3.235 \(\int x^2 \left (a+b x+c x^2+d x^3\right )^p \left (3 a+b (4+p) x+c (5+2 p) x^2+d (6+3 p) x^3\right ) \, dx\)

Optimal. Leaf size=23 \[ x^3 \left (a+b x+c x^2+d x^3\right )^{p+1} \]

[Out]

x^3*(a + b*x + c*x^2 + d*x^3)^(1 + p)

_______________________________________________________________________________________

Rubi [A]  time = 0.0232666, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.02 \[ x^3 \left (a+b x+c x^2+d x^3\right )^{p+1} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x + c*x^2 + d*x^3)^p*(3*a + b*(4 + p)*x + c*(5 + 2*p)*x^2 + d*(6 + 3*p)*x^3),x]

[Out]

x^3*(a + b*x + c*x^2 + d*x^3)^(1 + p)

_______________________________________________________________________________________

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(x**2*(d*x**3+c*x**2+b*x+a)**p*(3*a+b*(4+p)*x+c*(5+2*p)*x**2+d*(6+3*p)*x**3),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.063995, size = 21, normalized size = 0.91 \[ x^3 (a+x (b+x (c+d x)))^{p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x + c*x^2 + d*x^3)^p*(3*a + b*(4 + p)*x + c*(5 + 2*p)*x^2 + d*(6 + 3*p)*x^3),x]

[Out]

x^3*(a + x*(b + x*(c + d*x)))^(1 + p)

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 24, normalized size = 1. \[{x}^{3} \left ( d{x}^{3}+c{x}^{2}+bx+a \right ) ^{1+p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(d*x^3+c*x^2+b*x+a)^p*(3*a+b*(4+p)*x+c*(5+2*p)*x^2+d*(6+3*p)*x^3),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.939416, size = 53, normalized size = 2.3 \[{\left (d x^{6} + c x^{5} + b x^{4} + a x^{3}\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*d*(p + 2)*x^3 + c*(2*p + 5)*x^2 + b*(p + 4)*x + 3*a)*(d*x^3 + c*x^2 + b*x + a)^p*x^2,x, algorithm="maxima")

[Out]

(d*x^6 + c*x^5 + b*x^4 + a*x^3)*(d*x^3 + c*x^2 + b*x + a)^p

_______________________________________________________________________________________

Fricas [A]  time = 0.293899, size = 53, normalized size = 2.3 \[{\left (d x^{6} + c x^{5} + b x^{4} + a x^{3}\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*d*(p + 2)*x^3 + c*(2*p + 5)*x^2 + b*(p + 4)*x + 3*a)*(d*x^3 + c*x^2 + b*x + a)^p*x^2,x, algorithm="fricas")

[Out]

(d*x^6 + c*x^5 + b*x^4 + a*x^3)*(d*x^3 + c*x^2 + b*x + a)^p

_______________________________________________________________________________________

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*(d*x**3+c*x**2+b*x+a)**p*(3*a+b*(4+p)*x+c*(5+2*p)*x**2+d*(6+3*p)*x**3),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.469378, size = 131, normalized size = 5.7 \[ d x^{6} e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} + c x^{5} e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} + b x^{4} e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} + a x^{3} e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*d*(p + 2)*x^3 + c*(2*p + 5)*x^2 + b*(p + 4)*x + 3*a)*(d*x^3 + c*x^2 + b*x + a)^p*x^2,x, algorithm="giac")

[Out]

d*x^6*e^(p*ln(d*x^3 + c*x^2 + b*x + a)) + c*x^5*e^(p*ln(d*x^3 + c*x^2 + b*x + a)
) + b*x^4*e^(p*ln(d*x^3 + c*x^2 + b*x + a)) + a*x^3*e^(p*ln(d*x^3 + c*x^2 + b*x
+ a))