Integrand size = 20, antiderivative size = 61 \[ \int \frac {x \left (a+b x^2\right )^2}{c+d x^2} \, dx=-\frac {b (b c-a d) x^2}{2 d^2}+\frac {\left (a+b x^2\right )^2}{4 d}+\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 d^3} \]
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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} \[ \int \frac {x \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 d^3}-\frac {b x^2 (b c-a d)}{2 d^2}+\frac {\left (a+b x^2\right )^2}{4 d} \]
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Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2}{c+d x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b (b c-a d) x^2}{2 d^2}+\frac {\left (a+b x^2\right )^2}{4 d}+\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 d^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.80 \[ \int \frac {x \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {b d x^2 \left (-2 b c+4 a d+b d x^2\right )+2 (b c-a d)^2 \log \left (c+d x^2\right )}{4 d^3} \]
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Time = 2.72 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {b \left (\frac {1}{2} b d \,x^{4}+2 a d \,x^{2}-c b \,x^{2}\right )}{2 d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 d^{3}}\) | \(64\) |
norman | \(\frac {b^{2} x^{4}}{4 d}+\frac {b \left (2 a d -b c \right ) x^{2}}{2 d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 d^{3}}\) | \(65\) |
parallelrisch | \(\frac {b^{2} d^{2} x^{4}+4 x^{2} a b \,d^{2}-2 x^{2} b^{2} c d +2 \ln \left (d \,x^{2}+c \right ) a^{2} d^{2}-4 \ln \left (d \,x^{2}+c \right ) a b c d +2 \ln \left (d \,x^{2}+c \right ) b^{2} c^{2}}{4 d^{3}}\) | \(83\) |
risch | \(\frac {b^{2} x^{4}}{4 d}+\frac {x^{2} a b}{d}-\frac {x^{2} b^{2} c}{2 d^{2}}+\frac {a^{2}}{d}-\frac {a b c}{d^{2}}+\frac {b^{2} c^{2}}{4 d^{3}}+\frac {\ln \left (d \,x^{2}+c \right ) a^{2}}{2 d}-\frac {\ln \left (d \,x^{2}+c \right ) a b c}{d^{2}}+\frac {\ln \left (d \,x^{2}+c \right ) b^{2} c^{2}}{2 d^{3}}\) | \(111\) |
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Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08 \[ \int \frac {x \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {b^{2} d^{2} x^{4} - 2 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, d^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.80 \[ \int \frac {x \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {b^{2} x^{4}}{4 d} + x^{2} \left (\frac {a b}{d} - \frac {b^{2} c}{2 d^{2}}\right ) + \frac {\left (a d - b c\right )^{2} \log {\left (c + d x^{2} \right )}}{2 d^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07 \[ \int \frac {x \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {b^{2} d x^{4} - 2 \, {\left (b^{2} c - 2 \, a b d\right )} x^{2}}{4 \, d^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, d^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {b^{2} d x^{4} - 2 \, b^{2} c x^{2} + 4 \, a b d x^{2}}{4 \, d^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, d^{3}} \]
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Time = 4.98 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {b^2\,x^4}{4\,d}-x^2\,\left (\frac {b^2\,c}{2\,d^2}-\frac {a\,b}{d}\right )+\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,d^3} \]
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