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

Optimal. Leaf size=102 \[ \frac{c x \left (a+b x^3\right )^{p+1} \, _2F_1\left (1,p+\frac{4}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{a}+\frac{d x^2 \left (a+b x^3\right )^{p+1} \, _2F_1\left (1,p+\frac{5}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a}+\frac{e \left (a+b x^3\right )^{p+1}}{3 b (p+1)} \]

[Out]

(e*(a + b*x^3)^(1 + p))/(3*b*(1 + p)) + (c*x*(a + b*x^3)^(1 + p)*Hypergeometric2
F1[1, 4/3 + p, 4/3, -((b*x^3)/a)])/a + (d*x^2*(a + b*x^3)^(1 + p)*Hypergeometric
2F1[1, 5/3 + p, 5/3, -((b*x^3)/a)])/(2*a)

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Rubi [A]  time = 0.160852, antiderivative size = 120, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ c x \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{3},-p;\frac{4}{3};-\frac{b x^3}{a}\right )+\frac{1}{2} d x^2 \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{2}{3},-p;\frac{5}{3};-\frac{b x^3}{a}\right )+\frac{e \left (a+b x^3\right )^{p+1}}{3 b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(e*(a + b*x^3)^(1 + p))/(3*b*(1 + p)) + (c*x*(a + b*x^3)^p*Hypergeometric2F1[1/3
, -p, 4/3, -((b*x^3)/a)])/(1 + (b*x^3)/a)^p + (d*x^2*(a + b*x^3)^p*Hypergeometri
c2F1[2/3, -p, 5/3, -((b*x^3)/a)])/(2*(1 + (b*x^3)/a)^p)

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Rubi in Sympy [A]  time = 23.0659, size = 95, normalized size = 0.93 \[ c x \left (1 + \frac{b x^{3}}{a}\right )^{- p} \left (a + b x^{3}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )} + \frac{d x^{2} \left (1 + \frac{b x^{3}}{a}\right )^{- p} \left (a + b x^{3}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{2} + \frac{e \left (a + b x^{3}\right )^{p + 1}}{3 b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x*(1 + b*x**3/a)**(-p)*(a + b*x**3)**p*hyper((-p, 1/3), (4/3,), -b*x**3/a) + d
*x**2*(1 + b*x**3/a)**(-p)*(a + b*x**3)**p*hyper((-p, 2/3), (5/3,), -b*x**3/a)/2
+ e*(a + b*x**3)**(p + 1)/(3*b*(p + 1))

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Mathematica [A]  time = 0.156256, size = 129, normalized size = 1.26 \[ \frac{\left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \left (6 b c (p+1) x \, _2F_1\left (\frac{1}{3},-p;\frac{4}{3};-\frac{b x^3}{a}\right )+3 b d (p+1) x^2 \, _2F_1\left (\frac{2}{3},-p;\frac{5}{3};-\frac{b x^3}{a}\right )+2 e \left (b x^3 \left (\frac{b x^3}{a}+1\right )^p+a \left (\left (\frac{b x^3}{a}+1\right )^p-1\right )\right )\right )}{6 b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^3)^p*(2*e*(b*x^3*(1 + (b*x^3)/a)^p + a*(-1 + (1 + (b*x^3)/a)^p)) + 6*b
*c*(1 + p)*x*Hypergeometric2F1[1/3, -p, 4/3, -((b*x^3)/a)] + 3*b*d*(1 + p)*x^2*H
ypergeometric2F1[2/3, -p, 5/3, -((b*x^3)/a)]))/(6*b*(1 + p)*(1 + (b*x^3)/a)^p)

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int \left ( e{x}^{2}+dx+c \right ) \left ( b{x}^{3}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((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. \[ \int{\left (e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x^2 + d*x + c)*(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((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. \[ \int{\left (e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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