Optimal. Leaf size=166 \[ -\frac{2^{2 p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (1-2 p;-p,-p;2 (1-p);-\frac{b-\sqrt{b^2-4 a c}}{2 c x^2},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2}\right )}{(1-2 p) x^2} \]
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Rubi [A] time = 0.130781, antiderivative size = 166, 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 = {1114, 758, 133} \[ -\frac{2^{2 p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (1-2 p;-p,-p;2 (1-p);-\frac{b-\sqrt{b^2-4 a c}}{2 c x^2},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2}\right )}{(1-2 p) x^2} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 758
Rule 133
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^p}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^p}{x^2} \, dx,x,x^2\right )\\ &=-\left (\left (2^{-1+2 p} \left (\frac{1}{x^2}\right )^{2 p} \left (\frac{b-\sqrt{b^2-4 a c}+2 c x^2}{c x^2}\right )^{-p} \left (\frac{b+\sqrt{b^2-4 a c}+2 c x^2}{c x^2}\right )^{-p} \left (a+b x^2+c x^4\right )^p\right ) \operatorname{Subst}\left (\int x^{2-2 (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^2}\right )\right )\\ &=-\frac{2^{-1+2 p} \left (\frac{b-\sqrt{b^2-4 a c}+2 c x^2}{c x^2}\right )^{-p} \left (\frac{b+\sqrt{b^2-4 a c}+2 c x^2}{c x^2}\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (1-2 p;-p,-p;2 (1-p);-\frac{b-\sqrt{b^2-4 a c}}{2 c x^2},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2}\right )}{(1-2 p) x^2}\\ \end{align*}
Mathematica [A] time = 0.210106, size = 163, normalized size = 0.98 \[ \frac{2^{2 p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (1-2 p;-p,-p;2-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2},\frac{\sqrt{b^2-4 a c}-b}{2 c x^2}\right )}{(2 p-1) x^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{p}}{{x}^{3}}}\, 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^{4} + b x^{2} + a\right )}^{p}}{x^{3}}\,{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^{4} + b x^{2} + a\right )}^{p}}{x^{3}}, 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^{4} + b x^{2} + a\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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