Optimal. Leaf size=52 \[ \frac {2}{b \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {344, 294, 223,
212} \begin {gather*} \frac {2}{b \sqrt {x} \sqrt {a+\frac {b}{x}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 294
Rule 344
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{5/2}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{b}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 4.62, size = 64, normalized size = 1.23 \begin {gather*} \frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {2}{b \sqrt {b+a x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )}{b^{3/2}}\right )}{\sqrt {b+a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 53, normalized size = 1.02
method | result | size |
default | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (\arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {a x +b}-\sqrt {b}\right )}{b^{\frac {3}{2}} \left (a x +b \right )}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 62, normalized size = 1.19 \begin {gather*} \frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{b^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {b}{x}} b \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 138, normalized size = 2.65 \begin {gather*} \left [\frac {{\left (a x + b\right )} \sqrt {b} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, b \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a b^{2} x + b^{3}}, \frac {2 \, {\left ({\left (a x + b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + b \sqrt {x} \sqrt {\frac {a x + b}{x}}\right )}}{a b^{2} x + b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 146 vs.
\(2 (42) = 84\).
time = 6.55, size = 146, normalized size = 2.81 \begin {gather*} \frac {a b^{2} x \log {\left (\frac {a x}{b} \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} - \frac {2 a b^{2} x \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} + \frac {2 b^{3} \sqrt {\frac {a x}{b} + 1}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} + \frac {b^{3} \log {\left (\frac {a x}{b} \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} - \frac {2 b^{3} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.27, size = 37, normalized size = 0.71 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {2}{\sqrt {a x + b} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^{5/2}\,{\left (a+\frac {b}{x}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________