Optimal. Leaf size=24 \[ -\frac {a^2}{x}-\frac {4 a b}{\sqrt {x}}+b^2 \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {a^2}{x}-\frac {4 a b}{\sqrt {x}}+b^2 \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^2}{x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^3} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2}{x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^2}{x}-\frac {4 a b}{\sqrt {x}}+b^2 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 24, normalized size = 1.00 \[ -\frac {a^2}{x}-\frac {4 a b}{\sqrt {x}}+b^2 \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.24, size = 27, normalized size = 1.12 \[ \frac {2 \, b^{2} x \log \left (\sqrt {x}\right ) - 4 \, a b \sqrt {x} - a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 24, normalized size = 1.00 \[ b^{2} \log \left ({\left | x \right |}\right ) - \frac {4 \, a b \sqrt {x} + a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 23, normalized size = 0.96 \[ b^{2} \ln \relax (x )-\frac {4 a b}{\sqrt {x}}-\frac {a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.98, size = 23, normalized size = 0.96 \[ b^{2} \log \relax (x) - \frac {4 \, a b \sqrt {x} + a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.08, size = 26, normalized size = 1.08 \[ 2\,b^2\,\ln \left (\sqrt {x}\right )-\frac {a^2+4\,a\,b\,\sqrt {x}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.45, size = 20, normalized size = 0.83 \[ - \frac {a^{2}}{x} - \frac {4 a b}{\sqrt {x}} + b^{2} \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________