3.180 \(\int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^n \, dx\)

Optimal. Leaf size=25 \[ \frac{\left (a+b x+c x^2+d x^3\right )^{n+1}}{n+1} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.01828, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ \frac{\left (a+b x+c x^2+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.83432, size = 20, normalized size = 0.8 \[ \frac{\left (a + b x + c x^{2} + d x^{3}\right )^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x + c*x**2 + d*x**3)**(n + 1)/(n + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0489968, size = 23, normalized size = 0.92 \[ \frac{(a+x (b+x (c+d x)))^{n+1}}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

(a + x*(b + x*(c + d*x)))^(1 + n)/(1 + n)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 26, normalized size = 1. \[{\frac{ \left ( d{x}^{3}+c{x}^{2}+bx+a \right ) ^{1+n}}{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.29664, size = 51, normalized size = 2.04 \[ \frac{{\left (d x^{3} + c x^{2} + b x + a\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.289081, size = 132, normalized size = 5.28 \[ \frac{d x^{3} e^{\left (n{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} + c x^{2} e^{\left (n{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} + b x e^{\left (n{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )} + a e^{\left (n{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x^3*e^(n*ln(d*x^3 + c*x^2 + b*x + a)) + c*x^2*e^(n*ln(d*x^3 + c*x^2 + b*x + a
)) + b*x*e^(n*ln(d*x^3 + c*x^2 + b*x + a)) + a*e^(n*ln(d*x^3 + c*x^2 + b*x + a))
)/(n + 1)