Optimal. Leaf size=88 \[ \frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{3/2}}-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1357, 720, 724, 206} \[ \frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{3/2}}-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6} \]
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 1357
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3+c x^6}}{x^7} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6}-\frac {\left (b^2-4 a c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{24 a}\\ &=-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6}+\frac {\left (b^2-4 a c\right ) \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 )}{12 a}\\ &=-\frac {\left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{12 a x^6}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{24 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 89, normalized size = 1.01 \[ \frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )-\frac {2 \sqrt {a} \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{x^6}}{24 a^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 215, normalized size = 2.44 \[ \left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {a} x^{6} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (a b x^{3} + 2 \, a^{2}\right )}}{48 \, a^{2} x^{6}}, -\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (a b x^{3} + 2 \, a^{2}\right )}}{24 \, a^{2} x^{6}}\right ] \]
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 \[ \int \frac {\sqrt {c x^{6} + b x^{3} + a}}{x^{7}}\,{d x} \]
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 \[ \int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{x^{7}}\, dx \]
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 \[ \int \frac {\sqrt {c\,x^6+b\,x^3+a}}{x^7} \,d x \]
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 \[ \int \frac {\sqrt {a + b x^{3} + c x^{6}}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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