Optimal. Leaf size=70 \[ -\frac {A}{4 b x^4}-\frac {b B-A c}{2 b^2 x^2}-\frac {c (b B-A c) \log (x)}{b^3}+\frac {c (b B-A c) \log \left (b+c x^2\right )}{2 b^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 70, 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 (b B-A c) \log \left (b+c x^2\right )}{2 b^3}-\frac {c \log (x) (b B-A c)}{b^3}-\frac {b B-A c}{2 b^2 x^2}-\frac {A}{4 b x^4} \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^3 \left (b x^2+c x^4\right )} \, dx &=\int \frac {A+B x^2}{x^5 \left (b+c x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^3 (b+c x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{b x^3}+\frac {b B-A c}{b^2 x^2}-\frac {c (b B-A c)}{b^3 x}+\frac {c^2 (b B-A c)}{b^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{4 b x^4}-\frac {b B-A c}{2 b^2 x^2}-\frac {c (b B-A c) \log (x)}{b^3}+\frac {c (b B-A c) \log \left (b+c x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 70, normalized size = 1.00 \begin {gather*} \frac {-b \left (A b+2 b B x^2-2 A c x^2\right )+4 c (-b B+A c) x^4 \log (x)+2 c (b B-A c) x^4 \log \left (b+c x^2\right )}{4 b^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 64, normalized size = 0.91
method | result | size |
default | \(-\frac {c \left (A c -B b \right ) \ln \left (c \,x^{2}+b \right )}{2 b^{3}}-\frac {A}{4 b \,x^{4}}-\frac {-A c +B b}{2 b^{2} x^{2}}+\frac {c \left (A c -B b \right ) \ln \left (x \right )}{b^{3}}\) | \(64\) |
norman | \(\frac {-\frac {A}{4 b}+\frac {\left (A c -B b \right ) x^{2}}{2 b^{2}}}{x^{4}}+\frac {c \left (A c -B b \right ) \ln \left (x \right )}{b^{3}}-\frac {c \left (A c -B b \right ) \ln \left (c \,x^{2}+b \right )}{2 b^{3}}\) | \(66\) |
risch | \(\frac {-\frac {A}{4 b}+\frac {\left (A c -B b \right ) x^{2}}{2 b^{2}}}{x^{4}}+\frac {c^{2} \ln \left (x \right ) A}{b^{3}}-\frac {c \ln \left (x \right ) B}{b^{2}}-\frac {c^{2} \ln \left (c \,x^{2}+b \right ) A}{2 b^{3}}+\frac {c \ln \left (c \,x^{2}+b \right ) B}{2 b^{2}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 70, normalized size = 1.00 \begin {gather*} \frac {{\left (B b c - A c^{2}\right )} \log \left (c x^{2} + b\right )}{2 \, b^{3}} - \frac {{\left (B b c - A c^{2}\right )} \log \left (x^{2}\right )}{2 \, b^{3}} - \frac {2 \, {\left (B b - A c\right )} x^{2} + A b}{4 \, b^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.92, size = 73, normalized size = 1.04 \begin {gather*} \frac {2 \, {\left (B b c - A c^{2}\right )} x^{4} \log \left (c x^{2} + b\right ) - 4 \, {\left (B b c - A c^{2}\right )} x^{4} \log \left (x\right ) - A b^{2} - 2 \, {\left (B b^{2} - A b c\right )} x^{2}}{4 \, b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 61, normalized size = 0.87 \begin {gather*} \frac {- A b + x^{2} \cdot \left (2 A c - 2 B b\right )}{4 b^{2} x^{4}} - \frac {c \left (- A c + B b\right ) \log {\left (x \right )}}{b^{3}} + \frac {c \left (- A c + B b\right ) \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.78, size = 100, normalized size = 1.43 \begin {gather*} -\frac {{\left (B b c - A c^{2}\right )} \log \left (x^{2}\right )}{2 \, b^{3}} + \frac {{\left (B b c^{2} - A c^{3}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{3} c} + \frac {3 \, B b c x^{4} - 3 \, A c^{2} x^{4} - 2 \, B b^{2} x^{2} + 2 \, A b c x^{2} - A b^{2}}{4 \, b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 70, normalized size = 1.00 \begin {gather*} \frac {\ln \left (x\right )\,\left (A\,c^2-B\,b\,c\right )}{b^3}-\frac {\ln \left (c\,x^2+b\right )\,\left (A\,c^2-B\,b\,c\right )}{2\,b^3}-\frac {\frac {A}{4\,b}-\frac {x^2\,\left (A\,c-B\,b\right )}{2\,b^2}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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