Optimal. Leaf size=148 \[ -\frac {c^2 (3 b B-5 A c) \log \left (b+c x^2\right )}{b^6}+\frac {2 c^2 \log (x) (3 b B-5 A c)}{b^6}+\frac {c^2 (3 b B-4 A c)}{2 b^5 \left (b+c x^2\right )}+\frac {3 c (b B-2 A c)}{2 b^5 x^2}+\frac {c^2 (b B-A c)}{4 b^4 \left (b+c x^2\right )^2}-\frac {b B-3 A c}{4 b^4 x^4}-\frac {A}{6 b^3 x^6} \]
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Rubi [A] time = 0.17, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1584, 446, 77} \[ \frac {c^2 (3 b B-4 A c)}{2 b^5 \left (b+c x^2\right )}+\frac {c^2 (b B-A c)}{4 b^4 \left (b+c x^2\right )^2}-\frac {c^2 (3 b B-5 A c) \log \left (b+c x^2\right )}{b^6}+\frac {2 c^2 \log (x) (3 b B-5 A c)}{b^6}+\frac {3 c (b B-2 A c)}{2 b^5 x^2}-\frac {b B-3 A c}{4 b^4 x^4}-\frac {A}{6 b^3 x^6} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x \left (b x^2+c x^4\right )^3} \, dx &=\int \frac {A+B x^2}{x^7 \left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^4 (b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A}{b^3 x^4}+\frac {b B-3 A c}{b^4 x^3}-\frac {3 c (b B-2 A c)}{b^5 x^2}+\frac {2 c^2 (3 b B-5 A c)}{b^6 x}-\frac {c^3 (b B-A c)}{b^4 (b+c x)^3}-\frac {c^3 (3 b B-4 A c)}{b^5 (b+c x)^2}-\frac {2 c^3 (3 b B-5 A c)}{b^6 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{6 b^3 x^6}-\frac {b B-3 A c}{4 b^4 x^4}+\frac {3 c (b B-2 A c)}{2 b^5 x^2}+\frac {c^2 (b B-A c)}{4 b^4 \left (b+c x^2\right )^2}+\frac {c^2 (3 b B-4 A c)}{2 b^5 \left (b+c x^2\right )}+\frac {2 c^2 (3 b B-5 A c) \log (x)}{b^6}-\frac {c^2 (3 b B-5 A c) \log \left (b+c x^2\right )}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 135, normalized size = 0.91 \[ \frac {-\frac {2 A b^3}{x^6}+\frac {3 b^2 c^2 (b B-A c)}{\left (b+c x^2\right )^2}-\frac {3 b^2 (b B-3 A c)}{x^4}+\frac {6 b c^2 (3 b B-4 A c)}{b+c x^2}+12 c^2 (5 A c-3 b B) \log \left (b+c x^2\right )+24 c^2 \log (x) (3 b B-5 A c)+\frac {18 b c (b B-2 A c)}{x^2}}{12 b^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 267, normalized size = 1.80 \[ \frac {12 \, {\left (3 \, B b^{2} c^{3} - 5 \, A b c^{4}\right )} x^{8} + 18 \, {\left (3 \, B b^{3} c^{2} - 5 \, A b^{2} c^{3}\right )} x^{6} - 2 \, A b^{5} + 4 \, {\left (3 \, B b^{4} c - 5 \, A b^{3} c^{2}\right )} x^{4} - {\left (3 \, B b^{5} - 5 \, A b^{4} c\right )} x^{2} - 12 \, {\left ({\left (3 \, B b c^{4} - 5 \, A c^{5}\right )} x^{10} + 2 \, {\left (3 \, B b^{2} c^{3} - 5 \, A b c^{4}\right )} x^{8} + {\left (3 \, B b^{3} c^{2} - 5 \, A b^{2} c^{3}\right )} x^{6}\right )} \log \left (c x^{2} + b\right ) + 24 \, {\left ({\left (3 \, B b c^{4} - 5 \, A c^{5}\right )} x^{10} + 2 \, {\left (3 \, B b^{2} c^{3} - 5 \, A b c^{4}\right )} x^{8} + {\left (3 \, B b^{3} c^{2} - 5 \, A b^{2} c^{3}\right )} x^{6}\right )} \log \relax (x)}{12 \, {\left (b^{6} c^{2} x^{10} + 2 \, b^{7} c x^{8} + b^{8} x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 201, normalized size = 1.36 \[ \frac {{\left (3 \, B b c^{2} - 5 \, A c^{3}\right )} \log \left (x^{2}\right )}{b^{6}} - \frac {{\left (3 \, B b c^{3} - 5 \, A c^{4}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{b^{6} c} + \frac {18 \, B b c^{4} x^{4} - 30 \, A c^{5} x^{4} + 42 \, B b^{2} c^{3} x^{2} - 68 \, A b c^{4} x^{2} + 25 \, B b^{3} c^{2} - 39 \, A b^{2} c^{3}}{4 \, {\left (c x^{2} + b\right )}^{2} b^{6}} - \frac {66 \, B b c^{2} x^{6} - 110 \, A c^{3} x^{6} - 18 \, B b^{2} c x^{4} + 36 \, A b c^{2} x^{4} + 3 \, B b^{3} x^{2} - 9 \, A b^{2} c x^{2} + 2 \, A b^{3}}{12 \, b^{6} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 180, normalized size = 1.22 \[ -\frac {A \,c^{3}}{4 \left (c \,x^{2}+b \right )^{2} b^{4}}+\frac {B \,c^{2}}{4 \left (c \,x^{2}+b \right )^{2} b^{3}}-\frac {2 A \,c^{3}}{\left (c \,x^{2}+b \right ) b^{5}}-\frac {10 A \,c^{3} \ln \relax (x )}{b^{6}}+\frac {5 A \,c^{3} \ln \left (c \,x^{2}+b \right )}{b^{6}}+\frac {3 B \,c^{2}}{2 \left (c \,x^{2}+b \right ) b^{4}}+\frac {6 B \,c^{2} \ln \relax (x )}{b^{5}}-\frac {3 B \,c^{2} \ln \left (c \,x^{2}+b \right )}{b^{5}}-\frac {3 A \,c^{2}}{b^{5} x^{2}}+\frac {3 B c}{2 b^{4} x^{2}}+\frac {3 A c}{4 b^{4} x^{4}}-\frac {B}{4 b^{3} x^{4}}-\frac {A}{6 b^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 170, normalized size = 1.15 \[ \frac {12 \, {\left (3 \, B b c^{3} - 5 \, A c^{4}\right )} x^{8} + 18 \, {\left (3 \, B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{6} - 2 \, A b^{4} + 4 \, {\left (3 \, B b^{3} c - 5 \, A b^{2} c^{2}\right )} x^{4} - {\left (3 \, B b^{4} - 5 \, A b^{3} c\right )} x^{2}}{12 \, {\left (b^{5} c^{2} x^{10} + 2 \, b^{6} c x^{8} + b^{7} x^{6}\right )}} - \frac {{\left (3 \, B b c^{2} - 5 \, A c^{3}\right )} \log \left (c x^{2} + b\right )}{b^{6}} + \frac {{\left (3 \, B b c^{2} - 5 \, A c^{3}\right )} \log \left (x^{2}\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 155, normalized size = 1.05 \[ \frac {\ln \left (c\,x^2+b\right )\,\left (5\,A\,c^3-3\,B\,b\,c^2\right )}{b^6}-\frac {\frac {A}{6\,b}-\frac {x^2\,\left (5\,A\,c-3\,B\,b\right )}{12\,b^2}+\frac {3\,c^2\,x^6\,\left (5\,A\,c-3\,B\,b\right )}{2\,b^4}+\frac {c^3\,x^8\,\left (5\,A\,c-3\,B\,b\right )}{b^5}+\frac {c\,x^4\,\left (5\,A\,c-3\,B\,b\right )}{3\,b^3}}{b^2\,x^6+2\,b\,c\,x^8+c^2\,x^{10}}-\frac {\ln \relax (x)\,\left (10\,A\,c^3-6\,B\,b\,c^2\right )}{b^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.33, size = 165, normalized size = 1.11 \[ \frac {- 2 A b^{4} + x^{8} \left (- 60 A c^{4} + 36 B b c^{3}\right ) + x^{6} \left (- 90 A b c^{3} + 54 B b^{2} c^{2}\right ) + x^{4} \left (- 20 A b^{2} c^{2} + 12 B b^{3} c\right ) + x^{2} \left (5 A b^{3} c - 3 B b^{4}\right )}{12 b^{7} x^{6} + 24 b^{6} c x^{8} + 12 b^{5} c^{2} x^{10}} + \frac {2 c^{2} \left (- 5 A c + 3 B b\right ) \log {\relax (x )}}{b^{6}} - \frac {c^{2} \left (- 5 A c + 3 B b\right ) \log {\left (\frac {b}{c} + x^{2} \right )}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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