Optimal. Leaf size=138 \[ -\frac {\left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (-\frac {5}{3};-p,-p;-\frac {2}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{5 x^5} \]
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Rubi [A]
time = 0.06, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1399, 524}
\begin {gather*} -\frac {\left (\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (-\frac {5}{3};-p,-p;-\frac {2}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 1399
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3+c x^6\right )^p}{x^6} \, dx &=\left (\left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p\right ) \int \frac {\left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^p \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^p}{x^6} \, dx\\ &=-\frac {\left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (-\frac {5}{3};-p,-p;-\frac {2}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{5 x^5}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 166, normalized size = 1.20 \begin {gather*} -\frac {\left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (-\frac {5}{3};-p,-p;-\frac {2}{3};-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )}{5 x^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{x^{6}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^6+b\,x^3+a\right )}^p}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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