Optimal. Leaf size=79 \[ -\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{9/2}}+\frac {7 b^2 x}{2 c^4}-\frac {7 b x^3}{6 c^3}-\frac {x^7}{2 c \left (b+c x^2\right )}+\frac {7 x^5}{10 c^2} \]
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Rubi [A] time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1584, 288, 302, 205} \[ \frac {7 b^2 x}{2 c^4}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{9/2}}-\frac {7 b x^3}{6 c^3}-\frac {x^7}{2 c \left (b+c x^2\right )}+\frac {7 x^5}{10 c^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 288
Rule 302
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{12}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^8}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {x^7}{2 c \left (b+c x^2\right )}+\frac {7 \int \frac {x^6}{b+c x^2} \, dx}{2 c}\\ &=-\frac {x^7}{2 c \left (b+c x^2\right )}+\frac {7 \int \left (\frac {b^2}{c^3}-\frac {b x^2}{c^2}+\frac {x^4}{c}-\frac {b^3}{c^3 \left (b+c x^2\right )}\right ) \, dx}{2 c}\\ &=\frac {7 b^2 x}{2 c^4}-\frac {7 b x^3}{6 c^3}+\frac {7 x^5}{10 c^2}-\frac {x^7}{2 c \left (b+c x^2\right )}-\frac {\left (7 b^3\right ) \int \frac {1}{b+c x^2} \, dx}{2 c^4}\\ &=\frac {7 b^2 x}{2 c^4}-\frac {7 b x^3}{6 c^3}+\frac {7 x^5}{10 c^2}-\frac {x^7}{2 c \left (b+c x^2\right )}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 71, normalized size = 0.90 \[ \frac {x \left (\frac {15 b^3}{b+c x^2}+90 b^2-20 b c x^2+6 c^2 x^4\right )}{30 c^4}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 190, normalized size = 2.41 \[ \left [\frac {12 \, c^{3} x^{7} - 28 \, b c^{2} x^{5} + 140 \, b^{2} c x^{3} + 210 \, b^{3} x + 105 \, {\left (b^{2} c x^{2} + b^{3}\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right )}{60 \, {\left (c^{5} x^{2} + b c^{4}\right )}}, \frac {6 \, c^{3} x^{7} - 14 \, b c^{2} x^{5} + 70 \, b^{2} c x^{3} + 105 \, b^{3} x - 105 \, {\left (b^{2} c x^{2} + b^{3}\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right )}{30 \, {\left (c^{5} x^{2} + b c^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 73, normalized size = 0.92 \[ -\frac {7 \, b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{4}} + \frac {b^{3} x}{2 \, {\left (c x^{2} + b\right )} c^{4}} + \frac {3 \, c^{8} x^{5} - 10 \, b c^{7} x^{3} + 45 \, b^{2} c^{6} x}{15 \, c^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 0.86 \[ \frac {x^{5}}{5 c^{2}}-\frac {2 b \,x^{3}}{3 c^{3}}+\frac {b^{3} x}{2 \left (c \,x^{2}+b \right ) c^{4}}-\frac {7 b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, c^{4}}+\frac {3 b^{2} x}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 71, normalized size = 0.90 \[ \frac {b^{3} x}{2 \, {\left (c^{5} x^{2} + b c^{4}\right )}} - \frac {7 \, b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{4}} + \frac {3 \, c^{2} x^{5} - 10 \, b c x^{3} + 45 \, b^{2} x}{15 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 66, normalized size = 0.84 \[ \frac {x^5}{5\,c^2}-\frac {2\,b\,x^3}{3\,c^3}+\frac {3\,b^2\,x}{c^4}-\frac {7\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{2\,c^{9/2}}+\frac {b^3\,x}{2\,\left (c^5\,x^2+b\,c^4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 124, normalized size = 1.57 \[ \frac {b^{3} x}{2 b c^{4} + 2 c^{5} x^{2}} + \frac {3 b^{2} x}{c^{4}} - \frac {2 b x^{3}}{3 c^{3}} + \frac {7 \sqrt {- \frac {b^{5}}{c^{9}}} \log {\left (x - \frac {c^{4} \sqrt {- \frac {b^{5}}{c^{9}}}}{b^{2}} \right )}}{4} - \frac {7 \sqrt {- \frac {b^{5}}{c^{9}}} \log {\left (x + \frac {c^{4} \sqrt {- \frac {b^{5}}{c^{9}}}}{b^{2}} \right )}}{4} + \frac {x^{5}}{5 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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