Optimal. Leaf size=108 \[ -\frac {b \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c^2}+\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 c}+\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{48 c^{5/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 108, 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 = {1371, 654, 626,
635, 212} \begin {gather*} \frac {b \left (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 )}{48 c^{5/2}}-\frac {b \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c^2}+\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 1371
Rubi steps
\begin {align*} \int x^5 \sqrt {a+b x^3+c x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int x \sqrt {a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 c}-\frac {b \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{6 c}\\ &=-\frac {b \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c^2}+\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 c}+\frac {\left (b \left (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 )}{48 c^2}\\ &=-\frac {b \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c^2}+\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 c}+\frac {\left (b \left (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 )}{24 c^2}\\ &=-\frac {b \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{24 c^2}+\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 c}+\frac {b \left (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 )}{48 c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 101, normalized size = 0.94 \begin {gather*} \frac {\sqrt {a+b x^3+c x^6} \left (-3 b^2+2 b c x^3+8 c \left (a+c x^6\right )\right )}{72 c^2}-\frac {\left (b^3-4 a b c\right ) \log \left (c^2 \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right )\right )}{48 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x^{5} \sqrt {c \,x^{6}+b \,x^{3}+a}\, dx\]
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 = 0.38, size = 237, normalized size = 2.19 \begin {gather*} \left [-\frac {3 \, {\left (b^{3} - 4 \, a b 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 \, {\left (8 \, c^{3} x^{6} + 2 \, b c^{2} x^{3} - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, c^{3}}, -\frac {3 \, {\left (b^{3} - 4 \, a b 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 \, {\left (8 \, c^{3} x^{6} + 2 \, b c^{2} x^{3} - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, c^{3}}\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^{5} \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 [A]
time = 3.93, size = 98, normalized size = 0.91 \begin {gather*} \frac {1}{72} \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, {\left (4 \, x^{3} + \frac {b}{c}\right )} x^{3} - \frac {3 \, b^{2} - 8 \, a c}{c^{2}}\right )} - \frac {{\left (b^{3} - 4 \, a b c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{3} - \sqrt {c x^{6} + b x^{3} + a}\right )} \sqrt {c} - b \right |}\right )}{48 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.39, size = 87, normalized size = 0.81 \begin {gather*} \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}}{72\,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 )}{48\,c^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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