3.464 \(\int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^p \, dx\)

Optimal. Leaf size=107 \[ \frac{c \left (a+b x^3\right )^{p+1}}{3 b (p+1)}+\frac{d x^4 \left (a+b x^3\right )^{p+1} \, _2F_1\left (1,p+\frac{7}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 a}+\frac{e x^5 \left (a+b x^3\right )^{p+1} \, _2F_1\left (1,p+\frac{8}{3};\frac{8}{3};-\frac{b x^3}{a}\right )}{5 a} \]

[Out]

(c*(a + b*x^3)^(1 + p))/(3*b*(1 + p)) + (d*x^4*(a + b*x^3)^(1 + p)*Hypergeometri
c2F1[1, 7/3 + p, 7/3, -((b*x^3)/a)])/(4*a) + (e*x^5*(a + b*x^3)^(1 + p)*Hypergeo
metric2F1[1, 8/3 + p, 8/3, -((b*x^3)/a)])/(5*a)

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Rubi [A]  time = 0.225218, antiderivative size = 125, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{c \left (a+b x^3\right )^{p+1}}{3 b (p+1)}+\frac{1}{4} d x^4 \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{4}{3},-p;\frac{7}{3};-\frac{b x^3}{a}\right )+\frac{1}{5} e x^5 \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{3},-p;\frac{8}{3};-\frac{b x^3}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2*(c + d*x + e*x^2)*(a + b*x^3)^p,x]

[Out]

(c*(a + b*x^3)^(1 + p))/(3*b*(1 + p)) + (d*x^4*(a + b*x^3)^p*Hypergeometric2F1[4
/3, -p, 7/3, -((b*x^3)/a)])/(4*(1 + (b*x^3)/a)^p) + (e*x^5*(a + b*x^3)^p*Hyperge
ometric2F1[5/3, -p, 8/3, -((b*x^3)/a)])/(5*(1 + (b*x^3)/a)^p)

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Rubi in Sympy [A]  time = 26.8163, size = 99, normalized size = 0.93 \[ \frac{d x^{4} \left (1 + \frac{b x^{3}}{a}\right )^{- p} \left (a + b x^{3}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{4} + \frac{e x^{5} \left (1 + \frac{b x^{3}}{a}\right )^{- p} \left (a + b x^{3}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{5} + \frac{c \left (a + b x^{3}\right )^{p + 1}}{3 b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(e*x**2+d*x+c)*(b*x**3+a)**p,x)

[Out]

d*x**4*(1 + b*x**3/a)**(-p)*(a + b*x**3)**p*hyper((-p, 4/3), (7/3,), -b*x**3/a)/
4 + e*x**5*(1 + b*x**3/a)**(-p)*(a + b*x**3)**p*hyper((-p, 5/3), (8/3,), -b*x**3
/a)/5 + c*(a + b*x**3)**(p + 1)/(3*b*(p + 1))

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Mathematica [A]  time = 0.149455, size = 131, normalized size = 1.22 \[ \frac{\left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \left (20 c \left (b x^3 \left (\frac{b x^3}{a}+1\right )^p+a \left (\left (\frac{b x^3}{a}+1\right )^p-1\right )\right )+15 b d (p+1) x^4 \, _2F_1\left (\frac{4}{3},-p;\frac{7}{3};-\frac{b x^3}{a}\right )+12 b e (p+1) x^5 \, _2F_1\left (\frac{5}{3},-p;\frac{8}{3};-\frac{b x^3}{a}\right )\right )}{60 b (p+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(c + d*x + e*x^2)*(a + b*x^3)^p,x]

[Out]

((a + b*x^3)^p*(20*c*(b*x^3*(1 + (b*x^3)/a)^p + a*(-1 + (1 + (b*x^3)/a)^p)) + 15
*b*d*(1 + p)*x^4*Hypergeometric2F1[4/3, -p, 7/3, -((b*x^3)/a)] + 12*b*e*(1 + p)*
x^5*Hypergeometric2F1[5/3, -p, 8/3, -((b*x^3)/a)]))/(60*b*(1 + p)*(1 + (b*x^3)/a
)^p)

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( e{x}^{2}+dx+c \right ) \left ( b{x}^{3}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(e*x^2+d*x+c)*(b*x^3+a)^p,x)

[Out]

int(x^2*(e*x^2+d*x+c)*(b*x^3+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((e*x^2 + d*x + c)*(b*x^3 + a)^p*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{4} + d x^{3} + c x^{2}\right )}{\left (b x^{3} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d*x + c)*(b*x^3 + a)^p*x^2,x, algorithm="fricas")

[Out]

integral((e*x^4 + d*x^3 + c*x^2)*(b*x^3 + a)^p, x)

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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**2*(e*x**2+d*x+c)*(b*x**3+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{p} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d*x + c)*(b*x^3 + a)^p*x^2,x, algorithm="giac")

[Out]

integrate((e*x^2 + d*x + c)*(b*x^3 + a)^p*x^2, x)