Optimal. Leaf size=25 \[ \frac {\left (b x^2+c x^4\right )^{5/2}}{5 c x^5} \]
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Rubi [A] time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2014} \[ \frac {\left (b x^2+c x^4\right )^{5/2}}{5 c x^5} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx &=\frac {\left (b x^2+c x^4\right )^{5/2}}{5 c x^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2}}{5 c x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 39, normalized size = 1.56 \[ \frac {{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{5 \, c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 27, normalized size = 1.08 \[ \frac {{\left (c x^{2} + b\right )}^{\frac {5}{2}} \mathrm {sgn}\relax (x)}{5 \, c} - \frac {b^{\frac {5}{2}} \mathrm {sgn}\relax (x)}{5 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 1.16 \[ \frac {\left (c \,x^{2}+b \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{5 c \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 32, normalized size = 1.28 \[ \frac {{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {c x^{2} + b}}{5 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.15, size = 30, normalized size = 1.20 \[ \frac {{\left (c\,x^2+b\right )}^2\,\sqrt {c\,x^4+b\,x^2}}{5\,c\,x} \]
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 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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