Optimal. Leaf size=99 \[ \frac {b^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}+\frac {b^2 (A b-a B)}{a^4 x}-\frac {b (A b-a B)}{3 a^3 x^3}+\frac {A b-a B}{5 a^2 x^5}-\frac {A}{7 a x^7} \]
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Rubi [A] time = 0.07, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {b^2 (A b-a B)}{a^4 x}+\frac {b^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}-\frac {b (A b-a B)}{3 a^3 x^3}+\frac {A b-a B}{5 a^2 x^5}-\frac {A}{7 a x^7} \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^8 \left (a+b x^2\right )} \, dx &=-\frac {A}{7 a x^7}-\frac {(7 A b-7 a B) \int \frac {1}{x^6 \left (a+b x^2\right )} \, dx}{7 a}\\ &=-\frac {A}{7 a x^7}+\frac {A b-a B}{5 a^2 x^5}+\frac {(b (A b-a B)) \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{a^2}\\ &=-\frac {A}{7 a x^7}+\frac {A b-a B}{5 a^2 x^5}-\frac {b (A b-a B)}{3 a^3 x^3}-\frac {\left (b^2 (A b-a B)\right ) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{a^3}\\ &=-\frac {A}{7 a x^7}+\frac {A b-a B}{5 a^2 x^5}-\frac {b (A b-a B)}{3 a^3 x^3}+\frac {b^2 (A b-a B)}{a^4 x}+\frac {\left (b^3 (A b-a B)\right ) \int \frac {1}{a+b x^2} \, dx}{a^4}\\ &=-\frac {A}{7 a x^7}+\frac {A b-a B}{5 a^2 x^5}-\frac {b (A b-a B)}{3 a^3 x^3}+\frac {b^2 (A b-a B)}{a^4 x}+\frac {b^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 101, normalized size = 1.02 \begin {gather*} -\frac {b^{5/2} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}-\frac {b^2 (a B-A b)}{a^4 x}+\frac {b (a B-A b)}{3 a^3 x^3}+\frac {A b-a B}{5 a^2 x^5}-\frac {A}{7 a x^7} \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^8 \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 = 234, normalized size = 2.36 \begin {gather*} \left [-\frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{7} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 210 \, {\left (B a b^{2} - A b^{3}\right )} x^{6} - 70 \, {\left (B a^{2} b - A a b^{2}\right )} x^{4} + 30 \, A a^{3} + 42 \, {\left (B a^{3} - A a^{2} b\right )} x^{2}}{210 \, a^{4} x^{7}}, -\frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{7} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{6} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x^{2}}{105 \, a^{4} x^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 106, normalized size = 1.07 \begin {gather*} -\frac {{\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {105 \, B a b^{2} x^{6} - 105 \, A b^{3} x^{6} - 35 \, B a^{2} b x^{4} + 35 \, A a b^{2} x^{4} + 21 \, B a^{3} x^{2} - 21 \, A a^{2} b x^{2} + 15 \, A a^{3}}{105 \, a^{4} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 120, normalized size = 1.21 \begin {gather*} \frac {A \,b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}-\frac {B \,b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}+\frac {A \,b^{3}}{a^{4} x}-\frac {B \,b^{2}}{a^{3} x}-\frac {A \,b^{2}}{3 a^{3} x^{3}}+\frac {B b}{3 a^{2} x^{3}}+\frac {A b}{5 a^{2} x^{5}}-\frac {B}{5 a \,x^{5}}-\frac {A}{7 a \,x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.48, size = 103, normalized size = 1.04 \begin {gather*} -\frac {{\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{6} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x^{2}}{105 \, a^{4} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 89, normalized size = 0.90 \begin {gather*} \frac {b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b-B\,a\right )}{a^{9/2}}-\frac {\frac {A}{7\,a}-\frac {x^2\,\left (A\,b-B\,a\right )}{5\,a^2}-\frac {b^2\,x^6\,\left (A\,b-B\,a\right )}{a^4}+\frac {b\,x^4\,\left (A\,b-B\,a\right )}{3\,a^3}}{x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.55, size = 187, normalized size = 1.89 \begin {gather*} \frac {\sqrt {- \frac {b^{5}}{a^{9}}} \left (- A b + B a\right ) \log {\left (- \frac {a^{5} \sqrt {- \frac {b^{5}}{a^{9}}} \left (- A b + B a\right )}{- A b^{4} + B a b^{3}} + x \right )}}{2} - \frac {\sqrt {- \frac {b^{5}}{a^{9}}} \left (- A b + B a\right ) \log {\left (\frac {a^{5} \sqrt {- \frac {b^{5}}{a^{9}}} \left (- A b + B a\right )}{- A b^{4} + B a b^{3}} + x \right )}}{2} + \frac {- 15 A a^{3} + x^{6} \left (105 A b^{3} - 105 B a b^{2}\right ) + x^{4} \left (- 35 A a b^{2} + 35 B a^{2} b\right ) + x^{2} \left (21 A a^{2} b - 21 B a^{3}\right )}{105 a^{4} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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