Optimal. Leaf size=47 \[ \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {c x}{2 a \left (a+b x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {21, 199, 205} \[ \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {c x}{2 a \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 199
Rule 205
Rubi steps
\begin {align*} \int \frac {a c+b c x^2}{\left (a+b x^2\right )^3} \, dx &=c \int \frac {1}{\left (a+b x^2\right )^2} \, dx\\ &=\frac {c x}{2 a \left (a+b x^2\right )}+\frac {c \int \frac {1}{a+b x^2} \, dx}{2 a}\\ &=\frac {c x}{2 a \left (a+b x^2\right )}+\frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 47, normalized size = 1.00 \[ c \left (\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x}{2 a \left (a+b x^2\right )}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 128, normalized size = 2.72 \[ \left [\frac {2 \, a b c x - {\left (b c x^{2} + a c\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{4 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}, \frac {a b c x + {\left (b c x^{2} + a c\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{2 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.38, size = 37, normalized size = 0.79 \[ \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} + \frac {c x}{2 \, {\left (b x^{2} + a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 38, normalized size = 0.81 \[ \frac {c x}{2 \left (b \,x^{2}+a \right ) a}+\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.40, size = 37, normalized size = 0.79 \[ \frac {c x}{2 \, {\left (a b x^{2} + a^{2}\right )}} + \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 35, normalized size = 0.74 \[ \frac {c\,x}{2\,a\,\left (b\,x^2+a\right )}+\frac {c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.23, size = 80, normalized size = 1.70 \[ c \left (\frac {x}{2 a^{2} + 2 a b x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________