Optimal. Leaf size=328 \[ \frac{c x \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{c x \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a^2 \left (\sqrt{b^2-4 a c}+b\right )}+\frac{b \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a^2 x}-\frac{\left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (-\frac{3}{2},-q;-\frac{1}{2};-\frac{e x^2}{d}\right )}{3 a x^3} \]
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Rubi [A] time = 0.620463, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1303, 365, 364, 1692, 430, 429} \[ \frac{c x \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{c x \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a^2 \left (\sqrt{b^2-4 a c}+b\right )}+\frac{b \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a^2 x}-\frac{\left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (-\frac{3}{2},-q;-\frac{1}{2};-\frac{e x^2}{d}\right )}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 1303
Rule 365
Rule 364
Rule 1692
Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^q}{x^4 \left (a+b x^2+c x^4\right )} \, dx &=\int \left (\frac{\left (d+e x^2\right )^q}{a x^4}-\frac{b \left (d+e x^2\right )^q}{a^2 x^2}+\frac{\left (b^2-a c+b c x^2\right ) \left (d+e x^2\right )^q}{a^2 \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=\frac{\int \frac{\left (b^2-a c+b c x^2\right ) \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx}{a^2}+\frac{\int \frac{\left (d+e x^2\right )^q}{x^4} \, dx}{a}-\frac{b \int \frac{\left (d+e x^2\right )^q}{x^2} \, dx}{a^2}\\ &=\frac{\int \left (\frac{\left (b c+\frac{c \left (b^2-2 a c\right )}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^2}+\frac{\left (b c-\frac{c \left (b^2-2 a c\right )}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^2}\right ) \, dx}{a^2}+\frac{\left (\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}{x^4} \, dx}{a}-\frac{\left (b \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}{x^2} \, dx}{a^2}\\ &=-\frac{\left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (-\frac{3}{2},-q;-\frac{1}{2};-\frac{e x^2}{d}\right )}{3 a x^3}+\frac{b \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a^2 x}+\frac{\left (c \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{\left (d+e x^2\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^2} \, dx}{a^2}+\frac{\left (c \left (b+\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{\left (d+e x^2\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^2} \, dx}{a^2}\\ &=-\frac{\left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (-\frac{3}{2},-q;-\frac{1}{2};-\frac{e x^2}{d}\right )}{3 a x^3}+\frac{b \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a^2 x}+\frac{\left (c \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \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}{a^2}+\frac{\left (c \left (b+\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \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}{a^2}\\ &=\frac{c \left (b+\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) 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 )}{a^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{c \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) 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 )}{a^2 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{\left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (-\frac{3}{2},-q;-\frac{1}{2};-\frac{e x^2}{d}\right )}{3 a x^3}+\frac{b \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a^2 x}\\ \end{align*}
Mathematica [F] time = 0.516567, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^q}{x^4 \left (a+b x^2+c x^4\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( e{x}^{2}+d \right ) ^{q}}{{x}^{4} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}\, 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}}{{\left (c x^{4} + b x^{2} + a\right )} x^{4}}\,{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^{8} + b x^{6} + a x^{4}}, 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}}{{\left (c x^{4} + b x^{2} + a\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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