Optimal. Leaf size=97 \[ -\frac {b (3 A b-2 a B) \log \left (a+b x^2\right )}{2 a^4}+\frac {b \log (x) (3 A b-2 a B)}{a^4}+\frac {b (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac {2 A b-a B}{2 a^3 x^2}-\frac {A}{4 a^2 x^4} \]
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Rubi [A] time = 0.10, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 77} \[ \frac {b (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac {2 A b-a B}{2 a^3 x^2}-\frac {b (3 A b-2 a B) \log \left (a+b x^2\right )}{2 a^4}+\frac {b \log (x) (3 A b-2 a B)}{a^4}-\frac {A}{4 a^2 x^4} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^3 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A}{a^2 x^3}+\frac {-2 A b+a B}{a^3 x^2}-\frac {b (-3 A b+2 a B)}{a^4 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)^2}+\frac {b^2 (-3 A b+2 a B)}{a^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{4 a^2 x^4}+\frac {2 A b-a B}{2 a^3 x^2}+\frac {b (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac {b (3 A b-2 a B) \log (x)}{a^4}-\frac {b (3 A b-2 a B) \log \left (a+b x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 85, normalized size = 0.88 \[ -\frac {\frac {a^2 A}{x^4}+\frac {2 a b (a B-A b)}{a+b x^2}+\frac {2 a (a B-2 A b)}{x^2}+2 b (3 A b-2 a B) \log \left (a+b x^2\right )-4 b \log (x) (3 A b-2 a B)}{4 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 154, normalized size = 1.59 \[ -\frac {2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{4} + A a^{3} + {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x^{2} - 2 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{6} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{4}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{6} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{4}\right )} \log \relax (x)}{4 \, {\left (a^{4} b x^{6} + a^{5} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 150, normalized size = 1.55 \[ -\frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4}} + \frac {{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} b} - \frac {2 \, B a b^{2} x^{2} - 3 \, A b^{3} x^{2} + 3 \, B a^{2} b - 4 \, A a b^{2}}{2 \, {\left (b x^{2} + a\right )} a^{4}} + \frac {6 \, B a b x^{4} - 9 \, A b^{2} x^{4} - 2 \, B a^{2} x^{2} + 4 \, A a b x^{2} - A a^{2}}{4 \, a^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 114, normalized size = 1.18 \[ \frac {A \,b^{2}}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {3 A \,b^{2} \ln \relax (x )}{a^{4}}-\frac {3 A \,b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{4}}-\frac {B b}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {2 B b \ln \relax (x )}{a^{3}}+\frac {B b \ln \left (b \,x^{2}+a \right )}{a^{3}}+\frac {A b}{a^{3} x^{2}}-\frac {B}{2 a^{2} x^{2}}-\frac {A}{4 a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 106, normalized size = 1.09 \[ -\frac {2 \, {\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} + A a^{2} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}}{4 \, {\left (a^{3} b x^{6} + a^{4} x^{4}\right )}} + \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 100, normalized size = 1.03 \[ \frac {\frac {x^2\,\left (3\,A\,b-2\,B\,a\right )}{4\,a^2}-\frac {A}{4\,a}+\frac {b\,x^4\,\left (3\,A\,b-2\,B\,a\right )}{2\,a^3}}{b\,x^6+a\,x^4}-\frac {\ln \left (b\,x^2+a\right )\,\left (3\,A\,b^2-2\,B\,a\,b\right )}{2\,a^4}+\frac {\ln \relax (x)\,\left (3\,A\,b^2-2\,B\,a\,b\right )}{a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.02, size = 100, normalized size = 1.03 \[ \frac {- A a^{2} + x^{4} \left (6 A b^{2} - 4 B a b\right ) + x^{2} \left (3 A a b - 2 B a^{2}\right )}{4 a^{4} x^{4} + 4 a^{3} b x^{6}} - \frac {b \left (- 3 A b + 2 B a\right ) \log {\relax (x )}}{a^{4}} + \frac {b \left (- 3 A b + 2 B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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