Optimal. Leaf size=93 \[ \frac {(a C+A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {x (A b-a C)-a \left (B-\frac {a D}{b}\right )}{2 a b \left (a+b x^2\right )}+\frac {D \log \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A] time = 0.07, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1814, 635, 205, 260} \[ \frac {(a C+A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}-\frac {a \left (B-\frac {a D}{b}\right )-x (A b-a C)}{2 a b \left (a+b x^2\right )}+\frac {D \log \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 1814
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx &=-\frac {a \left (B-\frac {a D}{b}\right )-(A b-a C) x}{2 a b \left (a+b x^2\right )}-\frac {\int \frac {-\frac {A b+a C}{b}-\frac {2 a D x}{b}}{a+b x^2} \, dx}{2 a}\\ &=-\frac {a \left (B-\frac {a D}{b}\right )-(A b-a C) x}{2 a b \left (a+b x^2\right )}+\frac {(A b+a C) \int \frac {1}{a+b x^2} \, dx}{2 a b}+\frac {D \int \frac {x}{a+b x^2} \, dx}{b}\\ &=-\frac {a \left (B-\frac {a D}{b}\right )-(A b-a C) x}{2 a b \left (a+b x^2\right )}+\frac {(A b+a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {D \log \left (a+b x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 83, normalized size = 0.89 \[ \frac {\frac {\sqrt {b} (a C+A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}+\frac {a^2 D-a b (B+C x)+A b^2 x}{a \left (a+b x^2\right )}+D \log \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 257, normalized size = 2.76 \[ \left [\frac {2 \, D a^{3} - 2 \, B a^{2} b - {\left (C a^{2} + A a b + {\left (C a b + A b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, {\left (C a^{2} b - A a b^{2}\right )} x + 2 \, {\left (D a^{2} b x^{2} + D a^{3}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}, \frac {D a^{3} - B a^{2} b + {\left (C a^{2} + A a b + {\left (C a b + A b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (C a^{2} b - A a b^{2}\right )} x + {\left (D a^{2} b x^{2} + D a^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 88, normalized size = 0.95 \[ \frac {D \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {{\left (C a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} - \frac {{\left (C a - A b\right )} x - \frac {D a^{2} - B a b}{b}}{2 \, {\left (b x^{2} + a\right )} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 97, normalized size = 1.04 \[ \frac {A \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a}+\frac {C \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b}+\frac {D \ln \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {\frac {\left (A b -a C \right ) x}{2 a b}-\frac {b B -a D}{2 b^{2}}}{b \,x^{2}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 89, normalized size = 0.96 \[ \frac {D a^{2} - B a b - {\left (C a b - A b^{2}\right )} x}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}} + \frac {D \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {{\left (C a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 110, normalized size = 1.18 \[ \frac {\left (\ln \left (b\,x^2+a\right )+\frac {a}{b\,x^2+a}\right )\,D}{2\,b^2}-\frac {B}{2\,b\,\left (b\,x^2+a\right )}+\frac {A\,x}{2\,a\,\left (b\,x^2+a\right )}-\frac {C\,x}{2\,b\,\left (b\,x^2+a\right )}+\frac {A\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}}+\frac {C\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.09, size = 233, normalized size = 2.51 \[ \left (\frac {D}{2 b^{2}} - \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right ) \log {\left (x + \frac {- 2 D a^{2} + 4 a^{2} b^{2} \left (\frac {D}{2 b^{2}} - \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right )}{A b^{2} + C a b} \right )} + \left (\frac {D}{2 b^{2}} + \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right ) \log {\left (x + \frac {- 2 D a^{2} + 4 a^{2} b^{2} \left (\frac {D}{2 b^{2}} + \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right )}{A b^{2} + C a b} \right )} + \frac {- B a b + D a^{2} + x \left (A b^{2} - C a b\right )}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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