Optimal. Leaf size=157 \[ \frac{2^{2 p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (-2 p;-p,-p;1-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^3},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3}\right )}{3 p} \]
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Rubi [A] time = 0.141581, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1357, 758, 133} \[ \frac{2^{2 p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (-2 p;-p,-p;1-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^3},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3}\right )}{3 p} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 758
Rule 133
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3+c x^6\right )^p}{x} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^p}{x} \, dx,x,x^3\right )\\ &=-\left (\frac{1}{3} \left (2^{2 p} \left (\frac{1}{x^3}\right )^{2 p} \left (\frac{b-\sqrt{b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac{b+\sqrt{b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p\right ) \operatorname{Subst}\left (\int x^{1-2 (1+p)} \left (1+\frac{\left (b-\sqrt{b^2-4 a c}\right ) x}{2 c}\right )^p \left (1+\frac{\left (b+\sqrt{b^2-4 a c}\right ) x}{2 c}\right )^p \, dx,x,\frac{1}{x^3}\right )\right )\\ &=\frac{2^{-1+2 p} \left (\frac{b-\sqrt{b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac{b+\sqrt{b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (-2 p;-p,-p;1-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^3},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3}\right )}{3 p}\\ \end{align*}
Mathematica [A] time = 0.204122, size = 157, normalized size = 1. \[ \frac{2^{2 p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (-2 p;-p,-p;1-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right )}{3 p} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.014, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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