Optimal. Leaf size=25 \[ -\frac {\left (b x^2+c x^4\right )^{3/2}}{3 b x^6} \]
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Rubi [A] time = 0.04, 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} \begin {gather*} -\frac {\left (b x^2+c x^4\right )^{3/2}}{3 b x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^5} \, dx &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{3 b x^6}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -\frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2}}{3 b x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 35, normalized size = 1.40 \begin {gather*} \frac {\left (-b-c x^2\right ) \sqrt {b x^2+c x^4}}{3 b x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 28, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + b\right )}}{3 \, b x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 63, normalized size = 2.52 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} c^{\frac {3}{2}} \mathrm {sgn}\relax (x) + b^{2} c^{\frac {3}{2}} \mathrm {sgn}\relax (x)\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{3}} \end {gather*}
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 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3 b \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 41, normalized size = 1.64 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}} c}{3 \, b x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}}}{3 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.15, size = 28, normalized size = 1.12 \begin {gather*} -\frac {\left (c\,x^2+b\right )\,\sqrt {c\,x^4+b\,x^2}}{3\,b\,x^4} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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