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 = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 429
Rule 430
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*}
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Mathematica [B] time = 0.24, 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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fricas [F] time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x^{4} + c\right )}^{q}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{4} + c\right )}^{q}}{{\left (b x^{4} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{4}+c \right )^{q}}{\left (b \,x^{4}+a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{4} + c\right )}^{q}}{{\left (b x^{4} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d\,x^4+c\right )}^q}{{\left (b\,x^4+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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