Optimal. Leaf size=80 \[ -\frac {b^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}}-\frac {b (A b-a B)}{a^3 x}+\frac {A b-a B}{3 a^2 x^3}-\frac {A}{5 a x^5} \]
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Rubi [A] time = 0.05, antiderivative size = 80, 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 = {453, 325, 205} \[ -\frac {b^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}}+\frac {A b-a B}{3 a^2 x^3}-\frac {b (A b-a B)}{a^3 x}-\frac {A}{5 a x^5} \]
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^6 \left (a+b x^2\right )} \, dx &=-\frac {A}{5 a x^5}-\frac {(5 A b-5 a B) \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{5 a}\\ &=-\frac {A}{5 a x^5}+\frac {A b-a B}{3 a^2 x^3}+\frac {(b (A b-a B)) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{a^2}\\ &=-\frac {A}{5 a x^5}+\frac {A b-a B}{3 a^2 x^3}-\frac {b (A b-a B)}{a^3 x}-\frac {\left (b^2 (A b-a B)\right ) \int \frac {1}{a+b x^2} \, dx}{a^3}\\ &=-\frac {A}{5 a x^5}+\frac {A b-a B}{3 a^2 x^3}-\frac {b (A b-a B)}{a^3 x}-\frac {b^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 78, normalized size = 0.98 \[ \frac {b^{3/2} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}}+\frac {b (a B-A b)}{a^3 x}+\frac {A b-a B}{3 a^2 x^3}-\frac {A}{5 a x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 184, normalized size = 2.30 \[ \left [-\frac {15 \, {\left (B a b - A b^{2}\right )} x^{5} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 30 \, {\left (B a b - A b^{2}\right )} x^{4} + 6 \, A a^{2} + 10 \, {\left (B a^{2} - A a b\right )} x^{2}}{30 \, a^{3} x^{5}}, \frac {15 \, {\left (B a b - A b^{2}\right )} x^{5} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 15 \, {\left (B a b - A b^{2}\right )} x^{4} - 3 \, A a^{2} - 5 \, {\left (B a^{2} - A a b\right )} x^{2}}{15 \, a^{3} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 81, normalized size = 1.01 \[ \frac {{\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} + \frac {15 \, B a b x^{4} - 15 \, A b^{2} x^{4} - 5 \, B a^{2} x^{2} + 5 \, A a b x^{2} - 3 \, A a^{2}}{15 \, a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 1.20 \[ -\frac {A \,b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}+\frac {B \,b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}-\frac {A \,b^{2}}{a^{3} x}+\frac {B b}{a^{2} x}+\frac {A b}{3 a^{2} x^{3}}-\frac {B}{3 a \,x^{3}}-\frac {A}{5 a \,x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.46, size = 79, normalized size = 0.99 \[ \frac {{\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} + \frac {15 \, {\left (B a b - A b^{2}\right )} x^{4} - 3 \, A a^{2} - 5 \, {\left (B a^{2} - A a b\right )} x^{2}}{15 \, a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 70, normalized size = 0.88 \[ -\frac {\frac {A}{5\,a}-\frac {x^2\,\left (A\,b-B\,a\right )}{3\,a^2}+\frac {b\,x^4\,\left (A\,b-B\,a\right )}{a^3}}{x^5}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b-B\,a\right )}{a^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.49, size = 163, normalized size = 2.04 \[ - \frac {\sqrt {- \frac {b^{3}}{a^{7}}} \left (- A b + B a\right ) \log {\left (- \frac {a^{4} \sqrt {- \frac {b^{3}}{a^{7}}} \left (- A b + B a\right )}{- A b^{3} + B a b^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{3}}{a^{7}}} \left (- A b + B a\right ) \log {\left (\frac {a^{4} \sqrt {- \frac {b^{3}}{a^{7}}} \left (- A b + B a\right )}{- A b^{3} + B a b^{2}} + x \right )}}{2} + \frac {- 3 A a^{2} + x^{4} \left (- 15 A b^{2} + 15 B a b\right ) + x^{2} \left (5 A a b - 5 B a^{2}\right )}{15 a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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