Optimal. Leaf size=150 \[ -\frac{x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (a e^2-b c^2 (4 p+5)\right ) \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right )}{b (4 p+5)}+\frac{2}{3} c e x^3 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right )+\frac{e^2 x \left (a+b x^4\right )^{p+1}}{b (4 p+5)} \]
[Out]
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Rubi [A] time = 0.269505, antiderivative size = 142, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (c^2-\frac{a e^2}{4 b p+5 b}\right ) \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right )+\frac{2}{3} c e x^3 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right )+\frac{e^2 x \left (a+b x^4\right )^{p+1}}{b (4 p+5)} \]
Antiderivative was successfully verified.
[In] Int[(c + e*x^2)^2*(a + b*x^4)^p,x]
[Out]
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Rubi in Sympy [A] time = 25.4957, size = 124, normalized size = 0.83 \[ c^{2} x \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )} + \frac{2 c e x^{3} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{3} + \frac{e^{2} x^{5} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+c)**2*(b*x**4+a)**p,x)
[Out]
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Mathematica [A] time = 0.0547916, size = 106, normalized size = 0.71 \[ \frac{1}{15} x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (15 c^2 \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right )+e x^2 \left (10 c \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right )+3 e x^2 \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(c + e*x^2)^2*(a + b*x^4)^p,x]
[Out]
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Maple [F] time = 0.062, size = 0, normalized size = 0. \[ \int \left ( e{x}^{2}+c \right ) ^{2} \left ( b{x}^{4}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+c)^2*(b*x^4+a)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{2} + c\right )}^{2}{\left (b x^{4} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + c)^2*(b*x^4 + a)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{2} x^{4} + 2 \, c e x^{2} + c^{2}\right )}{\left (b x^{4} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + c)^2*(b*x^4 + a)^p,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((e*x**2+c)**2*(b*x**4+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{2} + c\right )}^{2}{\left (b x^{4} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + c)^2*(b*x^4 + a)^p,x, algorithm="giac")
[Out]