Optimal. Leaf size=44 \[ \frac {c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6}-\frac {\left (a+c x^4\right )^{3/2}}{10 a x^{10}} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac {c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6}-\frac {\left (a+c x^4\right )^{3/2}}{10 a x^{10}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^4}}{x^{11}} \, dx &=-\frac {\left (a+c x^4\right )^{3/2}}{10 a x^{10}}-\frac {(2 c) \int \frac {\sqrt {a+c x^4}}{x^7} \, dx}{5 a}\\ &=-\frac {\left (a+c x^4\right )^{3/2}}{10 a x^{10}}+\frac {c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.70 \[ \frac {\left (a+c x^4\right )^{3/2} \left (2 c x^4-3 a\right )}{30 a^2 x^{10}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 38, normalized size = 0.86 \[ \frac {{\left (2 \, c^{2} x^{8} - a c x^{4} - 3 \, a^{2}\right )} \sqrt {c x^{4} + a}}{30 \, a^{2} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 120, normalized size = 2.73 \[ \frac {2 \, {\left (15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{6} c^{\frac {5}{2}} + 5 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a c^{\frac {5}{2}} + 5 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} a^{2} c^{\frac {5}{2}} - a^{3} c^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.64 \[ -\frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}} \left (-2 c \,x^{4}+3 a \right )}{30 a^{2} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 35, normalized size = 0.80 \[ \frac {\frac {5 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} c}{x^{6}} - \frac {3 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}}}{x^{10}}}{30 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 37, normalized size = 0.84 \[ -\frac {\sqrt {c\,x^4+a}\,\left (3\,a^2+a\,c\,x^4-2\,c^2\,x^8\right )}{30\,a^2\,x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.35, size = 66, normalized size = 1.50 \[ - \frac {\sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{10 x^{8}} - \frac {c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{30 a x^{4}} + \frac {c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{15 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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