Optimal. Leaf size=107 \[ \frac{c x^2 \left (a+b x^3\right )^{p+1} \, _2F_1\left (1,p+\frac{5}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a}+\frac{d \left (a+b x^3\right )^{p+1}}{3 b (p+1)}+\frac{e x^4 \left (a+b x^3\right )^{p+1} \, _2F_1\left (1,p+\frac{7}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 a} \]
[Out]
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Rubi [A] time = 0.191935, antiderivative size = 125, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{1}{2} c x^2 \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{2}{3},-p;\frac{5}{3};-\frac{b x^3}{a}\right )+\frac{d \left (a+b x^3\right )^{p+1}}{3 b (p+1)}+\frac{1}{4} e x^4 \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{4}{3},-p;\frac{7}{3};-\frac{b x^3}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[x*(c + d*x + e*x^2)*(a + b*x^3)^p,x]
[Out]
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Rubi in Sympy [A] time = 23.8972, size = 99, normalized size = 0.93 \[ \frac{c x^{2} \left (1 + \frac{b x^{3}}{a}\right )^{- p} \left (a + b x^{3}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{2} + \frac{e x^{4} \left (1 + \frac{b x^{3}}{a}\right )^{- p} \left (a + b x^{3}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{4} + \frac{d \left (a + b x^{3}\right )^{p + 1}}{3 b \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(e*x**2+d*x+c)*(b*x**3+a)**p,x)
[Out]
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Mathematica [A] time = 0.124217, size = 131, normalized size = 1.22 \[ \frac{\left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \left (6 b c (p+1) x^2 \, _2F_1\left (\frac{2}{3},-p;\frac{5}{3};-\frac{b x^3}{a}\right )+4 d \left (b x^3 \left (\frac{b x^3}{a}+1\right )^p+a \left (\left (\frac{b x^3}{a}+1\right )^p-1\right )\right )+3 b e (p+1) x^4 \, _2F_1\left (\frac{4}{3},-p;\frac{7}{3};-\frac{b x^3}{a}\right )\right )}{12 b (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[x*(c + d*x + e*x^2)*(a + b*x^3)^p,x]
[Out]
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Maple [F] time = 0.057, size = 0, normalized size = 0. \[ \int x \left ( e{x}^{2}+dx+c \right ) \left ( b{x}^{3}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(e*x^2+d*x+c)*(b*x^3+a)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)*(b*x^3 + a)^p*x,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{3} + d x^{2} + c x\right )}{\left (b x^{3} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)*(b*x^3 + a)^p*x,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*(e*x**2+d*x+c)*(b*x**3+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)*(b*x^3 + a)^p*x,x, algorithm="giac")
[Out]