Integrand size = 15, antiderivative size = 59 \[ \int \frac {\left (1+(a+b x)^2\right )^2}{x} \, dx=a \left (2+a^2\right ) b x+\frac {1}{2} \left (2+a^2\right ) (a+b x)^2+\frac {1}{3} a (a+b x)^3+\frac {1}{4} (a+b x)^4+\left (1+a^2\right )^2 \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {378, 711} \[ \int \frac {\left (1+(a+b x)^2\right )^2}{x} \, dx=\frac {1}{2} \left (a^2+2\right ) (a+b x)^2+a \left (a^2+2\right ) b x+\left (a^2+1\right )^2 \log (x)+\frac {1}{4} (a+b x)^4+\frac {1}{3} a (a+b x)^3 \]
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Rule 378
Rule 711
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{-a+x} \, dx,x,a+b x\right ) \\ & = \text {Subst}\left (\int \left (a \left (2+a^2\right )-\frac {\left (1+a^2\right )^2}{a-x}+\left (2+a^2\right ) x+a x^2+x^3\right ) \, dx,x,a+b x\right ) \\ & = a \left (2+a^2\right ) b x+\frac {1}{2} \left (2+a^2\right ) (a+b x)^2+\frac {1}{3} a (a+b x)^3+\frac {1}{4} (a+b x)^4+\left (1+a^2\right )^2 \log (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08 \[ \int \frac {\left (1+(a+b x)^2\right )^2}{x} \, dx=a \left (2+a^2\right ) (a+b x)+\frac {1}{2} \left (2+a^2\right ) (a+b x)^2+\frac {1}{3} a (a+b x)^3+\frac {1}{4} (a+b x)^4+\left (1+a^2\right )^2 \log (b x) \]
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Time = 0.90 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\left (3 a^{2} b^{2}+b^{2}\right ) x^{2}+\left (4 a^{3} b +4 a b \right ) x +\frac {b^{4} x^{4}}{4}+\frac {4 a \,b^{3} x^{3}}{3}+\left (a^{4}+2 a^{2}+1\right ) \ln \left (x \right )\) | \(61\) |
default | \(\frac {b^{4} x^{4}}{4}+\frac {4 a \,b^{3} x^{3}}{3}+3 a^{2} b^{2} x^{2}+4 a^{3} b x +b^{2} x^{2}+4 a b x +\left (a^{4}+2 a^{2}+1\right ) \ln \left (x \right )\) | \(62\) |
risch | \(\frac {b^{4} x^{4}}{4}+\frac {4 a \,b^{3} x^{3}}{3}+3 a^{2} b^{2} x^{2}+b^{2} x^{2}+4 a^{3} b x +4 a b x +a^{4} \ln \left (x \right )+2 a^{2} \ln \left (x \right )+\ln \left (x \right )\) | \(64\) |
parallelrisch | \(\frac {b^{4} x^{4}}{4}+\frac {4 a \,b^{3} x^{3}}{3}+3 a^{2} b^{2} x^{2}+b^{2} x^{2}+4 a^{3} b x +4 a b x +a^{4} \ln \left (x \right )+2 a^{2} \ln \left (x \right )+\ln \left (x \right )\) | \(64\) |
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Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1+(a+b x)^2\right )^2}{x} \, dx=\frac {1}{4} \, b^{4} x^{4} + \frac {4}{3} \, a b^{3} x^{3} + {\left (3 \, a^{2} + 1\right )} b^{2} x^{2} + 4 \, {\left (a^{3} + a\right )} b x + {\left (a^{4} + 2 \, a^{2} + 1\right )} \log \left (x\right ) \]
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Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int \frac {\left (1+(a+b x)^2\right )^2}{x} \, dx=\frac {4 a b^{3} x^{3}}{3} + \frac {b^{4} x^{4}}{4} + x^{2} \cdot \left (3 a^{2} b^{2} + b^{2}\right ) + x \left (4 a^{3} b + 4 a b\right ) + \left (a^{2} + 1\right )^{2} \log {\left (x \right )} \]
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Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1+(a+b x)^2\right )^2}{x} \, dx=\frac {1}{4} \, b^{4} x^{4} + \frac {4}{3} \, a b^{3} x^{3} + {\left (3 \, a^{2} + 1\right )} b^{2} x^{2} + 4 \, {\left (a^{3} + a\right )} b x + {\left (a^{4} + 2 \, a^{2} + 1\right )} \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05 \[ \int \frac {\left (1+(a+b x)^2\right )^2}{x} \, dx=\frac {1}{4} \, b^{4} x^{4} + \frac {4}{3} \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + b^{2} x^{2} + 4 \, a b x + {\left (a^{4} + 2 \, a^{2} + 1\right )} \log \left ({\left | x \right |}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.93 \[ \int \frac {\left (1+(a+b x)^2\right )^2}{x} \, dx=\ln \left (x\right )\,\left (a^4+2\,a^2+1\right )+\frac {b^4\,x^4}{4}+\frac {4\,a\,b^3\,x^3}{3}+b^2\,x^2\,\left (3\,a^2+1\right )+4\,a\,b\,x\,\left (a^2+1\right ) \]
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