Integrand size = 20, antiderivative size = 68 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^3} \, dx=\frac {A b-a B}{4 a b \left (a+b x^2\right )^2}+\frac {A}{2 a^2 \left (a+b x^2\right )}+\frac {A \log (x)}{a^3}-\frac {A \log \left (a+b x^2\right )}{2 a^3} \]
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Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 78} \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^3} \, dx=-\frac {A \log \left (a+b x^2\right )}{2 a^3}+\frac {A \log (x)}{a^3}+\frac {A}{2 a^2 \left (a+b x^2\right )}+\frac {A b-a B}{4 a b \left (a+b x^2\right )^2} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x (a+b x)^3} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{a^3 x}+\frac {-A b+a B}{a (a+b x)^3}-\frac {A b}{a^2 (a+b x)^2}-\frac {A b}{a^3 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {A b-a B}{4 a b \left (a+b x^2\right )^2}+\frac {A}{2 a^2 \left (a+b x^2\right )}+\frac {A \log (x)}{a^3}-\frac {A \log \left (a+b x^2\right )}{2 a^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^3} \, dx=\frac {\frac {a \left (3 a A b-a^2 B+2 A b^2 x^2\right )}{b \left (a+b x^2\right )^2}+4 A \log (x)-2 A \log \left (a+b x^2\right )}{4 a^3} \]
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Time = 2.52 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {\frac {A b \,x^{2}}{2 a^{2}}+\frac {3 A b -B a}{4 a b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {A \ln \left (x \right )}{a^{3}}-\frac {A \ln \left (b \,x^{2}+a \right )}{2 a^{3}}\) | \(61\) |
default | \(\frac {A \ln \left (x \right )}{a^{3}}-\frac {A \ln \left (b \,x^{2}+a \right )-\frac {a^{2} \left (A b -B a \right )}{2 b \left (b \,x^{2}+a \right )^{2}}-\frac {A a}{b \,x^{2}+a}}{2 a^{3}}\) | \(63\) |
norman | \(\frac {-\frac {\left (2 A b -B a \right ) x^{2}}{2 a^{2}}-\frac {b \left (3 A b -B a \right ) x^{4}}{4 a^{3}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {A \ln \left (x \right )}{a^{3}}-\frac {A \ln \left (b \,x^{2}+a \right )}{2 a^{3}}\) | \(69\) |
parallelrisch | \(\frac {4 A \ln \left (x \right ) x^{4} b^{2}-2 A \ln \left (b \,x^{2}+a \right ) x^{4} b^{2}-3 A \,b^{2} x^{4}+B a b \,x^{4}+8 A \ln \left (x \right ) x^{2} a b -4 A \ln \left (b \,x^{2}+a \right ) x^{2} a b -4 a A b \,x^{2}+2 a^{2} B \,x^{2}+4 a^{2} A \ln \left (x \right )-2 A \ln \left (b \,x^{2}+a \right ) a^{2}}{4 a^{3} \left (b \,x^{2}+a \right )^{2}}\) | \(125\) |
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Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.75 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^3} \, dx=\frac {2 \, A a b^{2} x^{2} - B a^{3} + 3 \, A a^{2} b - 2 \, {\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\right )} \log \left (x\right )}{4 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^3} \, dx=\frac {A \log {\left (x \right )}}{a^{3}} - \frac {A \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3}} + \frac {3 A a b + 2 A b^{2} x^{2} - B a^{2}}{4 a^{4} b + 8 a^{3} b^{2} x^{2} + 4 a^{2} b^{3} x^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^3} \, dx=\frac {2 \, A b^{2} x^{2} - B a^{2} + 3 \, A a b}{4 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} - \frac {A \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {A \log \left (x^{2}\right )}{2 \, a^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^3} \, dx=\frac {A \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {A \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3}} + \frac {3 \, A b^{3} x^{4} + 8 \, A a b^{2} x^{2} - B a^{3} + 6 \, A a^{2} b}{4 \, {\left (b x^{2} + a\right )}^{2} a^{3} b} \]
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Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^3} \, dx=\frac {\frac {3\,A\,b-B\,a}{4\,a\,b}+\frac {A\,b\,x^2}{2\,a^2}}{a^2+2\,a\,b\,x^2+b^2\,x^4}-\frac {A\,\ln \left (b\,x^2+a\right )}{2\,a^3}+\frac {A\,\ln \left (x\right )}{a^3} \]
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