Optimal. Leaf size=113 \[ -\frac {b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}}-\frac {b^2 x (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac {b (3 A b-2 a B)}{a^4 x}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {A}{5 a^2 x^5} \]
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Rubi [A] time = 0.19, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {456, 1802, 205} \begin {gather*} -\frac {b^2 x (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac {b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {b (3 A b-2 a B)}{a^4 x}-\frac {A}{5 a^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 456
Rule 1802
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^2} \, dx &=-\frac {b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac {1}{2} b^2 \int \frac {-\frac {2 A}{a b^2}+\frac {2 (A b-a B) x^2}{a^2 b^2}-\frac {2 (A b-a B) x^4}{a^3 b}+\frac {(A b-a B) x^6}{a^4}}{x^6 \left (a+b x^2\right )} \, dx\\ &=-\frac {b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac {1}{2} b^2 \int \left (-\frac {2 A}{a^2 b^2 x^6}-\frac {2 (-2 A b+a B)}{a^3 b^2 x^4}+\frac {2 (-3 A b+2 a B)}{a^4 b x^2}+\frac {7 A b-5 a B}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {A}{5 a^2 x^5}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {b (3 A b-2 a B)}{a^4 x}-\frac {b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac {\left (b^2 (7 A b-5 a B)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^4}\\ &=-\frac {A}{5 a^2 x^5}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {b (3 A b-2 a B)}{a^4 x}-\frac {b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac {b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 112, normalized size = 0.99 \begin {gather*} \frac {b^{3/2} (5 a B-7 A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}}+\frac {b^2 x (a B-A b)}{2 a^4 \left (a+b x^2\right )}+\frac {b (2 a B-3 A b)}{a^4 x}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {A}{5 a^2 x^5} \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^6 \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 = 308, normalized size = 2.73 \begin {gather*} \left [\frac {30 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 20 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 12 \, A a^{3} - 4 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} - 15 \, {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{60 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}, \frac {15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 6 \, A a^{3} - 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \, {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 112, normalized size = 0.99 \begin {gather*} \frac {{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} + \frac {B a b^{2} x - A b^{3} x}{2 \, {\left (b x^{2} + a\right )} a^{4}} + \frac {30 \, B a b x^{4} - 45 \, A b^{2} x^{4} - 5 \, B a^{2} x^{2} + 10 \, A a b x^{2} - 3 \, A a^{2}}{15 \, a^{4} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 136, normalized size = 1.20 \begin {gather*} -\frac {A \,b^{3} x}{2 \left (b \,x^{2}+a \right ) a^{4}}-\frac {7 A \,b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{4}}+\frac {B \,b^{2} x}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 B \,b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{3}}-\frac {3 A \,b^{2}}{a^{4} x}+\frac {2 B b}{a^{3} x}+\frac {2 A b}{3 a^{3} x^{3}}-\frac {B}{3 a^{2} x^{3}}-\frac {A}{5 a^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.31, size = 119, normalized size = 1.05 \begin {gather*} \frac {15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 6 \, A a^{3} - 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}} + \frac {{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 104, normalized size = 0.92 \begin {gather*} -\frac {\frac {A}{5\,a}-\frac {x^2\,\left (7\,A\,b-5\,B\,a\right )}{15\,a^2}+\frac {b^2\,x^6\,\left (7\,A\,b-5\,B\,a\right )}{2\,a^4}+\frac {b\,x^4\,\left (7\,A\,b-5\,B\,a\right )}{3\,a^3}}{b\,x^7+a\,x^5}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (7\,A\,b-5\,B\,a\right )}{2\,a^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.68, size = 218, normalized size = 1.93 \begin {gather*} - \frac {\sqrt {- \frac {b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right ) \log {\left (- \frac {a^{5} \sqrt {- \frac {b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right )}{- 7 A b^{3} + 5 B a b^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right ) \log {\left (\frac {a^{5} \sqrt {- \frac {b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right )}{- 7 A b^{3} + 5 B a b^{2}} + x \right )}}{4} + \frac {- 6 A a^{3} + x^{6} \left (- 105 A b^{3} + 75 B a b^{2}\right ) + x^{4} \left (- 70 A a b^{2} + 50 B a^{2} b\right ) + x^{2} \left (14 A a^{2} b - 10 B a^{3}\right )}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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