Optimal. Leaf size=103 \[ -\frac {(b c-a d) (a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} b^{3/2}}-\frac {x \left (\frac {3 b c^2}{a}+\frac {a d^2}{b}-2 c d\right )}{2 a \left (a+b x^2\right )}-\frac {c^2}{a x \left (a+b x^2\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {462, 385, 205} \begin {gather*} -\frac {(b c-a d) (a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} b^{3/2}}-\frac {x \left (\frac {3 b c^2}{a}+\frac {a d^2}{b}-2 c d\right )}{2 a \left (a+b x^2\right )}-\frac {c^2}{a x \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 385
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{x^2 \left (a+b x^2\right )^2} \, dx &=-\frac {c^2}{a x \left (a+b x^2\right )}+\frac {\int \frac {-c (3 b c-2 a d)+a d^2 x^2}{\left (a+b x^2\right )^2} \, dx}{a}\\ &=-\frac {c^2}{a x \left (a+b x^2\right )}-\frac {\left (\frac {d^2}{b}+\frac {c (3 b c-2 a d)}{a^2}\right ) x}{2 \left (a+b x^2\right )}-\frac {((b c-a d) (3 b c+a d)) \int \frac {1}{a+b x^2} \, dx}{2 a^2 b}\\ &=-\frac {c^2}{a x \left (a+b x^2\right )}-\frac {\left (\frac {d^2}{b}+\frac {c (3 b c-2 a d)}{a^2}\right ) x}{2 \left (a+b x^2\right )}-\frac {(b c-a d) (3 b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} b^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 91, normalized size = 0.88 \begin {gather*} -\frac {x (a d-b c)^2}{2 a^2 b \left (a+b x^2\right )}-\frac {c^2}{a^2 x}+\frac {\left (a^2 d^2+2 a b c d-3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c+d x^2\right )^2}{x^2 \left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.98, size = 308, normalized size = 2.99 \begin {gather*} \left [-\frac {4 \, a^{2} b^{2} c^{2} + 2 \, {\left (3 \, a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} - {\left ({\left (3 \, b^{3} c^{2} - 2 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 2 \, a^{2} b c d - a^{3} d^{2}\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^{3} x^{3} + a^{4} b^{2} x\right )}}, -\frac {2 \, a^{2} b^{2} c^{2} + {\left (3 \, a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} + {\left ({\left (3 \, b^{3} c^{2} - 2 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 2 \, a^{2} b c d - a^{3} d^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{2 \, {\left (a^{3} b^{3} x^{3} + a^{4} b^{2} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.33, size = 103, normalized size = 1.00 \begin {gather*} -\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2} b} - \frac {3 \, b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + a^{2} d^{2} x^{2} + 2 \, a b c^{2}}{2 \, {\left (b x^{3} + a x\right )} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 131, normalized size = 1.27 \begin {gather*} \frac {c d x}{\left (b \,x^{2}+a \right ) a}+\frac {c d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}-\frac {b \,c^{2} x}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 b \,c^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{2}}-\frac {d^{2} x}{2 \left (b \,x^{2}+a \right ) b}+\frac {d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b}-\frac {c^{2}}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.29, size = 101, normalized size = 0.98 \begin {gather*} -\frac {2 \, a b c^{2} + {\left (3 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{2 \, {\left (a^{2} b^{2} x^{3} + a^{3} b x\right )}} - \frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.14, size = 128, normalized size = 1.24 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,\left (a\,d-b\,c\right )\,\left (a\,d+3\,b\,c\right )}{\sqrt {a}\,\left (a^2\,d^2+2\,a\,b\,c\,d-3\,b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d+3\,b\,c\right )}{2\,a^{5/2}\,b^{3/2}}-\frac {\frac {c^2}{a}+\frac {x^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+3\,b^2\,c^2\right )}{2\,a^2\,b}}{b\,x^3+a\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.87, size = 238, normalized size = 2.31 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right ) \left (a d + 3 b c\right ) \log {\left (- \frac {a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right ) \left (a d + 3 b c\right ) \log {\left (\frac {a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}} + x \right )}}{4} + \frac {- 2 a b c^{2} + x^{2} \left (- a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}\right )}{2 a^{3} b x + 2 a^{2} b^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________