3.2660 \(\int x^m \left (a+b x^n\right )^3 \, dx\)

Optimal. Leaf size=75 \[ \frac{a^3 x^{m+1}}{m+1}+\frac{3 a^2 b x^{m+n+1}}{m+n+1}+\frac{3 a b^2 x^{m+2 n+1}}{m+2 n+1}+\frac{b^3 x^{m+3 n+1}}{m+3 n+1} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0886964, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^3 x^{m+1}}{m+1}+\frac{3 a^2 b x^{m+n+1}}{m+n+1}+\frac{3 a b^2 x^{m+2 n+1}}{m+2 n+1}+\frac{b^3 x^{m+3 n+1}}{m+3 n+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^n)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.7855, size = 70, normalized size = 0.93 \[ \frac{a^{3} x^{m + 1}}{m + 1} + \frac{3 a^{2} b x^{m + n + 1}}{m + n + 1} + \frac{3 a b^{2} x^{m + 2 n + 1}}{m + 2 n + 1} + \frac{b^{3} x^{m + 3 n + 1}}{m + 3 n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(a+b*x**n)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0673276, size = 67, normalized size = 0.89 \[ x^{m+1} \left (\frac{a^3}{m+1}+\frac{3 a^2 b x^n}{m+n+1}+\frac{3 a b^2 x^{2 n}}{m+2 n+1}+\frac{b^3 x^{3 n}}{m+3 n+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^n)^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.027, size = 92, normalized size = 1.2 \[{\frac{{a}^{3}x{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}+{\frac{{b}^{3}x{{\rm e}^{m\ln \left ( x \right ) }} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+m+3\,n}}+3\,{\frac{a{b}^{2}x{{\rm e}^{m\ln \left ( x \right ) }} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+m+2\,n}}+3\,{\frac{{a}^{2}bx{{\rm e}^{m\ln \left ( x \right ) }}{{\rm e}^{n\ln \left ( x \right ) }}}{1+m+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^n)^3,x)

[Out]

a^3/(1+m)*x*exp(m*ln(x))+b^3/(1+m+3*n)*x*exp(m*ln(x))*exp(n*ln(x))^3+3*a*b^2/(1+
m+2*n)*x*exp(m*ln(x))*exp(n*ln(x))^2+3*a^2*b/(1+m+n)*x*exp(m*ln(x))*exp(n*ln(x))

_______________________________________________________________________________________

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((b*x^n + a)^3*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.243464, size = 489, normalized size = 6.52 \[ \frac{{\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3} + 2 \,{\left (b^{3} m + b^{3}\right )} n^{2} + 3 \,{\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n\right )} x x^{m} x^{3 \, n} + 3 \,{\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2} + 3 \,{\left (a b^{2} m + a b^{2}\right )} n^{2} + 4 \,{\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n\right )} x x^{m} x^{2 \, n} + 3 \,{\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b + 6 \,{\left (a^{2} b m + a^{2} b\right )} n^{2} + 5 \,{\left (a^{2} b m^{2} + 2 \, a^{2} b m + a^{2} b\right )} n\right )} x x^{m} x^{n} +{\left (a^{3} m^{3} + 6 \, a^{3} n^{3} + 3 \, a^{3} m^{2} + 3 \, a^{3} m + a^{3} + 11 \,{\left (a^{3} m + a^{3}\right )} n^{2} + 6 \,{\left (a^{3} m^{2} + 2 \, a^{3} m + a^{3}\right )} n\right )} x x^{m}}{m^{4} + 6 \,{\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \,{\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \,{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b^3*m^3 + 3*b^3*m^2 + 3*b^3*m + b^3 + 2*(b^3*m + b^3)*n^2 + 3*(b^3*m^2 + 2*b^3
*m + b^3)*n)*x*x^m*x^(3*n) + 3*(a*b^2*m^3 + 3*a*b^2*m^2 + 3*a*b^2*m + a*b^2 + 3*
(a*b^2*m + a*b^2)*n^2 + 4*(a*b^2*m^2 + 2*a*b^2*m + a*b^2)*n)*x*x^m*x^(2*n) + 3*(
a^2*b*m^3 + 3*a^2*b*m^2 + 3*a^2*b*m + a^2*b + 6*(a^2*b*m + a^2*b)*n^2 + 5*(a^2*b
*m^2 + 2*a^2*b*m + a^2*b)*n)*x*x^m*x^n + (a^3*m^3 + 6*a^3*n^3 + 3*a^3*m^2 + 3*a^
3*m + a^3 + 11*(a^3*m + a^3)*n^2 + 6*(a^3*m^2 + 2*a^3*m + a^3)*n)*x*x^m)/(m^4 +
6*(m + 1)*n^3 + 4*m^3 + 11*(m^2 + 2*m + 1)*n^2 + 6*m^2 + 6*(m^3 + 3*m^2 + 3*m +
1)*n + 4*m + 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(x**m*(a+b*x**n)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234546, size = 988, normalized size = 13.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^3*m^3*x*e^(m*ln(x) + 3*n*ln(x)) + 3*b^3*m^2*n*x*e^(m*ln(x) + 3*n*ln(x)) + 2*b
^3*m*n^2*x*e^(m*ln(x) + 3*n*ln(x)) + 3*a*b^2*m^3*x*e^(m*ln(x) + 2*n*ln(x)) + 12*
a*b^2*m^2*n*x*e^(m*ln(x) + 2*n*ln(x)) + 9*a*b^2*m*n^2*x*e^(m*ln(x) + 2*n*ln(x))
+ 3*a^2*b*m^3*x*e^(m*ln(x) + n*ln(x)) + 15*a^2*b*m^2*n*x*e^(m*ln(x) + n*ln(x)) +
18*a^2*b*m*n^2*x*e^(m*ln(x) + n*ln(x)) + a^3*m^3*x*e^(m*ln(x)) + 6*a^3*m^2*n*x*
e^(m*ln(x)) + 11*a^3*m*n^2*x*e^(m*ln(x)) + 6*a^3*n^3*x*e^(m*ln(x)) + 3*b^3*m^2*x
*e^(m*ln(x) + 3*n*ln(x)) + 6*b^3*m*n*x*e^(m*ln(x) + 3*n*ln(x)) + 2*b^3*n^2*x*e^(
m*ln(x) + 3*n*ln(x)) + 9*a*b^2*m^2*x*e^(m*ln(x) + 2*n*ln(x)) + 24*a*b^2*m*n*x*e^
(m*ln(x) + 2*n*ln(x)) + 9*a*b^2*n^2*x*e^(m*ln(x) + 2*n*ln(x)) + 9*a^2*b*m^2*x*e^
(m*ln(x) + n*ln(x)) + 30*a^2*b*m*n*x*e^(m*ln(x) + n*ln(x)) + 18*a^2*b*n^2*x*e^(m
*ln(x) + n*ln(x)) + 3*a^3*m^2*x*e^(m*ln(x)) + 12*a^3*m*n*x*e^(m*ln(x)) + 11*a^3*
n^2*x*e^(m*ln(x)) + 3*b^3*m*x*e^(m*ln(x) + 3*n*ln(x)) + 3*b^3*n*x*e^(m*ln(x) + 3
*n*ln(x)) + 9*a*b^2*m*x*e^(m*ln(x) + 2*n*ln(x)) + 12*a*b^2*n*x*e^(m*ln(x) + 2*n*
ln(x)) + 9*a^2*b*m*x*e^(m*ln(x) + n*ln(x)) + 15*a^2*b*n*x*e^(m*ln(x) + n*ln(x))
+ 3*a^3*m*x*e^(m*ln(x)) + 6*a^3*n*x*e^(m*ln(x)) + b^3*x*e^(m*ln(x) + 3*n*ln(x))
+ 3*a*b^2*x*e^(m*ln(x) + 2*n*ln(x)) + 3*a^2*b*x*e^(m*ln(x) + n*ln(x)) + a^3*x*e^
(m*ln(x)))/(m^4 + 6*m^3*n + 11*m^2*n^2 + 6*m*n^3 + 4*m^3 + 18*m^2*n + 22*m*n^2 +
6*n^3 + 6*m^2 + 18*m*n + 11*n^2 + 4*m + 6*n + 1)