Optimal. Leaf size=92 \[ -\frac {A}{6 b x^6}-\frac {b B-A c}{4 b^2 x^4}+\frac {c (b B-A c)}{2 b^3 x^2}+\frac {c^2 (b B-A c) \log (x)}{b^4}-\frac {c^2 (b B-A c) \log \left (b+c x^2\right )}{2 b^4} \]
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Rubi [A]
time = 0.06, antiderivative size = 92, 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 {c^2 (b B-A c) \log \left (b+c x^2\right )}{2 b^4}+\frac {c^2 \log (x) (b B-A c)}{b^4}+\frac {c (b B-A c)}{2 b^3 x^2}-\frac {b B-A c}{4 b^2 x^4}-\frac {A}{6 b x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^5 \left (b x^2+c x^4\right )} \, dx &=\int \frac {A+B x^2}{x^7 \left (b+c x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^4 (b+c x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{b x^4}+\frac {b B-A c}{b^2 x^3}-\frac {c (b B-A c)}{b^3 x^2}+\frac {c^2 (b B-A c)}{b^4 x}-\frac {c^3 (b B-A c)}{b^4 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{6 b x^6}-\frac {b B-A c}{4 b^2 x^4}+\frac {c (b B-A c)}{2 b^3 x^2}+\frac {c^2 (b B-A c) \log (x)}{b^4}-\frac {c^2 (b B-A c) \log \left (b+c x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 96, normalized size = 1.04 \begin {gather*} -\frac {A}{6 b x^6}+\frac {-b B+A c}{4 b^2 x^4}+\frac {c (b B-A c)}{2 b^3 x^2}+\frac {\left (b B c^2-A c^3\right ) \log (x)}{b^4}+\frac {\left (-b B c^2+A c^3\right ) \log \left (b+c x^2\right )}{2 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 86, normalized size = 0.93
method | result | size |
default | \(\frac {\left (A c -B b \right ) c^{2} \ln \left (c \,x^{2}+b \right )}{2 b^{4}}-\frac {A}{6 b \,x^{6}}-\frac {-A c +B b}{4 b^{2} x^{4}}-\frac {c \left (A c -B b \right )}{2 b^{3} x^{2}}-\frac {c^{2} \left (A c -B b \right ) \ln \left (x \right )}{b^{4}}\) | \(86\) |
norman | \(\frac {-\frac {A}{6 b}+\frac {\left (A c -B b \right ) x^{2}}{4 b^{2}}-\frac {c \left (A c -B b \right ) x^{4}}{2 b^{3}}}{x^{6}}-\frac {c^{2} \left (A c -B b \right ) \ln \left (x \right )}{b^{4}}+\frac {\left (A c -B b \right ) c^{2} \ln \left (c \,x^{2}+b \right )}{2 b^{4}}\) | \(88\) |
risch | \(\frac {-\frac {A}{6 b}+\frac {\left (A c -B b \right ) x^{2}}{4 b^{2}}-\frac {c \left (A c -B b \right ) x^{4}}{2 b^{3}}}{x^{6}}-\frac {c^{3} \ln \left (x \right ) A}{b^{4}}+\frac {c^{2} \ln \left (x \right ) B}{b^{3}}+\frac {c^{3} \ln \left (-c \,x^{2}-b \right ) A}{2 b^{4}}-\frac {c^{2} \ln \left (-c \,x^{2}-b \right ) B}{2 b^{3}}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 96, normalized size = 1.04 \begin {gather*} -\frac {{\left (B b c^{2} - A c^{3}\right )} \log \left (c x^{2} + b\right )}{2 \, b^{4}} + \frac {{\left (B b c^{2} - A c^{3}\right )} \log \left (x^{2}\right )}{2 \, b^{4}} + \frac {6 \, {\left (B b c - A c^{2}\right )} x^{4} - 2 \, A b^{2} - 3 \, {\left (B b^{2} - A b c\right )} x^{2}}{12 \, b^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 98, normalized size = 1.07 \begin {gather*} -\frac {6 \, {\left (B b c^{2} - A c^{3}\right )} x^{6} \log \left (c x^{2} + b\right ) - 12 \, {\left (B b c^{2} - A c^{3}\right )} x^{6} \log \left (x\right ) - 6 \, {\left (B b^{2} c - A b c^{2}\right )} x^{4} + 2 \, A b^{3} + 3 \, {\left (B b^{3} - A b^{2} c\right )} x^{2}}{12 \, b^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.48, size = 88, normalized size = 0.96 \begin {gather*} \frac {- 2 A b^{2} + x^{4} \left (- 6 A c^{2} + 6 B b c\right ) + x^{2} \cdot \left (3 A b c - 3 B b^{2}\right )}{12 b^{3} x^{6}} + \frac {c^{2} \left (- A c + B b\right ) \log {\left (x \right )}}{b^{4}} - \frac {c^{2} \left (- 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 = 0.68, size = 126, normalized size = 1.37 \begin {gather*} \frac {{\left (B b c^{2} - A c^{3}\right )} \log \left (x^{2}\right )}{2 \, b^{4}} - \frac {{\left (B b c^{3} - A c^{4}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4} c} - \frac {11 \, B b c^{2} x^{6} - 11 \, A c^{3} x^{6} - 6 \, B b^{2} c x^{4} + 6 \, A b c^{2} x^{4} + 3 \, B b^{3} x^{2} - 3 \, A b^{2} c x^{2} + 2 \, A b^{3}}{12 \, b^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 92, normalized size = 1.00 \begin {gather*} \frac {\ln \left (c\,x^2+b\right )\,\left (A\,c^3-B\,b\,c^2\right )}{2\,b^4}-\frac {\frac {A}{6\,b}-\frac {x^2\,\left (A\,c-B\,b\right )}{4\,b^2}+\frac {c\,x^4\,\left (A\,c-B\,b\right )}{2\,b^3}}{x^6}-\frac {\ln \left (x\right )\,\left (A\,c^3-B\,b\,c^2\right )}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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