Optimal. Leaf size=104 \[ \frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{24 c^{5/2}}-\frac {b \sqrt {a+b x^3+c x^6}}{4 c^2}+\frac {x^3 \sqrt {a+b x^3+c x^6}}{6 c} \]
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Rubi [A] time = 0.09, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1357, 742, 640, 621, 206} \[ \frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{24 c^{5/2}}-\frac {b \sqrt {a+b x^3+c x^6}}{4 c^2}+\frac {x^3 \sqrt {a+b x^3+c x^6}}{6 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 742
Rule 1357
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {a+b x^3+c x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )\\ &=\frac {x^3 \sqrt {a+b x^3+c x^6}}{6 c}+\frac {\operatorname {Subst}\left (\int \frac {-a-\frac {3 b x}{2}}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{6 c}\\ &=-\frac {b \sqrt {a+b x^3+c x^6}}{4 c^2}+\frac {x^3 \sqrt {a+b x^3+c x^6}}{6 c}+\frac {\left (3 b^2-4 a c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{24 c^2}\\ &=-\frac {b \sqrt {a+b x^3+c x^6}}{4 c^2}+\frac {x^3 \sqrt {a+b x^3+c x^6}}{6 c}+\frac {\left (3 b^2-4 a c\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^3}{\sqrt {a+b x^3+c x^6}}\right )}{12 c^2}\\ &=-\frac {b \sqrt {a+b x^3+c x^6}}{4 c^2}+\frac {x^3 \sqrt {a+b x^3+c x^6}}{6 c}+\frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{24 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 88, normalized size = 0.85 \[ \frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )+2 \sqrt {c} \left (2 c x^3-3 b\right ) \sqrt {a+b x^3+c x^6}}{24 c^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 203, normalized size = 1.95 \[ \left [-\frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c^{2} x^{3} - 3 \, b c\right )}}{48 \, c^{3}}, -\frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c^{2} x^{3} - 3 \, b c\right )}}{24 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{8}}{\sqrt {c x^{6} + b x^{3} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {x^{8}}{\sqrt {c \,x^{6}+b \,x^{3}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^8}{\sqrt {c\,x^6+b\,x^3+a}} \,d x \]
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 \[ \int \frac {x^{8}}{\sqrt {a + b x^{3} + c x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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