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 = 0.0831709, 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 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} F_1\left (\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^4)^q/(a + b*x^4)^2,x]
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Rubi in Sympy [A] time = 20.2223, size = 44, normalized size = 0.77 \[ \frac{x \left (1 + \frac{d x^{4}}{c}\right )^{- q} \left (c + d x^{4}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{4},2,- q,\frac{5}{4},- \frac{b x^{4}}{a},- \frac{d x^{4}}{c} \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**4+c)**q/(b*x**4+a)**2,x)
[Out]
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Mathematica [B] time = 0.307696, 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.
[In] Integrate[(c + d*x^4)^q/(a + b*x^4)^2,x]
[Out]
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Maple [F] time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d{x}^{4}+c \right ) ^{q}}{ \left ( b{x}^{4}+a \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^4+c)^q/(b*x^4+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{4} + c\right )}^{q}}{{\left (b x^{4} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^4 + c)^q/(b*x^4 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x^{4} + c\right )}^{q}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^4 + c)^q/(b*x^4 + a)^2,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)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{4} + c\right )}^{q}}{{\left (b x^{4} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^4 + c)^q/(b*x^4 + a)^2,x, algorithm="giac")
[Out]