Optimal. Leaf size=124 \[ \frac {b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{2 a^5}-\frac {b^2 \log (x) (4 A b-3 a B)}{a^5}-\frac {b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac {b (3 A b-2 a B)}{2 a^4 x^2}+\frac {2 A b-a B}{4 a^3 x^4}-\frac {A}{6 a^2 x^6} \]
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Rubi [A] time = 0.13, antiderivative size = 124, 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} \[ -\frac {b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )}+\frac {b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{2 a^5}-\frac {b^2 \log (x) (4 A b-3 a B)}{a^5}-\frac {b (3 A b-2 a B)}{2 a^4 x^2}+\frac {2 A b-a B}{4 a^3 x^4}-\frac {A}{6 a^2 x^6} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^4 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A}{a^2 x^4}+\frac {-2 A b+a B}{a^3 x^3}-\frac {b (-3 A b+2 a B)}{a^4 x^2}+\frac {b^2 (-4 A b+3 a B)}{a^5 x}-\frac {b^3 (-A b+a B)}{a^4 (a+b x)^2}-\frac {b^3 (-4 A b+3 a B)}{a^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{6 a^2 x^6}+\frac {2 A b-a B}{4 a^3 x^4}-\frac {b (3 A b-2 a B)}{2 a^4 x^2}-\frac {b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac {b^2 (4 A b-3 a B) \log (x)}{a^5}+\frac {b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{2 a^5}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 110, normalized size = 0.89 \[ \frac {-\frac {2 a^3 A}{x^6}-\frac {3 a^2 (a B-2 A b)}{x^4}+\frac {6 a b^2 (a B-A b)}{a+b x^2}+6 b^2 (4 A b-3 a B) \log \left (a+b x^2\right )+12 b^2 \log (x) (3 a B-4 A b)+\frac {6 a b (2 a B-3 A b)}{x^2}}{12 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 184, normalized size = 1.48 \[ \frac {6 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} - 2 \, A a^{4} + 3 \, {\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{4} - {\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x^{2} - 6 \, {\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{8} + {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{8} + {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6}\right )} \log \relax (x)}{12 \, {\left (a^{5} b x^{8} + a^{6} x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 178, normalized size = 1.44 \[ \frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5}} - \frac {{\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{5} b} + \frac {3 \, B a b^{3} x^{2} - 4 \, A b^{4} x^{2} + 4 \, B a^{2} b^{2} - 5 \, A a b^{3}}{2 \, {\left (b x^{2} + a\right )} a^{5}} - \frac {33 \, B a b^{2} x^{6} - 44 \, A b^{3} x^{6} - 12 \, B a^{2} b x^{4} + 18 \, A a b^{2} x^{4} + 3 \, B a^{3} x^{2} - 6 \, A a^{2} b x^{2} + 2 \, A a^{3}}{12 \, a^{5} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 143, normalized size = 1.15 \[ -\frac {A \,b^{3}}{2 \left (b \,x^{2}+a \right ) a^{4}}-\frac {4 A \,b^{3} \ln \relax (x )}{a^{5}}+\frac {2 A \,b^{3} \ln \left (b \,x^{2}+a \right )}{a^{5}}+\frac {B \,b^{2}}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {3 B \,b^{2} \ln \relax (x )}{a^{4}}-\frac {3 B \,b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{4}}-\frac {3 A \,b^{2}}{2 a^{4} x^{2}}+\frac {B b}{a^{3} x^{2}}+\frac {A b}{2 a^{3} x^{4}}-\frac {B}{4 a^{2} x^{4}}-\frac {A}{6 a^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 136, normalized size = 1.10 \[ \frac {6 \, {\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} x^{6} + 3 \, {\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{4} - 2 \, A a^{3} - {\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} x^{2}}{12 \, {\left (a^{4} b x^{8} + a^{5} x^{6}\right )}} - \frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{5}} + \frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 126, normalized size = 1.02 \[ \frac {\ln \left (b\,x^2+a\right )\,\left (4\,A\,b^3-3\,B\,a\,b^2\right )}{2\,a^5}-\frac {\frac {A}{6\,a}-\frac {x^2\,\left (4\,A\,b-3\,B\,a\right )}{12\,a^2}+\frac {b^2\,x^6\,\left (4\,A\,b-3\,B\,a\right )}{2\,a^4}+\frac {b\,x^4\,\left (4\,A\,b-3\,B\,a\right )}{4\,a^3}}{b\,x^8+a\,x^6}-\frac {\ln \relax (x)\,\left (4\,A\,b^3-3\,B\,a\,b^2\right )}{a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.10, size = 129, normalized size = 1.04 \[ \frac {- 2 A a^{3} + x^{6} \left (- 24 A b^{3} + 18 B a b^{2}\right ) + x^{4} \left (- 12 A a b^{2} + 9 B a^{2} b\right ) + x^{2} \left (4 A a^{2} b - 3 B a^{3}\right )}{12 a^{5} x^{6} + 12 a^{4} b x^{8}} + \frac {b^{2} \left (- 4 A b + 3 B a\right ) \log {\relax (x )}}{a^{5}} - \frac {b^{2} \left (- 4 A b + 3 B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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