Optimal. Leaf size=93 \[ \frac{b x \left (c+d x^4\right )^{q+1}}{d (4 q+5)}-\frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} (b c-a d (4 q+5)) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )}{d (4 q+5)} \]
[Out]
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Rubi [A] time = 0.111122, antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (a-\frac{b c}{4 d q+5 d}\right ) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )+\frac{b x \left (c+d x^4\right )^{q+1}}{d (4 q+5)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)*(c + d*x^4)^q,x]
[Out]
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Rubi in Sympy [A] time = 11.3714, size = 73, normalized size = 0.78 \[ \frac{b x \left (c + d x^{4}\right )^{q + 1}}{d \left (4 q + 5\right )} - \frac{x \left (1 + \frac{d x^{4}}{c}\right )^{- q} \left (c + d x^{4}\right )^{q} \left (- a d \left (4 q + 5\right ) + b c\right ){{}_{2}F_{1}\left (\begin{matrix} - q, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{d \left (4 q + 5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)*(d*x**4+c)**q,x)
[Out]
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Mathematica [A] time = 0.0347268, size = 75, normalized size = 0.81 \[ \frac{1}{5} x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (5 a \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )+b x^4 \, _2F_1\left (\frac{5}{4},-q;\frac{9}{4};-\frac{d x^4}{c}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)*(c + d*x^4)^q,x]
[Out]
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Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int \left ( b{x}^{4}+a \right ) \left ( d{x}^{4}+c \right ) ^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)*(d*x^4+c)^q,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}{\left (d x^{4} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(d*x^4 + c)^q,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}{\left (d x^{4} + c\right )}^{q}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(d*x^4 + c)^q,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((b*x**4+a)*(d*x**4+c)**q,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}{\left (d x^{4} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(d*x^4 + c)^q,x, algorithm="giac")
[Out]