Optimal. Leaf size=75 \[ \frac {2 \sqrt {a^2+2 a b \sqrt {x}+b^2 x}}{b^2}-\frac {2 a \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b^2 \sqrt {a^2+2 a b \sqrt {x}+b^2 x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1341, 640, 608, 31} \[ \frac {2 \sqrt {a^2+2 a b \sqrt {x}+b^2 x}}{b^2}-\frac {2 a \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b^2 \sqrt {a^2+2 a b \sqrt {x}+b^2 x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 608
Rule 640
Rule 1341
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a^2+2 a b \sqrt {x}+b^2 x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sqrt {a^2+2 a b \sqrt {x}+b^2 x}}{b^2}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {2 \sqrt {a^2+2 a b \sqrt {x}+b^2 x}}{b^2}-\frac {\left (2 a \left (a+b \sqrt {x}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2+2 a b \sqrt {x}+b^2 x}}\\ &=\frac {2 \sqrt {a^2+2 a b \sqrt {x}+b^2 x}}{b^2}-\frac {2 a \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b^2 \sqrt {a^2+2 a b \sqrt {x}+b^2 x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 50, normalized size = 0.67 \[ \frac {2 \left (a+b \sqrt {x}\right ) \left (b \sqrt {x}-a \log \left (a+b \sqrt {x}\right )\right )}{b^2 \sqrt {\left (a+b \sqrt {x}\right )^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.40, size = 45, normalized size = 0.60 \[ -\frac {2 \, {\left | a \right |} \log \left ({\left | \sqrt {b^{2} x} \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (b) + {\left | a \right |} \right |}\right )}{b^{2}} + \frac {2 \, \sqrt {b^{2} x}}{b^{2} \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (b)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 50, normalized size = 0.67 \[ \frac {2 \sqrt {b^{2} x +2 a b \sqrt {x}+a^{2}}\, \left (-a \ln \left (b \sqrt {x}+a \right )+b \sqrt {x}\right )}{\left (b \sqrt {x}+a \right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.96, size = 23, normalized size = 0.31 \[ -\frac {2 \, a \log \left (b \sqrt {x} + a\right )}{b^{2}} + \frac {2 \, \sqrt {x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {b^2\,x+a^2+2\,a\,b\,\sqrt {x}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a^{2} + 2 a b \sqrt {x} + b^{2} x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________