Optimal. Leaf size=71 \[ -\frac {(3 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b}}-\frac {x (b c-a d)}{2 a^2 \left (a+b x^2\right )}-\frac {c}{a^2 x} \]
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Rubi [A] time = 0.05, 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 = {456, 453, 205} \begin {gather*} -\frac {x (b c-a d)}{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}}-\frac {c}{a^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 453
Rule 456
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*}
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Mathematica [A] time = 0.03, size = 70, normalized size = 0.99 \begin {gather*} \frac {(a d-3 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b}}+\frac {x (a d-b c)}{2 a^2 \left (a+b x^2\right )}-\frac {c}{a^2 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.95, 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.
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giac [A] time = 0.35, 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.
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maple [A] time = 0.01, size = 85, normalized size = 1.20 \begin {gather*} \frac {d x}{2 \left (b \,x^{2}+a \right ) a}+\frac {d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a}-\frac {b c x}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 b c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{2}}-\frac {c}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.23, 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.
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mupad [B] time = 0.19, 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.
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sympy [A] time = 0.49, 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.
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