Integrand size = 54, antiderivative size = 42 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx=\frac {c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {d \log \left (a+b x^2\right )}{2 b} \]
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Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1600, 649, 211, 266} \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx=\frac {c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {d \log \left (a+b x^2\right )}{2 b} \]
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Rule 211
Rule 266
Rule 649
Rule 1600
Rubi steps \begin{align*} \text {integral}& = \int \frac {a c+a d x+b c x^2+b d x^3}{\left (a+b x^2\right )^2} \, dx \\ & = \int \frac {c+d x}{a+b x^2} \, dx \\ & = c \int \frac {1}{a+b x^2} \, dx+d \int \frac {x}{a+b x^2} \, dx \\ & = \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {d \log \left (a+b x^2\right )}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx=\frac {c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {d \log \left (a+b x^2\right )}{2 b} \]
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Time = 0.78 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {d \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\) | \(32\) |
risch | \(\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) c \sqrt {-a b}}{2 a b}+\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) d}{2 b}-\frac {\ln \left (\sqrt {-a b}\, x +a \right ) c \sqrt {-a b}}{2 a b}+\frac {\ln \left (\sqrt {-a b}\, x +a \right ) d}{2 b}\) | \(90\) |
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Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.33 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx=\left [\frac {a d \log \left (b x^{2} + a\right ) - \sqrt {-a b} c \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b}, \frac {a d \log \left (b x^{2} + a\right ) + 2 \, \sqrt {a b} c \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{2 \, a b}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (37) = 74\).
Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.95 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx=\left (\frac {d}{2 b} - \frac {c \sqrt {- a b^{3}}}{2 a b^{2}}\right ) \log {\left (x + \frac {2 a b \left (\frac {d}{2 b} - \frac {c \sqrt {- a b^{3}}}{2 a b^{2}}\right ) - a d}{b c} \right )} + \left (\frac {d}{2 b} + \frac {c \sqrt {- a b^{3}}}{2 a b^{2}}\right ) \log {\left (x + \frac {2 a b \left (\frac {d}{2 b} + \frac {c \sqrt {- a b^{3}}}{2 a b^{2}}\right ) - a d}{b c} \right )} \]
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx=\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} + \frac {d \log \left (b x^{2} + a\right )}{2 \, b} \]
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx=\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} + \frac {d \log \left (b x^{2} + a\right )}{2 \, b} \]
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Time = 9.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.76 \[ \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx=\frac {d\,\ln \left (b\,x^2+a\right )}{2\,b}+\frac {c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \]
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