Optimal. Leaf size=57 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} F_1\left (\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^2} \]
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Rubi [A] time = 0.0276523, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {430, 429} \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} F_1\left (\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{\left (c+d x^4\right )^q}{\left (a+b x^4\right )^2} \, dx &=\left (\left (c+d x^4\right )^q \left (1+\frac{d x^4}{c}\right )^{-q}\right ) \int \frac{\left (1+\frac{d x^4}{c}\right )^q}{\left (a+b x^4\right )^2} \, dx\\ &=\frac{x \left (c+d x^4\right )^q \left (1+\frac{d x^4}{c}\right )^{-q} F_1\left (\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^2}\\ \end{align*}
Mathematica [B] time = 0.187785, size = 162, normalized size = 2.84 \[ \frac{5 a c x \left (c+d x^4\right )^q F_1\left (\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (a+b x^4\right )^2 \left (4 x^4 \left (a d q F_1\left (\frac{5}{4};2,1-q;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-2 b c F_1\left (\frac{5}{4};3,-q;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.413, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d{x}^{4}+c \right ) ^{q}}{ \left ( b{x}^{4}+a \right ) ^{2}}}\, 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 (d x^{4} + c\right )}^{q}}{{\left (b x^{4} + a\right )}^{2}}\,{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 (d x^{4} + c\right )}^{q}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, 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 (d x^{4} + c\right )}^{q}}{{\left (b x^{4} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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