Optimal. Leaf size=62 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{5/2} \sqrt {c}}+\frac {3 x}{8 b^2 \left (b+c x^2\right )}+\frac {x}{4 b \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1584, 199, 205} \[ \frac {3 x}{8 b^2 \left (b+c x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{5/2} \sqrt {c}}+\frac {x}{4 b \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^6}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {1}{\left (b+c x^2\right )^3} \, dx\\ &=\frac {x}{4 b \left (b+c x^2\right )^2}+\frac {3 \int \frac {1}{\left (b+c x^2\right )^2} \, dx}{4 b}\\ &=\frac {x}{4 b \left (b+c x^2\right )^2}+\frac {3 x}{8 b^2 \left (b+c x^2\right )}+\frac {3 \int \frac {1}{b+c x^2} \, dx}{8 b^2}\\ &=\frac {x}{4 b \left (b+c x^2\right )^2}+\frac {3 x}{8 b^2 \left (b+c x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{5/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.89 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{5/2} \sqrt {c}}+\frac {5 b x+3 c x^3}{8 b^2 \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 188, normalized size = 3.03 \[ \left [\frac {6 \, b c^{2} x^{3} + 10 \, b^{2} c x - 3 \, {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right )}{16 \, {\left (b^{3} c^{3} x^{4} + 2 \, b^{4} c^{2} x^{2} + b^{5} c\right )}}, \frac {3 \, b c^{2} x^{3} + 5 \, b^{2} c x + 3 \, {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right )}{8 \, {\left (b^{3} c^{3} x^{4} + 2 \, b^{4} c^{2} x^{2} + b^{5} c\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 45, normalized size = 0.73 \[ \frac {3 \, \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{2}} + \frac {3 \, c x^{3} + 5 \, b x}{8 \, {\left (c x^{2} + b\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 0.82 \[ \frac {x}{4 \left (c \,x^{2}+b \right )^{2} b}+\frac {3 x}{8 \left (c \,x^{2}+b \right ) b^{2}}+\frac {3 \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 58, normalized size = 0.94 \[ \frac {3 \, c x^{3} + 5 \, b x}{8 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )}} + \frac {3 \, \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.21, size = 55, normalized size = 0.89 \[ \frac {\frac {5\,x}{8\,b}+\frac {3\,c\,x^3}{8\,b^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{8\,b^{5/2}\,\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 105, normalized size = 1.69 \[ - \frac {3 \sqrt {- \frac {1}{b^{5} c}} \log {\left (- b^{3} \sqrt {- \frac {1}{b^{5} c}} + x \right )}}{16} + \frac {3 \sqrt {- \frac {1}{b^{5} c}} \log {\left (b^{3} \sqrt {- \frac {1}{b^{5} c}} + x \right )}}{16} + \frac {5 b x + 3 c x^{3}}{8 b^{4} + 16 b^{3} c x^{2} + 8 b^{2} c^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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