Optimal. Leaf size=25 \[ \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 205} \[ \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 205
Rubi steps
\begin {align*} \int \frac {a c+b c x^2}{\left (a+b x^2\right )^2} \, dx &=c \int \frac {1}{a+b x^2} \, dx\\ &=\frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 25, normalized size = 1.00 \[ \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 69, normalized size = 2.76 \[ \left [-\frac {\sqrt {-a b} c \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b}, \frac {\sqrt {a b} c \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.33, size = 16, normalized size = 0.64 \[ \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 17, normalized size = 0.68 \[ \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.23, size = 16, normalized size = 0.64 \[ \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 17, normalized size = 0.68 \[ \frac {c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.15, size = 54, normalized size = 2.16 \[ c \left (- \frac {\sqrt {- \frac {1}{a b}} \log {\left (- a \sqrt {- \frac {1}{a b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (a \sqrt {- \frac {1}{a b}} + x \right )}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________