Optimal. Leaf size=92 \[ \frac {2 \left (-2 a c+b^2+b c x^3\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{3 a^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 92, 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, 740, 12, 724, 206} \[ \frac {2 \left (-2 a c+b^2+b c x^3\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{3 a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 740
Rule 1357
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^3+c x^6\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {2 \operatorname {Subst}\left (\int \frac {-\frac {b^2}{2}+2 a c}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{3 a \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{3 a}\\ &=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {2 \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 )}{3 a}\\ &=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{3 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 92, normalized size = 1.00 \[ \frac {1}{3} \left (\frac {2 \left (-2 a c+b^2+b c x^3\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{a^{3/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 389, normalized size = 4.23 \[ \left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} + {\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )} \sqrt {a} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (a b c x^{3} + a b^{2} - 2 \, a^{2} c\right )}}{6 \, {\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{6} + a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{3}\right )}}, \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} + {\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (a b c x^{3} + a b^{2} - 2 \, a^{2} c\right )}}{3 \, {\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{6} + a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{3}\right )}}\right ] \]
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 \[ \int \frac {1}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}} x}\, dx \]
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 [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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