Optimal. Leaf size=74 \[ -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{16 c^{3/2}}+\frac {1}{8} x^6 \sqrt {a+c x^4}+\frac {a x^2 \sqrt {a+c x^4}}{16 c} \]
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Rubi [A] time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {275, 279, 321, 217, 206} \[ -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{16 c^{3/2}}+\frac {1}{8} x^6 \sqrt {a+c x^4}+\frac {a x^2 \sqrt {a+c x^4}}{16 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^5 \sqrt {a+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 \sqrt {a+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{8} x^6 \sqrt {a+c x^4}+\frac {1}{8} a \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+c x^2}} \, dx,x,x^2\right )\\ &=\frac {a x^2 \sqrt {a+c x^4}}{16 c}+\frac {1}{8} x^6 \sqrt {a+c x^4}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^2}} \, dx,x,x^2\right )}{16 c}\\ &=\frac {a x^2 \sqrt {a+c x^4}}{16 c}+\frac {1}{8} x^6 \sqrt {a+c x^4}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {a+c x^4}}\right )}{16 c}\\ &=\frac {a x^2 \sqrt {a+c x^4}}{16 c}+\frac {1}{8} x^6 \sqrt {a+c x^4}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{16 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 74, normalized size = 1.00 \[ \frac {\sqrt {a+c x^4} \left (\sqrt {c} x^2 \left (a+2 c x^4\right )-\frac {a^{3/2} \sinh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {\frac {c x^4}{a}+1}}\right )}{16 c^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 127, normalized size = 1.72 \[ \left [\frac {a^{2} \sqrt {c} \log \left (-2 \, c x^{4} + 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) + 2 \, {\left (2 \, c^{2} x^{6} + a c x^{2}\right )} \sqrt {c x^{4} + a}}{32 \, c^{2}}, \frac {a^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2}}{\sqrt {c x^{4} + a}}\right ) + {\left (2 \, c^{2} x^{6} + a c x^{2}\right )} \sqrt {c x^{4} + a}}{16 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 54, normalized size = 0.73 \[ \frac {1}{16} \, \sqrt {c x^{4} + a} {\left (2 \, x^{4} + \frac {a}{c}\right )} x^{2} + \frac {a^{2} \log \left ({\left | -\sqrt {c} x^{2} + \sqrt {c x^{4} + a} \right |}\right )}{16 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.85 \[ -\frac {\sqrt {c \,x^{4}+a}\, a \,x^{2}}{16 c}-\frac {a^{2} \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{16 c^{\frac {3}{2}}}+\frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}} x^{2}}{8 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.99, size = 120, normalized size = 1.62 \[ \frac {a^{2} \log \left (-\frac {\sqrt {c} - \frac {\sqrt {c x^{4} + a}}{x^{2}}}{\sqrt {c} + \frac {\sqrt {c x^{4} + a}}{x^{2}}}\right )}{32 \, c^{\frac {3}{2}}} + \frac {\frac {\sqrt {c x^{4} + a} a^{2} c}{x^{2}} + \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}} a^{2}}{x^{6}}}{16 \, {\left (c^{3} - \frac {2 \, {\left (c x^{4} + a\right )} c^{2}}{x^{4}} + \frac {{\left (c x^{4} + a\right )}^{2} c}{x^{8}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,\sqrt {c\,x^4+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.17, size = 95, normalized size = 1.28 \[ \frac {a^{\frac {3}{2}} x^{2}}{16 c \sqrt {1 + \frac {c x^{4}}{a}}} + \frac {3 \sqrt {a} x^{6}}{16 \sqrt {1 + \frac {c x^{4}}{a}}} - \frac {a^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{16 c^{\frac {3}{2}}} + \frac {c x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {c x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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