Integrand size = 20, antiderivative size = 61 \[ \int \frac {x \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^2 x^2}{2 b^2}-\frac {(b c-a d)^2}{2 b^3 \left (a+b x^2\right )}+\frac {d (b c-a d) \log \left (a+b x^2\right )}{b^3} \]
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Time = 0.04 (sec) , antiderivative size = 61, 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 = {455, 45} \[ \int \frac {x \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=-\frac {(b c-a d)^2}{2 b^3 \left (a+b x^2\right )}+\frac {d (b c-a d) \log \left (a+b x^2\right )}{b^3}+\frac {d^2 x^2}{2 b^2} \]
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Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^2}{(a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {d^2}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)^2}+\frac {2 d (b c-a d)}{b^2 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {d^2 x^2}{2 b^2}-\frac {(b c-a d)^2}{2 b^3 \left (a+b x^2\right )}+\frac {d (b c-a d) \log \left (a+b x^2\right )}{b^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {x \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {b d^2 x^2-\frac {(b c-a d)^2}{a+b x^2}+2 d (b c-a d) \log \left (a+b x^2\right )}{2 b^3} \]
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Time = 2.70 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {d^{2} x^{2}}{2 b^{2}}-\frac {\left (a d -b c \right ) \left (\frac {2 d \ln \left (b \,x^{2}+a \right )}{b}-\frac {-a d +b c}{b \left (b \,x^{2}+a \right )}\right )}{2 b^{2}}\) | \(63\) |
norman | \(\frac {-\frac {2 a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 b^{3}}+\frac {d^{2} x^{4}}{2 b}}{b \,x^{2}+a}-\frac {\left (a d -b c \right ) d \ln \left (b \,x^{2}+a \right )}{b^{3}}\) | \(73\) |
risch | \(\frac {d^{2} x^{2}}{2 b^{2}}-\frac {a^{2} d^{2}}{2 b^{3} \left (b \,x^{2}+a \right )}+\frac {a c d}{b^{2} \left (b \,x^{2}+a \right )}-\frac {c^{2}}{2 b \left (b \,x^{2}+a \right )}-\frac {d^{2} \ln \left (b \,x^{2}+a \right ) a}{b^{3}}+\frac {d \ln \left (b \,x^{2}+a \right ) c}{b^{2}}\) | \(97\) |
parallelrisch | \(-\frac {-b^{2} d^{2} x^{4}+2 \ln \left (b \,x^{2}+a \right ) x^{2} a b \,d^{2}-2 \ln \left (b \,x^{2}+a \right ) x^{2} b^{2} c d +2 \ln \left (b \,x^{2}+a \right ) a^{2} d^{2}-2 \ln \left (b \,x^{2}+a \right ) a b c d +2 a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 b^{3} \left (b \,x^{2}+a \right )}\) | \(114\) |
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Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.66 \[ \int \frac {x \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {b^{2} d^{2} x^{4} + a b d^{2} x^{2} - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} + 2 \, {\left (a b c d - a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \]
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Time = 0.42 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {- a^{2} d^{2} + 2 a b c d - b^{2} c^{2}}{2 a b^{3} + 2 b^{4} x^{2}} + \frac {d^{2} x^{2}}{2 b^{2}} - \frac {d \left (a d - b c\right ) \log {\left (a + b x^{2} \right )}}{b^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.20 \[ \int \frac {x \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^{2} x^{2}}{2 \, b^{2}} - \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} + \frac {{\left (b c d - a d^{2}\right )} \log \left (b x^{2} + a\right )}{b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.82 \[ \int \frac {x \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (b x^{2} + a\right )} d^{2}}{2 \, b^{3}} - \frac {{\left (b c d - a d^{2}\right )} \log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} - \frac {\frac {b^{3} c^{2}}{b x^{2} + a} - \frac {2 \, a b^{2} c d}{b x^{2} + a} + \frac {a^{2} b d^{2}}{b x^{2} + a}}{2 \, b^{4}} \]
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Time = 4.78 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.26 \[ \int \frac {x \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^2\,x^2}{2\,b^2}-\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{2\,b\,\left (b^3\,x^2+a\,b^2\right )}-\frac {\ln \left (b\,x^2+a\right )\,\left (a\,d^2-b\,c\,d\right )}{b^3} \]
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