Optimal. Leaf size=59 \[ \frac {\sqrt {b} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A b-a B}{a^2 x}-\frac {A}{3 a x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 59, 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 = {453, 325, 205} \begin {gather*} \frac {A b-a B}{a^2 x}+\frac {\sqrt {b} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}-\frac {A}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 325
Rule 453
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )} \, dx &=-\frac {A}{3 a x^3}-\frac {(3 A b-3 a B) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{3 a}\\ &=-\frac {A}{3 a x^3}+\frac {A b-a B}{a^2 x}+\frac {(b (A b-a B)) \int \frac {1}{a+b x^2} \, dx}{a^2}\\ &=-\frac {A}{3 a x^3}+\frac {A b-a B}{a^2 x}+\frac {\sqrt {b} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 1.02 \begin {gather*} -\frac {\sqrt {b} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A b-a B}{a^2 x}-\frac {A}{3 a x^3} \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 {A+B x^2}{x^4 \left (a+b x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.45, size = 135, normalized size = 2.29 \begin {gather*} \left [-\frac {3 \, {\left (B a - A b\right )} x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 6 \, {\left (B a - A b\right )} x^{2} + 2 \, A a}{6 \, a^{2} x^{3}}, -\frac {3 \, {\left (B a - A b\right )} x^{3} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 3 \, {\left (B a - A b\right )} x^{2} + A a}{3 \, a^{2} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 57, normalized size = 0.97 \begin {gather*} -\frac {{\left (B a b - A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} - \frac {3 \, B a x^{2} - 3 \, A b x^{2} + A a}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 1.22 \begin {gather*} \frac {A \,b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}-\frac {B b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {A b}{a^{2} x}-\frac {B}{a x}-\frac {A}{3 a \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 56, normalized size = 0.95 \begin {gather*} -\frac {{\left (B a b - A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} - \frac {3 \, {\left (B a - A b\right )} x^{2} + A a}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 53, normalized size = 0.90 \begin {gather*} \frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b-B\,a\right )}{a^{5/2}}-\frac {\frac {A}{3\,a}-\frac {x^2\,\left (A\,b-B\,a\right )}{a^2}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.42, size = 129, normalized size = 2.19 \begin {gather*} \frac {\sqrt {- \frac {b}{a^{5}}} \left (- A b + B a\right ) \log {\left (- \frac {a^{3} \sqrt {- \frac {b}{a^{5}}} \left (- A b + B a\right )}{- A b^{2} + B a b} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{5}}} \left (- A b + B a\right ) \log {\left (\frac {a^{3} \sqrt {- \frac {b}{a^{5}}} \left (- A b + B a\right )}{- A b^{2} + B a b} + x \right )}}{2} + \frac {- A a + x^{2} \left (3 A b - 3 B a\right )}{3 a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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