Optimal. Leaf size=175 \[ \frac{c \left (a+b x^4\right )^{p+1}}{4 b (p+1)}+\frac{1}{5} d x^5 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right )+\frac{1}{6} e x^6 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^4}{a}\right )+\frac{1}{7} f x^7 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{7}{4},-p;\frac{11}{4};-\frac{b x^4}{a}\right ) \]
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Rubi [A] time = 0.403432, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{c \left (a+b x^4\right )^{p+1}}{4 b (p+1)}+\frac{1}{5} d x^5 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right )+\frac{1}{6} e x^6 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^4}{a}\right )+\frac{1}{7} f x^7 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{7}{4},-p;\frac{11}{4};-\frac{b x^4}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^p,x]
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Rubi in Sympy [A] time = 46.9727, size = 139, normalized size = 0.79 \[ \frac{d x^{5} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{5} + \frac{e x^{6} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{6} + \frac{f x^{7} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{7} + \frac{c \left (a + b x^{4}\right )^{p + 1}}{4 b \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(f*x**3+e*x**2+d*x+c)*(b*x**4+a)**p,x)
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Mathematica [A] time = 0.239194, size = 160, normalized size = 0.91 \[ \frac{\left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (105 c \left (b x^4 \left (\frac{b x^4}{a}+1\right )^p+a \left (\left (\frac{b x^4}{a}+1\right )^p-1\right )\right )+84 b d (p+1) x^5 \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right )+70 b e (p+1) x^6 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^4}{a}\right )+60 b f (p+1) x^7 \, _2F_1\left (\frac{7}{4},-p;\frac{11}{4};-\frac{b x^4}{a}\right )\right )}{420 b (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^p,x]
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Maple [F] time = 0.077, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( f{x}^{3}+e{x}^{2}+dx+c \right ) \left ( b{x}^{4}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*x^3,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f x^{6} + e x^{5} + d x^{4} + c x^{3}\right )}{\left (b x^{4} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*x^3,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(x**3*(f*x**3+e*x**2+d*x+c)*(b*x**4+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{4} + a\right )}^{p} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*x^3,x, algorithm="giac")
[Out]