Optimal. Leaf size=69 \[ -\frac {A}{4 a x^4}+\frac {A b-a B}{2 a^2 x^2}+\frac {b (A b-a B) \log (x)}{a^3}-\frac {b (A b-a B) \log \left (a+b x^2\right )}{2 a^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 69, 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 = {457, 78}
\begin {gather*} -\frac {b (A b-a B) \log \left (a+b x^2\right )}{2 a^3}+\frac {b \log (x) (A b-a B)}{a^3}+\frac {A b-a B}{2 a^2 x^2}-\frac {A}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^3 (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{a x^3}+\frac {-A b+a B}{a^2 x^2}-\frac {b (-A b+a B)}{a^3 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{4 a x^4}+\frac {A b-a B}{2 a^2 x^2}+\frac {b (A b-a B) \log (x)}{a^3}-\frac {b (A b-a B) \log \left (a+b x^2\right )}{2 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 70, normalized size = 1.01 \begin {gather*} \frac {-a \left (a A-2 A b x^2+2 a B x^2\right )+4 b (A b-a B) x^4 \log (x)+2 b (-A b+a B) x^4 \log \left (a+b x^2\right )}{4 a^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 64, normalized size = 0.93
method | result | size |
default | \(-\frac {b \left (A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{3}}-\frac {A}{4 a \,x^{4}}-\frac {-A b +B a}{2 a^{2} x^{2}}+\frac {b \left (A b -B a \right ) \ln \left (x \right )}{a^{3}}\) | \(64\) |
norman | \(\frac {-\frac {A}{4 a}+\frac {\left (A b -B a \right ) x^{2}}{2 a^{2}}}{x^{4}}+\frac {b \left (A b -B a \right ) \ln \left (x \right )}{a^{3}}-\frac {b \left (A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{3}}\) | \(66\) |
risch | \(\frac {-\frac {A}{4 a}+\frac {\left (A b -B a \right ) x^{2}}{2 a^{2}}}{x^{4}}+\frac {b^{2} \ln \left (x \right ) A}{a^{3}}-\frac {b \ln \left (x \right ) B}{a^{2}}-\frac {b^{2} \ln \left (b \,x^{2}+a \right ) A}{2 a^{3}}+\frac {b \ln \left (b \,x^{2}+a \right ) B}{2 a^{2}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 70, normalized size = 1.01 \begin {gather*} \frac {{\left (B a b - A b^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3}} - \frac {{\left (B a b - A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {2 \, {\left (B a - A b\right )} x^{2} + A a}{4 \, a^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 73, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (B a b - A b^{2}\right )} x^{4} \log \left (b x^{2} + a\right ) - 4 \, {\left (B a b - A b^{2}\right )} x^{4} \log \left (x\right ) - A a^{2} - 2 \, {\left (B a^{2} - A a b\right )} x^{2}}{4 \, a^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.59, size = 61, normalized size = 0.88 \begin {gather*} \frac {- A a + x^{2} \cdot \left (2 A b - 2 B a\right )}{4 a^{2} x^{4}} - \frac {b \left (- A b + B a\right ) \log {\left (x \right )}}{a^{3}} + \frac {b \left (- A b + B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.21, size = 100, normalized size = 1.45 \begin {gather*} -\frac {{\left (B a b - A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {{\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b} + \frac {3 \, B a b x^{4} - 3 \, A b^{2} x^{4} - 2 \, B a^{2} x^{2} + 2 \, A a b x^{2} - A a^{2}}{4 \, a^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 70, normalized size = 1.01 \begin {gather*} \frac {\ln \left (x\right )\,\left (A\,b^2-B\,a\,b\right )}{a^3}-\frac {\ln \left (b\,x^2+a\right )\,\left (A\,b^2-B\,a\,b\right )}{2\,a^3}-\frac {\frac {A}{4\,a}-\frac {x^2\,\left (A\,b-B\,a\right )}{2\,a^2}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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