Optimal. Leaf size=67 \[ \frac {(A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}-\frac {x (A b-a B)}{2 b^2 \left (a+b x^2\right )}+\frac {B x}{b^2} \]
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Rubi [A] time = 0.05, antiderivative size = 67, 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 = {455, 388, 205} \begin {gather*} -\frac {x (A b-a B)}{2 b^2 \left (a+b x^2\right )}+\frac {(A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}+\frac {B x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 455
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=-\frac {(A b-a B) x}{2 b^2 \left (a+b x^2\right )}-\frac {\int \frac {-A b+a B-2 b B x^2}{a+b x^2} \, dx}{2 b^2}\\ &=\frac {B x}{b^2}-\frac {(A b-a B) x}{2 b^2 \left (a+b x^2\right )}+\frac {(A b-3 a B) \int \frac {1}{a+b x^2} \, dx}{2 b^2}\\ &=\frac {B x}{b^2}-\frac {(A b-a B) x}{2 b^2 \left (a+b x^2\right )}+\frac {(A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 68, normalized size = 1.01 \begin {gather*} -\frac {(3 a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}-\frac {x (A b-a B)}{2 b^2 \left (a+b x^2\right )}+\frac {B 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^2 \left (A+B x^2\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 = 0.46, size = 208, normalized size = 3.10 \begin {gather*} \left [\frac {4 \, B a b^{2} x^{3} + {\left (3 \, B a^{2} - A a b + {\left (3 \, B 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 (3 \, B a^{2} b - A a b^{2}\right )} x}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, \frac {2 \, B a b^{2} x^{3} - {\left (3 \, B a^{2} - A a b + {\left (3 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (3 \, B a^{2} b - A a b^{2}\right )} x}{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 = 0.34, size = 59, normalized size = 0.88 \begin {gather*} \frac {B x}{b^{2}} - \frac {{\left (3 \, B a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {B a x - A b 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 = 82, normalized size = 1.22 \begin {gather*} -\frac {A x}{2 \left (b \,x^{2}+a \right ) b}+\frac {A \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b}+\frac {B a x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {3 B a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}+\frac {B x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.36, size = 61, normalized size = 0.91 \begin {gather*} \frac {{\left (B a - A b\right )} x}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {B x}{b^{2}} - \frac {{\left (3 \, B a - A 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 [B] time = 0.12, size = 59, normalized size = 0.88 \begin {gather*} \frac {B\,x}{b^2}-\frac {x\,\left (\frac {A\,b}{2}-\frac {B\,a}{2}\right )}{b^3\,x^2+a\,b^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b-3\,B\,a\right )}{2\,\sqrt {a}\,b^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 114, normalized size = 1.70 \begin {gather*} \frac {B x}{b^{2}} + \frac {x \left (- A b + B a\right )}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {\sqrt {- \frac {1}{a b^{5}}} \left (- A b + 3 B a\right ) \log {\left (- a b^{2} \sqrt {- \frac {1}{a b^{5}}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a b^{5}}} \left (- A b + 3 B a\right ) \log {\left (a b^{2} \sqrt {- \frac {1}{a b^{5}}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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