Optimal. Leaf size=69 \[ \frac {(a C+A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}-\frac {a B-x (A c-a C)}{2 a c \left (a+c x^2\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 69, 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 = {1814, 12, 205} \[ \frac {(a C+A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}-\frac {a B-x (A c-a C)}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 1814
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{\left (a+c x^2\right )^2} \, dx &=-\frac {a B-(A c-a C) x}{2 a c \left (a+c x^2\right )}-\frac {\int \frac {-A-\frac {a C}{c}}{a+c x^2} \, dx}{2 a}\\ &=-\frac {a B-(A c-a C) x}{2 a c \left (a+c x^2\right )}+\frac {(A c+a C) \int \frac {1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {a B-(A c-a C) x}{2 a c \left (a+c x^2\right )}+\frac {(A c+a C) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 68, normalized size = 0.99 \[ \frac {(a C+A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}+\frac {-a B-a C x+A c x}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 195, normalized size = 2.83 \[ \left [-\frac {2 \, B a^{2} c + {\left (C a^{2} + A a c + {\left (C a c + A c^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (C a^{2} c - A a c^{2}\right )} x}{4 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, -\frac {B a^{2} c - {\left (C a^{2} + A a c + {\left (C a c + A c^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (C a^{2} c - A a c^{2}\right )} x}{2 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 60, normalized size = 0.87 \[ \frac {{\left (C a + A c\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} - \frac {C a x - A c x + B a}{2 \, {\left (c x^{2} + a\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 76, normalized size = 1.10 \[ \frac {A \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a}+\frac {C \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {-\frac {B}{2 c}+\frac {\left (A c -a C \right ) x}{2 a c}}{c \,x^{2}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 62, normalized size = 0.90 \[ -\frac {B a + {\left (C a - A c\right )} x}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} + \frac {{\left (C a + A c\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 60, normalized size = 0.87 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (A\,c+C\,a\right )}{2\,a^{3/2}\,c^{3/2}}-\frac {\frac {B}{2\,c}-\frac {x\,\left (A\,c-C\,a\right )}{2\,a\,c}}{c\,x^2+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 116, normalized size = 1.68 \[ - \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \left (A c + C a\right ) \log {\left (- a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \left (A c + C a\right ) \log {\left (a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x \right )}}{4} + \frac {- B a + x \left (A c - C a\right )}{2 a^{2} c + 2 a c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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