Optimal. Leaf size=57 \[ \frac {x \left (c+d x^4\right )^q \left (1+\frac {d x^4}{c}\right )^{-q} F_1\left (\frac {1}{4};1,-q;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a} \]
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Rubi [A]
time = 0.02, 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 = {441, 440}
\begin {gather*} \frac {x \left (c+d x^4\right )^q \left (\frac {d x^4}{c}+1\right )^{-q} F_1\left (\frac {1}{4};1,-q;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rubi steps
\begin {align*} \int \frac {\left (c+d x^4\right )^q}{a+b x^4} \, 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}{a+b x^4} \, 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};1,-q;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(57)=114\).
time = 0.32, size = 162, normalized size = 2.84 \begin {gather*} \frac {5 a c x \left (c+d x^4\right )^q F_1\left (\frac {1}{4};-q,1;\frac {5}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{\left (a+b x^4\right ) \left (5 a c F_1\left (\frac {1}{4};-q,1;\frac {5}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+4 x^4 \left (a d q F_1\left (\frac {5}{4};1-q,1;\frac {9}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )-b c F_1\left (\frac {5}{4};-q,2;\frac {9}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (d \,x^{4}+c \right )^{q}}{b \,x^{4}+a}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d\,x^4+c\right )}^q}{b\,x^4+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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