Optimal. Leaf size=109 \[ \frac {1}{3} \sqrt {a+b x^3+c x^6}-\frac {1}{3} \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )+\frac {b \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{6 \sqrt {c}} \]
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Rubi [A] time = 0.11, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1357, 734, 843, 621, 206, 724} \[ \frac {1}{3} \sqrt {a+b x^3+c x^6}-\frac {1}{3} \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )+\frac {b \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{6 \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 734
Rule 843
Rule 1357
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3+c x^6}}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \sqrt {a+b x^3+c x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {-2 a-b x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \sqrt {a+b x^3+c x^6}+\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )+\frac {1}{6} b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \sqrt {a+b x^3+c x^6}-\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^3}{\sqrt {a+b x^3+c x^6}}\right )+\frac {1}{3} b \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 )\\ &=\frac {1}{3} \sqrt {a+b x^3+c x^6}-\frac {1}{3} \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )+\frac {b \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{6 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 106, normalized size = 0.97 \[ \frac {1}{6} \left (2 \sqrt {a+b x^3+c x^6}-2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )+\frac {b \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 566, normalized size = 5.19 \[ \left [\frac {b \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 2 \, \sqrt {a} c \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, \sqrt {c x^{6} + b x^{3} + a} c}{12 \, c}, -\frac {b \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - \sqrt {a} c \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 2 \, \sqrt {c x^{6} + b x^{3} + a} c}{6 \, c}, \frac {4 \, \sqrt {-a} c \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + b \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, \sqrt {c x^{6} + b x^{3} + a} c}{12 \, c}, \frac {2 \, \sqrt {-a} c \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) - b \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) + 2 \, \sqrt {c x^{6} + b x^{3} + a} c}{6 \, c}\right ] \]
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 \[ \int \frac {\sqrt {c x^{6} + b x^{3} + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{x}\, dx \]
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 [B] time = 1.36, size = 88, normalized size = 0.81 \[ \frac {\sqrt {c\,x^6+b\,x^3+a}}{3}-\frac {\sqrt {a}\,\ln \left (\frac {b}{2}+\frac {a}{x^3}+\frac {\sqrt {a}\,\sqrt {c\,x^6+b\,x^3+a}}{x^3}\right )}{3}+\frac {b\,\ln \left (\sqrt {c\,x^6+b\,x^3+a}+\frac {c\,x^3+\frac {b}{2}}{\sqrt {c}}\right )}{6\,\sqrt {c}} \]
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 \[ \int \frac {\sqrt {a + b x^{3} + c x^{6}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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