Optimal. Leaf size=190 \[ -\frac{2 c x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]
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Rubi [A] time = 0.293837, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1174, 430, 429} \[ -\frac{2 c x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]
Antiderivative was successfully verified.
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Rule 1174
Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx &=\frac{(2 c) \int \frac{\left (d+e x^2\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^2} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c) \int \frac{\left (d+e x^2\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^2} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{\left (2 c \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q}\right ) \int \frac{\left (1+\frac{e x^2}{d}\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^2} \, dx}{\sqrt{b^2-4 a c}}-\frac{\left (2 c \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q}\right ) \int \frac{\left (1+\frac{e x^2}{d}\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^2} \, dx}{\sqrt{b^2-4 a c}}\\ &=-\frac{2 c x \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}-\frac{2 c x \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}\\ \end{align*}
Mathematica [F] time = 0.0357463, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( e{x}^{2}+d \right ) ^{q}}{c{x}^{4}+b{x}^{2}+a}}\, 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 (e x^{2} + d\right )}^{q}}{c x^{4} + b x^{2} + a}\,{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 (e x^{2} + d\right )}^{q}}{c x^{4} + b x^{2} + a}, 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 (e x^{2} + d\right )}^{q}}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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