Optimal. Leaf size=176 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (a^2 d^2 \left (16 q^2+56 q+45\right )-2 a b c d (4 q+9)+5 b^2 c^2\right ) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )}{d^2 (4 q+5) (4 q+9)}-\frac{b x \left (c+d x^4\right )^{q+1} (5 b c-a d (4 q+13))}{d^2 (4 q+5) (4 q+9)}+\frac{b x \left (a+b x^4\right ) \left (c+d x^4\right )^{q+1}}{d (4 q+9)} \]
[Out]
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Rubi [A] time = 0.316393, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (a^2 d^2 \left (16 q^2+56 q+45\right )-2 a b c d (4 q+9)+5 b^2 c^2\right ) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )}{d^2 (4 q+5) (4 q+9)}-\frac{b x \left (c+d x^4\right )^{q+1} (5 b c-a d (4 q+13))}{d^2 (4 q+5) (4 q+9)}+\frac{b x \left (a+b x^4\right ) \left (c+d x^4\right )^{q+1}}{d (4 q+9)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^2*(c + d*x^4)^q,x]
[Out]
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Rubi in Sympy [A] time = 32.8228, size = 153, normalized size = 0.87 \[ \frac{b x \left (a + b x^{4}\right ) \left (c + d x^{4}\right )^{q + 1}}{d \left (4 q + 9\right )} - \frac{b x \left (c + d x^{4}\right )^{q + 1} \left (- a d \left (4 q + 13\right ) + 5 b c\right )}{d^{2} \left (4 q + 5\right ) \left (4 q + 9\right )} + \frac{x \left (1 + \frac{d x^{4}}{c}\right )^{- q} \left (c + d x^{4}\right )^{q} \left (- a d \left (4 q + 5\right ) \left (- a d \left (4 q + 9\right ) + b c\right ) + b c \left (- a d \left (4 q + 13\right ) + 5 b c\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - q, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{d^{2} \left (4 q + 5\right ) \left (4 q + 9\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**2*(d*x**4+c)**q,x)
[Out]
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Mathematica [A] time = 0.0788624, size = 106, normalized size = 0.6 \[ \frac{1}{45} x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (45 a^2 \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )+b x^4 \left (18 a \, _2F_1\left (\frac{5}{4},-q;\frac{9}{4};-\frac{d x^4}{c}\right )+5 b x^4 \, _2F_1\left (\frac{9}{4},-q;\frac{13}{4};-\frac{d x^4}{c}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)^2*(c + d*x^4)^q,x]
[Out]
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Maple [F] time = 0.057, size = 0, normalized size = 0. \[ \int \left ( b{x}^{4}+a \right ) ^{2} \left ( d{x}^{4}+c \right ) ^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^2*(d*x^4+c)^q,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{2}{\left (d x^{4} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^2*(d*x^4 + c)^q,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (d x^{4} + c\right )}^{q}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^2*(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)**2*(d*x**4+c)**q,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{2}{\left (d x^{4} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^2*(d*x^4 + c)^q,x, algorithm="giac")
[Out]