Optimal. Leaf size=153 \[ \frac {\left (5 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{192 c^3}-\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{3/2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{384 c^{7/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1371, 756, 654,
626, 635, 212} \begin {gather*} -\frac {\left (b^2-4 a c\right ) \left (5 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 )}{384 c^{7/2}}+\frac {\left (5 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{192 c^3}-\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{3/2}}{12 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 756
Rule 1371
Rubi steps
\begin {align*} \int x^8 \sqrt {a+b x^3+c x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int x^2 \sqrt {a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac {x^3 \left (a+b x^3+c x^6\right )^{3/2}}{12 c}+\frac {\text {Subst}\left (\int \left (-a-\frac {5 b x}{2}\right ) \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{12 c}\\ &=-\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{3/2}}{12 c}+\frac {\left (5 b^2-4 a c\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{48 c^2}\\ &=\frac {\left (5 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{192 c^3}-\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{3/2}}{12 c}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{384 c^3}\\ &=\frac {\left (5 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{192 c^3}-\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{3/2}}{12 c}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \text {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 )}{192 c^3}\\ &=\frac {\left (5 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{192 c^3}-\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{3/2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (5 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 )}{384 c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 132, normalized size = 0.86 \begin {gather*} \frac {\sqrt {a+b x^3+c x^6} \left (15 b^3-52 a b c-10 b^2 c x^3+24 a c^2 x^3+8 b c^2 x^6+48 c^3 x^9\right )}{576 c^3}+\frac {\left (5 b^4-24 a b^2 c+16 a^2 c^2\right ) \log \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right )}{384 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{8} \sqrt {c \,x^{6}+b \,x^{3}+a}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 303, normalized size = 1.98 \begin {gather*} \left [\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\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 \, {\left (48 \, c^{4} x^{9} + 8 \, b c^{3} x^{6} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \, {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{2304 \, c^{4}}, \frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\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 \, {\left (48 \, c^{4} x^{9} + 8 \, b c^{3} x^{6} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \, {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{1152 \, c^{4}}\right ] \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 x^{8} \sqrt {a + b x^{3} + c x^{6}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.59, size = 193, normalized size = 1.26 \begin {gather*} \frac {x^3\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{12\,c}-\frac {a\,\left (\left (\frac {b}{4\,c}+\frac {x^3}{2}\right )\,\sqrt {c\,x^6+b\,x^3+a}+\frac {\ln \left (\sqrt {c\,x^6+b\,x^3+a}+\frac {c\,x^3+\frac {b}{2}}{\sqrt {c}}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{12\,c}-\frac {5\,b\,\left (\frac {\left (8\,c\,\left (c\,x^6+a\right )-3\,b^2+2\,b\,c\,x^3\right )\,\sqrt {c\,x^6+b\,x^3+a}}{24\,c^2}+\frac {\ln \left (2\,\sqrt {c\,x^6+b\,x^3+a}+\frac {2\,c\,x^3+b}{\sqrt {c}}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}\right )}{24\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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