Optimal. Leaf size=19 \[ \frac {\tan ^{-1}\left (\frac {c x^3}{a+b x^2}\right )}{c} \]
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Rubi [A] time = 0.10, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2094, 205} \[ \frac {\tan ^{-1}\left (\frac {c x^3}{a+b x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2094
Rubi steps
\begin {align*} \int \frac {x^2 \left (3 a+b x^2\right )}{a^2+2 a b x^2+b^2 x^4+c^2 x^6} \, dx &=\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a^2+9 a^2 c^2 x^2} \, dx,x,\frac {x^3}{3 a+3 b x^2}\right )\\ &=\frac {\tan ^{-1}\left (\frac {c x^3}{a+b x^2}\right )}{c}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 87, normalized size = 4.58 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^6 c^2+\text {$\#$1}^4 b^2+2 \text {$\#$1}^2 a b+a^2\& ,\frac {\text {$\#$1}^3 b \log (x-\text {$\#$1})+3 \text {$\#$1} a \log (x-\text {$\#$1})}{3 \text {$\#$1}^4 c^2+2 \text {$\#$1}^2 b^2+2 a b}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 83, normalized size = 4.37 \[ \frac {\arctan \left (\frac {c x}{b}\right ) - \arctan \left (\frac {b c^{2} x^{5} + a b^{2} x + {\left (b^{3} - a c^{2}\right )} x^{3}}{a^{2} c}\right ) + \arctan \left (\frac {b c^{2} x^{3} + {\left (b^{3} - a c^{2}\right )} x}{a b c}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.08, size = 87, normalized size = 4.58 \[ \frac {\arctan \left (\frac {c x}{b}\right ) + \arctan \left (-\frac {b c^{2} x^{5} + b^{3} x^{3} - a c^{2} x^{3} + a b^{2} x}{a^{2} c}\right ) - \arctan \left (-\frac {b c^{2} x^{3} + b^{3} x - a c^{2} x}{a b c}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 75, normalized size = 3.95 \[ \frac {\left (\RootOf \left (c^{2} \textit {\_Z}^{6}+b^{2} \textit {\_Z}^{4}+2 a b \,\textit {\_Z}^{2}+a^{2}\right )^{4} b +3 \RootOf \left (c^{2} \textit {\_Z}^{6}+b^{2} \textit {\_Z}^{4}+2 a b \,\textit {\_Z}^{2}+a^{2}\right )^{2} a \right ) \ln \left (-\RootOf \left (c^{2} \textit {\_Z}^{6}+b^{2} \textit {\_Z}^{4}+2 a b \,\textit {\_Z}^{2}+a^{2}\right )+x \right )}{6 \RootOf \left (c^{2} \textit {\_Z}^{6}+b^{2} \textit {\_Z}^{4}+2 a b \,\textit {\_Z}^{2}+a^{2}\right )^{5} c^{2}+4 \RootOf \left (c^{2} \textit {\_Z}^{6}+b^{2} \textit {\_Z}^{4}+2 a b \,\textit {\_Z}^{2}+a^{2}\right )^{3} b^{2}+4 \RootOf \left (c^{2} \textit {\_Z}^{6}+b^{2} \textit {\_Z}^{4}+2 a b \,\textit {\_Z}^{2}+a^{2}\right ) a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + 3 \, a\right )} x^{2}}{c^{2} x^{6} + b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.27, size = 252, normalized size = 13.26 \[ \frac {\mathrm {atan}\left (\frac {27\,a\,c^5\,x^3}{27\,a^2\,c^4-4\,a\,b^3\,c^2}-\frac {27\,b\,c^5\,x^5}{27\,a^2\,c^4-4\,a\,b^3\,c^2}-\frac {31\,b^3\,c^3\,x^3}{27\,a^2\,c^4-4\,a\,b^3\,c^2}+\frac {4\,b^6\,c\,x^3}{27\,a^3\,c^4-4\,a^2\,b^3\,c^2}+\frac {4\,b^5\,c\,x}{27\,a^2\,c^4-4\,a\,b^3\,c^2}+\frac {4\,b^4\,c^3\,x^5}{27\,a^3\,c^4-4\,a^2\,b^3\,c^2}-\frac {27\,a\,b^2\,c^3\,x}{27\,a^2\,c^4-4\,a\,b^3\,c^2}\right )+\mathrm {atan}\left (\frac {c\,x^3}{a}-\frac {c\,x}{b}+\frac {b^2\,x}{a\,c}\right )+\mathrm {atan}\left (\frac {c\,x}{b}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.05, size = 44, normalized size = 2.32 \[ \frac {- \frac {i \log {\left (- \frac {i a}{c} - \frac {i b x^{2}}{c} + x^{3} \right )}}{2} + \frac {i \log {\left (\frac {i a}{c} + \frac {i b x^{2}}{c} + x^{3} \right )}}{2}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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