Optimal. Leaf size=52 \[ -\frac {\left (b x^2+c x^4\right )^{3/2}}{5 b x^8}+\frac {2 c \left (b x^2+c x^4\right )^{3/2}}{15 b^2 x^6} \]
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Rubi [A]
time = 0.05, antiderivative size = 52, 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 = {2041, 2039}
\begin {gather*} \frac {2 c \left (b x^2+c x^4\right )^{3/2}}{15 b^2 x^6}-\frac {\left (b x^2+c x^4\right )^{3/2}}{5 b x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^7} \, dx &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{5 b x^8}-\frac {(2 c) \int \frac {\sqrt {b x^2+c x^4}}{x^5} \, dx}{5 b}\\ &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{5 b x^8}+\frac {2 c \left (b x^2+c x^4\right )^{3/2}}{15 b^2 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 46, normalized size = 0.88 \begin {gather*} \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (-3 b^2-b c x^2+2 c^2 x^4\right )}{15 b^2 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 39, normalized size = 0.75
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (-2 c \,x^{2}+3 b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{15 b^{2} x^{6}}\) | \(39\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (-2 c \,x^{2}+3 b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{15 b^{2} x^{6}}\) | \(39\) |
trager | \(-\frac {\left (-2 c^{2} x^{4}+b c \,x^{2}+3 b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{15 b^{2} x^{6}}\) | \(42\) |
risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (-2 c^{2} x^{4}+b c \,x^{2}+3 b^{2}\right )}{15 x^{6} b^{2}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 65, normalized size = 1.25 \begin {gather*} \frac {2 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{15 \, b^{2} x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}} c}{15 \, b x^{4}} - \frac {\sqrt {c x^{4} + b x^{2}}}{5 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 42, normalized size = 0.81 \begin {gather*} \frac {{\left (2 \, c^{2} x^{4} - b c x^{2} - 3 \, b^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{15 \, b^{2} x^{6}} \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^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs.
\(2 (44) = 88\).
time = 7.44, size = 120, normalized size = 2.31 \begin {gather*} \frac {4 \, {\left (15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b^{2} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) - b^{3} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )\right )}}{15 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.26, size = 41, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {c\,x^4+b\,x^2}\,\left (3\,b^2+b\,c\,x^2-2\,c^2\,x^4\right )}{15\,b^2\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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