Optimal. Leaf size=47 \[ \frac {2 \sqrt {b x^2+c x^4}}{c^2 x}-\frac {x^3}{c \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2015, 1588} \[ \frac {2 \sqrt {b x^2+c x^4}}{c^2 x}-\frac {x^3}{c \sqrt {b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1588
Rule 2015
Rubi steps
\begin {align*} \int \frac {x^6}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {x^3}{c \sqrt {b x^2+c x^4}}+\frac {2 \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx}{c}\\ &=-\frac {x^3}{c \sqrt {b x^2+c x^4}}+\frac {2 \sqrt {b x^2+c x^4}}{c^2 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.62 \[ \frac {x \left (2 b+c x^2\right )}{c^2 \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 39, normalized size = 0.83 \[ \frac {\sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + 2 \, b\right )}}{c^{3} x^{3} + b c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 52, normalized size = 1.11 \[ -\frac {2 \, \sqrt {b}}{{\left ({\left (\sqrt {c + \frac {b}{x^{2}}} - \frac {\sqrt {b}}{x}\right )}^{2} - c\right )} c} + \frac {b}{\sqrt {c + \frac {b}{x^{2}}} c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 0.79 \[ \frac {\left (c \,x^{2}+b \right ) \left (c \,x^{2}+2 b \right ) x^{3}}{\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 22, normalized size = 0.47 \[ \frac {c x^{2} + 2 \, b}{\sqrt {c x^{2} + b} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.23, size = 38, normalized size = 0.81 \[ \frac {\sqrt {c\,x^4+b\,x^2}\,\left (c\,x^2+2\,b\right )}{c^2\,x\,\left (c\,x^2+b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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