Integrand size = 23, antiderivative size = 35 \[ \int \frac {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {a c}{2 b^2 \left (a+b x^2\right )}+\frac {c \log \left (a+b x^2\right )}{2 b^2} \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {21, 272, 45} \[ \int \frac {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {a c}{2 b^2 \left (a+b x^2\right )}+\frac {c \log \left (a+b x^2\right )}{2 b^2} \]
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Rule 21
Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = c \int \frac {x^3}{\left (a+b x^2\right )^2} \, dx \\ & = \frac {1}{2} c \text {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} c \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {a c}{2 b^2 \left (a+b x^2\right )}+\frac {c \log \left (a+b x^2\right )}{2 b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \frac {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {c \left (\frac {a}{a+b x^2}+\log \left (a+b x^2\right )\right )}{2 b^2} \]
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Time = 2.54 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91
method | result | size |
default | \(c \left (\frac {a}{2 b^{2} \left (b \,x^{2}+a \right )}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{2}}\right )\) | \(32\) |
risch | \(\frac {a c}{2 b^{2} \left (b \,x^{2}+a \right )}+\frac {c \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(32\) |
parallelrisch | \(\frac {b c \ln \left (b \,x^{2}+a \right ) x^{2}+a c \ln \left (b \,x^{2}+a \right )+a c}{2 b^{2} \left (b \,x^{2}+a \right )}\) | \(44\) |
norman | \(\frac {\frac {a^{2} c}{2 b^{2}}+\frac {c a \,x^{2}}{2 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {c \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(46\) |
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none
Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \frac {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {a c + {\left (b c x^{2} + a c\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=c \left (\frac {a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {\log {\left (a + b x^{2} \right )}}{2 b^{2}}\right ) \]
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none
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {a c}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {c \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {c \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} + \frac {a c}{2 \, {\left (b x^{2} + a\right )} b^{2}} \]
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Time = 5.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {c\,\ln \left (b\,x^2+a\right )}{2\,b^2}+\frac {a\,c}{2\,b^2\,\left (b\,x^2+a\right )} \]
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