{a x}+1} \sinh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{4 b^{5/2} x^{5/2} \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.81, size = 151, normalized size = 1.82 \[ \left [\frac {3 \, a^{2} \sqrt {b} x^{2} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (3 \, a b x - 2 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{8 \, b^{3} x^{2}}, \frac {3 \, a^{2} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (3 \, a b x - 2 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{4 \, b^{3} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.32, size = 69, normalized size = 0.83 \[ \frac {\frac {3 \, a^{3} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {3 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{3} - 5 \, \sqrt {a x + b} a^{3} b}{a^{2} b^{2} x^{2}}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 74, normalized size = 0.89 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (3 a^{2} x^{2} \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )-3 \sqrt {a x +b}\, a \sqrt {b}\, x +2 \sqrt {a x +b}\, b^{\frac {3}{2}}\right )}{4 \sqrt {a x +b}\, b^{\frac {5}{2}} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.31, size = 122, normalized size = 1.47 \[ \frac {3 \, a^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{8 \, b^{\frac {5}{2}}} + \frac {3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} x^{\frac {3}{2}} - 5 \, \sqrt {a + \frac {b}{x}} a^{2} b \sqrt {x}}{4 \, {\left ({\left (a + \frac {b}{x}\right )}^{2} b^{2} x^{2} - 2 \, {\left (a + \frac {b}{x}\right )} b^{3} x + b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{7/2}\,\sqrt {a+\frac {b}{x}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 22.61, size = 102, normalized size = 1.23 \[ \frac {3 a^{\frac {3}{2}}}{4 b^{2} \sqrt {x} \sqrt {1 + \frac {b}{a x}}} + \frac {\sqrt {a}}{4 b x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} - \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{4 b^{\frac {5}{2}}} - \frac {1}{2 \sqrt {a} x^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________