3.237 \(\int \left (a+b x+c x^2+d x^3\right )^p \left (a+b (2+p) x+c (3+2 p) x^2+d (4+3 p) x^3\right ) \, dx\)

Optimal. Leaf size=21 \[ x \left (a+b x+c x^2+d x^3\right )^{p+1} \]

[Out]

x*(a + b*x + c*x^2 + d*x^3)^(1 + p)

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Rubi [A]  time = 0.0163447, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022 \[ x \left (a+b x+c x^2+d x^3\right )^{p+1} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2 + d*x^3)^p*(a + b*(2 + p)*x + c*(3 + 2*p)*x^2 + d*(4 + 3*p)*x^3),x]

[Out]

x*(a + b*x + c*x^2 + d*x^3)^(1 + p)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+c*x**2+b*x+a)**p*(a+b*(2+p)*x+c*(3+2*p)*x**2+d*(4+3*p)*x**3),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0527342, size = 19, normalized size = 0.9 \[ x (a+x (b+x (c+d x)))^{p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2 + d*x^3)^p*(a + b*(2 + p)*x + c*(3 + 2*p)*x^2 + d*(4 + 3*p)*x^3),x]

[Out]

x*(a + x*(b + x*(c + d*x)))^(1 + p)

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Maple [A]  time = 0.009, size = 22, normalized size = 1.1 \[ x \left ( d{x}^{3}+c{x}^{2}+bx+a \right ) ^{1+p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c*x^2+b*x+a)^p*(a+b*(2+p)*x+c*(3+2*p)*x^2+d*(4+3*p)*x^3),x)

[Out]

x*(d*x^3+c*x^2+b*x+a)^(1+p)

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Maxima [A]  time = 0.89127, size = 50, normalized size = 2.38 \[{\left (d x^{4} + c x^{3} + b x^{2} + a x\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*(3*p + 4)*x^3 + c*(2*p + 3)*x^2 + b*(p + 2)*x + a)*(d*x^3 + c*x^2 + b*x + a)^p,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.290751, size = 50, normalized size = 2.38 \[{\left (d x^{4} + c x^{3} + b x^{2} + a x\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*(3*p + 4)*x^3 + c*(2*p + 3)*x^2 + b*(p + 2)*x + a)*(d*x^3 + c*x^2 + b*x + a)^p,x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.320907, size = 128, normalized size = 6.1 \[ d x^{4} e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} + c x^{3} e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} + b x^{2} e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} + a x e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*(3*p + 4)*x^3 + c*(2*p + 3)*x^2 + b*(p + 2)*x + a)*(d*x^3 + c*x^2 + b*x + a)^p,x, algorithm="giac")

[Out]

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