Optimal. Leaf size=101 \[ -\frac {x \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac {(b B-3 a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}+\frac {C \log \left (a+b x^2\right )}{2 b^2}+\frac {D x}{b^2} \]
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Rubi [A] time = 0.12, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1804, 1810, 635, 205, 260} \begin {gather*} -\frac {x \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac {(b B-3 a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}+\frac {C \log \left (a+b x^2\right )}{2 b^2}+\frac {D x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 1804
Rule 1810
Rubi steps
\begin {align*} \int \frac {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx &=-\frac {x \left (a \left (B-\frac {a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}-\frac {\int \frac {-a \left (B-\frac {a D}{b}\right )-2 a C x-2 a D x^2}{a+b x^2} \, dx}{2 a b}\\ &=-\frac {x \left (a \left (B-\frac {a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}-\frac {\int \left (-\frac {2 a D}{b}-\frac {a (b B-3 a D)+2 a b C x}{b \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=\frac {D x}{b^2}-\frac {x \left (a \left (B-\frac {a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}+\frac {\int \frac {a (b B-3 a D)+2 a b C x}{a+b x^2} \, dx}{2 a b^2}\\ &=\frac {D x}{b^2}-\frac {x \left (a \left (B-\frac {a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}+\frac {C \int \frac {x}{a+b x^2} \, dx}{b}+\frac {(b B-3 a D) \int \frac {1}{a+b x^2} \, dx}{2 b^2}\\ &=\frac {D x}{b^2}-\frac {x \left (a \left (B-\frac {a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}+\frac {(b B-3 a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}+\frac {C \log \left (a+b x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 92, normalized size = 0.91 \begin {gather*} \frac {a C+a D x-A b-b B x}{2 b^2 \left (a+b x^2\right )}-\frac {(3 a D-b B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}+\frac {C \log \left (a+b x^2\right )}{2 b^2}+\frac {D x}{b^2} \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 {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.81, size = 287, normalized size = 2.84 \begin {gather*} \left [\frac {4 \, D a b^{2} x^{3} + 2 \, C a^{2} b - 2 \, A a b^{2} - {\left (3 \, D a^{2} - B a b + {\left (3 \, D a b - B 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 (3 \, D a^{2} b - B a b^{2}\right )} x + 2 \, {\left (C a b^{2} x^{2} + C a^{2} b\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, \frac {2 \, D a b^{2} x^{3} + C a^{2} b - A a b^{2} - {\left (3 \, D a^{2} - B a b + {\left (3 \, D a b - B b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (3 \, D a^{2} b - B a b^{2}\right )} x + {\left (C a b^{2} x^{2} + C a^{2} b\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 81, normalized size = 0.80 \begin {gather*} \frac {D x}{b^{2}} + \frac {C \log \left (b x^{2} + a\right )}{2 \, b^{2}} - \frac {{\left (3 \, D a - B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {C a - A b + {\left (D a - B b\right )} x}{2 \, {\left (b x^{2} + a\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 127, normalized size = 1.26 \begin {gather*} -\frac {B x}{2 \left (b \,x^{2}+a \right ) b}+\frac {B \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b}+\frac {D a x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {3 D a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}-\frac {A}{2 \left (b \,x^{2}+a \right ) b}+\frac {C a}{2 \left (b \,x^{2}+a \right ) b^{2}}+\frac {C \ln \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {D x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 84, normalized size = 0.83 \begin {gather*} \frac {C a - A b + {\left (D a - B b\right )} x}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {D x}{b^{2}} + \frac {C \log \left (b x^{2} + a\right )}{2 \, b^{2}} - \frac {{\left (3 \, D a - B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.76, size = 212, normalized size = 2.10 \begin {gather*} \frac {D x}{b^{2}} + \left (\frac {C}{2 b^{2}} - \frac {\sqrt {- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right ) \log {\left (x + \frac {2 C a - 4 a b^{2} \left (\frac {C}{2 b^{2}} - \frac {\sqrt {- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right )}{- B b + 3 D a} \right )} + \left (\frac {C}{2 b^{2}} + \frac {\sqrt {- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right ) \log {\left (x + \frac {2 C a - 4 a b^{2} \left (\frac {C}{2 b^{2}} + \frac {\sqrt {- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right )}{- B b + 3 D a} \right )} + \frac {- A b + C a + x \left (- B b + D a\right )}{2 a b^{2} + 2 b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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