Optimal. Leaf size=71 \[ -\frac {c}{a^2 x}-\frac {(b c-a d) x}{2 a^2 \left (a+b x^2\right )}-\frac {(3 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 71, 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 = {467, 464, 211}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (3 b c-a d)}{2 a^{5/2} \sqrt {b}}-\frac {x (b c-a d)}{2 a^2 \left (a+b x^2\right )}-\frac {c}{a^2 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 464
Rule 467
Rubi steps
\begin {align*} \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )^2} \, dx &=-\frac {(b c-a d) x}{2 a^2 \left (a+b x^2\right )}-\frac {1}{2} \int \frac {-\frac {2 c}{a}+\frac {(b c-a d) x^2}{a^2}}{x^2 \left (a+b x^2\right )} \, dx\\ &=-\frac {c}{a^2 x}-\frac {(b c-a d) x}{2 a^2 \left (a+b x^2\right )}-\frac {(3 b c-a d) \int \frac {1}{a+b x^2} \, dx}{2 a^2}\\ &=-\frac {c}{a^2 x}-\frac {(b c-a d) x}{2 a^2 \left (a+b x^2\right )}-\frac {(3 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 70, normalized size = 0.99 \begin {gather*} -\frac {c}{a^2 x}+\frac {(-b c+a d) x}{2 a^2 \left (a+b x^2\right )}+\frac {(-3 b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 60, normalized size = 0.85
method | result | size |
default | \(\frac {\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (a d -3 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{a^{2}}-\frac {c}{a^{2} x}\) | \(60\) |
risch | \(\frac {\frac {\left (a d -3 b c \right ) x^{2}}{2 a^{2}}-\frac {c}{a}}{x \left (b \,x^{2}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{5} \textit {\_Z}^{2} b +a^{2} d^{2}-6 a b c d +9 b^{2} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{5} b +2 a^{2} d^{2}-12 a b c d +18 b^{2} c^{2}\right ) x +\left (-a^{4} d +3 a^{3} b c \right ) \textit {\_R} \right )\right )}{4}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 65, normalized size = 0.92 \begin {gather*} -\frac {{\left (3 \, b c - a d\right )} x^{2} + 2 \, a c}{2 \, {\left (a^{2} b x^{3} + a^{3} x\right )}} - \frac {{\left (3 \, b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.09, size = 214, normalized size = 3.01 \begin {gather*} \left [-\frac {4 \, a^{2} b c + 2 \, {\left (3 \, a b^{2} c - a^{2} b d\right )} x^{2} - {\left ({\left (3 \, b^{2} c - a b d\right )} x^{3} + {\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{4 \, {\left (a^{3} b^{2} x^{3} + a^{4} b x\right )}}, -\frac {2 \, a^{2} b c + {\left (3 \, a b^{2} c - a^{2} b d\right )} x^{2} + {\left ({\left (3 \, b^{2} c - a b d\right )} x^{3} + {\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{2 \, {\left (a^{3} b^{2} x^{3} + a^{4} b x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.25, size = 114, normalized size = 1.61 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{5} b}} \left (a d - 3 b c\right ) \log {\left (- a^{3} \sqrt {- \frac {1}{a^{5} b}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{5} b}} \left (a d - 3 b c\right ) \log {\left (a^{3} \sqrt {- \frac {1}{a^{5} b}} + x \right )}}{4} + \frac {- 2 a c + x^{2} \left (a d - 3 b c\right )}{2 a^{3} x + 2 a^{2} b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.11, size = 64, normalized size = 0.90 \begin {gather*} -\frac {{\left (3 \, b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} - \frac {3 \, b c x^{2} - a d x^{2} + 2 \, a c}{2 \, {\left (b x^{3} + a x\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.11, size = 61, normalized size = 0.86 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d-3\,b\,c\right )}{2\,a^{5/2}\,\sqrt {b}}-\frac {\frac {c}{a}-\frac {x^2\,\left (a\,d-3\,b\,c\right )}{2\,a^2}}{b\,x^3+a\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________