Optimal. Leaf size=97 \[ -\frac {A}{2 b^3 x^2}+\frac {b B-A c}{4 b^2 \left (b+c x^2\right )^2}+\frac {b B-2 A c}{2 b^3 \left (b+c x^2\right )}+\frac {(b B-3 A c) \log (x)}{b^4}-\frac {(b B-3 A c) \log \left (b+c x^2\right )}{2 b^4} \]
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Rubi [A]
time = 0.09, antiderivative size = 97, 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 = {1598, 457, 78}
\begin {gather*} -\frac {(b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}+\frac {\log (x) (b B-3 A c)}{b^4}+\frac {b B-2 A c}{2 b^3 \left (b+c x^2\right )}-\frac {A}{2 b^3 x^2}+\frac {b B-A c}{4 b^2 \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^3 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {A+B x^2}{x^3 \left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^2 (b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{b^3 x^2}+\frac {b B-3 A c}{b^4 x}-\frac {c (b B-A c)}{b^2 (b+c x)^3}-\frac {c (b B-2 A c)}{b^3 (b+c x)^2}-\frac {c (b B-3 A c)}{b^4 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{2 b^3 x^2}+\frac {b B-A c}{4 b^2 \left (b+c x^2\right )^2}+\frac {b B-2 A c}{2 b^3 \left (b+c x^2\right )}+\frac {(b B-3 A c) \log (x)}{b^4}-\frac {(b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 86, normalized size = 0.89 \begin {gather*} \frac {-\frac {2 A b}{x^2}+\frac {b^2 (b B-A c)}{\left (b+c x^2\right )^2}+\frac {2 b (b B-2 A c)}{b+c x^2}+4 (b B-3 A c) \log (x)-2 (b B-3 A c) \log \left (b+c x^2\right )}{4 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 102, normalized size = 1.05
method | result | size |
norman | \(\frac {\frac {c \left (3 A c -B b \right ) x^{7}}{b^{3}}-\frac {A \,x^{3}}{2 b}+\frac {c^{2} \left (9 A c -3 B b \right ) x^{9}}{4 b^{4}}}{x^{5} \left (c \,x^{2}+b \right )^{2}}-\frac {\left (3 A c -B b \right ) \ln \left (x \right )}{b^{4}}+\frac {\left (3 A c -B b \right ) \ln \left (c \,x^{2}+b \right )}{2 b^{4}}\) | \(100\) |
default | \(\frac {c \left (\frac {\left (3 A c -B b \right ) \ln \left (c \,x^{2}+b \right )}{c}-\frac {b^{2} \left (A c -B b \right )}{2 c \left (c \,x^{2}+b \right )^{2}}-\frac {b \left (2 A c -B b \right )}{c \left (c \,x^{2}+b \right )}\right )}{2 b^{4}}-\frac {A}{2 b^{3} x^{2}}+\frac {\left (-3 A c +B b \right ) \ln \left (x \right )}{b^{4}}\) | \(102\) |
risch | \(\frac {-\frac {c \left (3 A c -B b \right ) x^{4}}{2 b^{3}}-\frac {3 \left (3 A c -B b \right ) x^{2}}{4 b^{2}}-\frac {A}{2 b}}{x^{2} \left (c \,x^{2}+b \right )^{2}}-\frac {3 \ln \left (x \right ) A c}{b^{4}}+\frac {\ln \left (x \right ) B}{b^{3}}+\frac {3 \ln \left (-c \,x^{2}-b \right ) A c}{2 b^{4}}-\frac {\ln \left (-c \,x^{2}-b \right ) B}{2 b^{3}}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 109, normalized size = 1.12 \begin {gather*} \frac {2 \, {\left (B b c - 3 \, A c^{2}\right )} x^{4} - 2 \, A b^{2} + 3 \, {\left (B b^{2} - 3 \, A b c\right )} x^{2}}{4 \, {\left (b^{3} c^{2} x^{6} + 2 \, b^{4} c x^{4} + b^{5} x^{2}\right )}} - \frac {{\left (B b - 3 \, A c\right )} \log \left (c x^{2} + b\right )}{2 \, b^{4}} + \frac {{\left (B b - 3 \, A c\right )} \log \left (x^{2}\right )}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs.
\(2 (89) = 178\).
time = 1.80, size = 197, normalized size = 2.03 \begin {gather*} \frac {2 \, {\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} - 2 \, A b^{3} + 3 \, {\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2} - 2 \, {\left ({\left (B b c^{2} - 3 \, A c^{3}\right )} x^{6} + 2 \, {\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} + {\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2}\right )} \log \left (c x^{2} + b\right ) + 4 \, {\left ({\left (B b c^{2} - 3 \, A c^{3}\right )} x^{6} + 2 \, {\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} + {\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.59, size = 107, normalized size = 1.10 \begin {gather*} \frac {- 2 A b^{2} + x^{4} \left (- 6 A c^{2} + 2 B b c\right ) + x^{2} \left (- 9 A b c + 3 B b^{2}\right )}{4 b^{5} x^{2} + 8 b^{4} c x^{4} + 4 b^{3} c^{2} x^{6}} + \frac {\left (- 3 A c + B b\right ) \log {\left (x \right )}}{b^{4}} - \frac {\left (- 3 A c + B b\right ) \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.36, size = 105, normalized size = 1.08 \begin {gather*} \frac {{\left (B b - 3 \, A c\right )} \log \left ({\left | x \right |}\right )}{b^{4}} - \frac {{\left (B b c - 3 \, A c^{2}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4} c} + \frac {2 \, {\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} - 2 \, A b^{3} + 3 \, {\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2}}{4 \, {\left (c x^{2} + b\right )}^{2} b^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 107, normalized size = 1.10 \begin {gather*} \frac {\ln \left (c\,x^2+b\right )\,\left (3\,A\,c-B\,b\right )}{2\,b^4}-\frac {\frac {A}{2\,b}+\frac {3\,x^2\,\left (3\,A\,c-B\,b\right )}{4\,b^2}+\frac {c\,x^4\,\left (3\,A\,c-B\,b\right )}{2\,b^3}}{b^2\,x^2+2\,b\,c\,x^4+c^2\,x^6}-\frac {\ln \left (x\right )\,\left (3\,A\,c-B\,b\right )}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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