Optimal. Leaf size=57 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} F_1\left (\frac{1}{4};1,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a} \]
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Rubi [A] time = 0.0838656, 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 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} F_1\left (\frac{1}{4};1,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^4)^q/(a + b*x^4),x]
[Out]
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Rubi in Sympy [A] time = 21.9034, size = 42, normalized size = 0.74 \[ \frac{x \left (1 + \frac{d x^{4}}{c}\right )^{- q} \left (c + d x^{4}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{4},1,- q,\frac{5}{4},- \frac{b x^{4}}{a},- \frac{d x^{4}}{c} \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**4+c)**q/(b*x**4+a),x)
[Out]
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Mathematica [B] time = 0.280766, size = 162, normalized size = 2.84 \[ \frac{5 a c x \left (c+d x^4\right )^q F_1\left (\frac{1}{4};-q,1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \left (4 x^4 \left (a d q F_1\left (\frac{5}{4};1-q,1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-b c F_1\left (\frac{5}{4};-q,2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};-q,1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(c + d*x^4)^q/(a + b*x^4),x]
[Out]
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Maple [F] time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d{x}^{4}+c \right ) ^{q}}{b{x}^{4}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^4+c)^q/(b*x^4+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{4} + c\right )}^{q}}{b x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^4 + c)^q/(b*x^4 + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x^{4} + c\right )}^{q}}{b x^{4} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^4 + c)^q/(b*x^4 + a),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**4+c)**q/(b*x**4+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{4} + c\right )}^{q}}{b x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^4 + c)^q/(b*x^4 + a),x, algorithm="giac")
[Out]