3.343 \(\int x^m (a+b x)^2 (A+B x) \, dx\)

Optimal. Leaf size=71 \[ \frac{a^2 A x^{m+1}}{m+1}+\frac{a x^{m+2} (a B+2 A b)}{m+2}+\frac{b x^{m+3} (2 a B+A b)}{m+3}+\frac{b^2 B x^{m+4}}{m+4} \]

[Out]

(a^2*A*x^(1 + m))/(1 + m) + (a*(2*A*b + a*B)*x^(2 + m))/(2 + m) + (b*(A*b + 2*a*
B)*x^(3 + m))/(3 + m) + (b^2*B*x^(4 + m))/(4 + m)

_______________________________________________________________________________________

Rubi [A]  time = 0.0981001, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{a^2 A x^{m+1}}{m+1}+\frac{a x^{m+2} (a B+2 A b)}{m+2}+\frac{b x^{m+3} (2 a B+A b)}{m+3}+\frac{b^2 B x^{m+4}}{m+4} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x)^2*(A + B*x),x]

[Out]

(a^2*A*x^(1 + m))/(1 + m) + (a*(2*A*b + a*B)*x^(2 + m))/(2 + m) + (b*(A*b + 2*a*
B)*x^(3 + m))/(3 + m) + (b^2*B*x^(4 + m))/(4 + m)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.8523, size = 63, normalized size = 0.89 \[ \frac{A a^{2} x^{m + 1}}{m + 1} + \frac{B b^{2} x^{m + 4}}{m + 4} + \frac{a x^{m + 2} \left (2 A b + B a\right )}{m + 2} + \frac{b x^{m + 3} \left (A b + 2 B a\right )}{m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x+a)**2*(B*x+A),x)

[Out]

A*a**2*x**(m + 1)/(m + 1) + B*b**2*x**(m + 4)/(m + 4) + a*x**(m + 2)*(2*A*b + B*
a)/(m + 2) + b*x**(m + 3)*(A*b + 2*B*a)/(m + 3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0736252, size = 65, normalized size = 0.92 \[ x^m \left (\frac{a^2 A x}{m+1}+\frac{b x^3 (2 a B+A b)}{m+3}+\frac{a x^2 (a B+2 A b)}{m+2}+\frac{b^2 B x^4}{m+4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x)^2*(A + B*x),x]

[Out]

x^m*((a^2*A*x)/(1 + m) + (a*(2*A*b + a*B)*x^2)/(2 + m) + (b*(A*b + 2*a*B)*x^3)/(
3 + m) + (b^2*B*x^4)/(4 + m))

_______________________________________________________________________________________

Maple [B]  time = 0.007, size = 246, normalized size = 3.5 \[{\frac{{x}^{1+m} \left ( B{b}^{2}{m}^{3}{x}^{3}+A{b}^{2}{m}^{3}{x}^{2}+2\,Bab{m}^{3}{x}^{2}+6\,B{b}^{2}{m}^{2}{x}^{3}+2\,Aab{m}^{3}x+7\,A{b}^{2}{m}^{2}{x}^{2}+B{a}^{2}{m}^{3}x+14\,Bab{m}^{2}{x}^{2}+11\,B{b}^{2}m{x}^{3}+A{a}^{2}{m}^{3}+16\,Aab{m}^{2}x+14\,A{b}^{2}m{x}^{2}+8\,B{a}^{2}{m}^{2}x+28\,Babm{x}^{2}+6\,B{b}^{2}{x}^{3}+9\,A{a}^{2}{m}^{2}+38\,Aabmx+8\,A{b}^{2}{x}^{2}+19\,B{a}^{2}mx+16\,B{x}^{2}ab+26\,A{a}^{2}m+24\,aAbx+12\,{a}^{2}Bx+24\,{a}^{2}A \right ) }{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x+a)^2*(B*x+A),x)

[Out]

x^(1+m)*(B*b^2*m^3*x^3+A*b^2*m^3*x^2+2*B*a*b*m^3*x^2+6*B*b^2*m^2*x^3+2*A*a*b*m^3
*x+7*A*b^2*m^2*x^2+B*a^2*m^3*x+14*B*a*b*m^2*x^2+11*B*b^2*m*x^3+A*a^2*m^3+16*A*a*
b*m^2*x+14*A*b^2*m*x^2+8*B*a^2*m^2*x+28*B*a*b*m*x^2+6*B*b^2*x^3+9*A*a^2*m^2+38*A
*a*b*m*x+8*A*b^2*x^2+19*B*a^2*m*x+16*B*a*b*x^2+26*A*a^2*m+24*A*a*b*x+12*B*a^2*x+
24*A*a^2)/(4+m)/(3+m)/(2+m)/(1+m)

_______________________________________________________________________________________

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 + A)*(b*x + a)^2*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.221886, size = 290, normalized size = 4.08 \[ \frac{{\left ({\left (B b^{2} m^{3} + 6 \, B b^{2} m^{2} + 11 \, B b^{2} m + 6 \, B b^{2}\right )} x^{4} +{\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 16 \, B a b + 8 \, A b^{2} + 7 \,{\left (2 \, B a b + A b^{2}\right )} m^{2} + 14 \,{\left (2 \, B a b + A b^{2}\right )} m\right )} x^{3} +{\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 12 \, B a^{2} + 24 \, A a b + 8 \,{\left (B a^{2} + 2 \, A a b\right )} m^{2} + 19 \,{\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{2} +{\left (A a^{2} m^{3} + 9 \, A a^{2} m^{2} + 26 \, A a^{2} m + 24 \, A a^{2}\right )} x\right )} x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2*x^m,x, algorithm="fricas")

[Out]

((B*b^2*m^3 + 6*B*b^2*m^2 + 11*B*b^2*m + 6*B*b^2)*x^4 + ((2*B*a*b + A*b^2)*m^3 +
16*B*a*b + 8*A*b^2 + 7*(2*B*a*b + A*b^2)*m^2 + 14*(2*B*a*b + A*b^2)*m)*x^3 + ((
B*a^2 + 2*A*a*b)*m^3 + 12*B*a^2 + 24*A*a*b + 8*(B*a^2 + 2*A*a*b)*m^2 + 19*(B*a^2
+ 2*A*a*b)*m)*x^2 + (A*a^2*m^3 + 9*A*a^2*m^2 + 26*A*a^2*m + 24*A*a^2)*x)*x^m/(m
^4 + 10*m^3 + 35*m^2 + 50*m + 24)

_______________________________________________________________________________________

Sympy [A]  time = 3.04478, size = 1020, normalized size = 14.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x+a)**2*(B*x+A),x)

[Out]

Piecewise((-A*a**2/(3*x**3) - A*a*b/x**2 - A*b**2/x - B*a**2/(2*x**2) - 2*B*a*b/
x + B*b**2*log(x), Eq(m, -4)), (-A*a**2/(2*x**2) - 2*A*a*b/x + A*b**2*log(x) - B
*a**2/x + 2*B*a*b*log(x) + B*b**2*x, Eq(m, -3)), (-A*a**2/x + 2*A*a*b*log(x) + A
*b**2*x + B*a**2*log(x) + 2*B*a*b*x + B*b**2*x**2/2, Eq(m, -2)), (A*a**2*log(x)
+ 2*A*a*b*x + A*b**2*x**2/2 + B*a**2*x + B*a*b*x**2 + B*b**2*x**3/3, Eq(m, -1)),
(A*a**2*m**3*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 9*A*a**2*m**2*x*x*
*m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 26*A*a**2*m*x*x**m/(m**4 + 10*m**3 +
35*m**2 + 50*m + 24) + 24*A*a**2*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24)
+ 2*A*a*b*m**3*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 16*A*a*b*m**2*
x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 38*A*a*b*m*x**2*x**m/(m**4 +
10*m**3 + 35*m**2 + 50*m + 24) + 24*A*a*b*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 +
50*m + 24) + A*b**2*m**3*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 7*A*
b**2*m**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 14*A*b**2*m*x**3*x*
*m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8*A*b**2*x**3*x**m/(m**4 + 10*m**3 +
35*m**2 + 50*m + 24) + B*a**2*m**3*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m +
24) + 8*B*a**2*m**2*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 19*B*a**
2*m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 12*B*a**2*x**2*x**m/(m**4
+ 10*m**3 + 35*m**2 + 50*m + 24) + 2*B*a*b*m**3*x**3*x**m/(m**4 + 10*m**3 + 35*
m**2 + 50*m + 24) + 14*B*a*b*m**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 2
4) + 28*B*a*b*m*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 16*B*a*b*x**3
*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + B*b**2*m**3*x**4*x**m/(m**4 + 10*
m**3 + 35*m**2 + 50*m + 24) + 6*B*b**2*m**2*x**4*x**m/(m**4 + 10*m**3 + 35*m**2
+ 50*m + 24) + 11*B*b**2*m*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 6*
B*b**2*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216361, size = 513, normalized size = 7.23 \[ \frac{B b^{2} m^{3} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, B a b m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + A b^{2} m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, B b^{2} m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + B a^{2} m^{3} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, A a b m^{3} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 14 \, B a b m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 7 \, A b^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 11 \, B b^{2} m x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + A a^{2} m^{3} x e^{\left (m{\rm ln}\left (x\right )\right )} + 8 \, B a^{2} m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 16 \, A a b m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 28 \, B a b m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 14 \, A b^{2} m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, B b^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 9 \, A a^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 19 \, B a^{2} m x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 38 \, A a b m x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 16 \, B a b x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 8 \, A b^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 26 \, A a^{2} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 12 \, B a^{2} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 24 \, A a b x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 24 \, A a^{2} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2*x^m,x, algorithm="giac")

[Out]

(B*b^2*m^3*x^4*e^(m*ln(x)) + 2*B*a*b*m^3*x^3*e^(m*ln(x)) + A*b^2*m^3*x^3*e^(m*ln
(x)) + 6*B*b^2*m^2*x^4*e^(m*ln(x)) + B*a^2*m^3*x^2*e^(m*ln(x)) + 2*A*a*b*m^3*x^2
*e^(m*ln(x)) + 14*B*a*b*m^2*x^3*e^(m*ln(x)) + 7*A*b^2*m^2*x^3*e^(m*ln(x)) + 11*B
*b^2*m*x^4*e^(m*ln(x)) + A*a^2*m^3*x*e^(m*ln(x)) + 8*B*a^2*m^2*x^2*e^(m*ln(x)) +
16*A*a*b*m^2*x^2*e^(m*ln(x)) + 28*B*a*b*m*x^3*e^(m*ln(x)) + 14*A*b^2*m*x^3*e^(m
*ln(x)) + 6*B*b^2*x^4*e^(m*ln(x)) + 9*A*a^2*m^2*x*e^(m*ln(x)) + 19*B*a^2*m*x^2*e
^(m*ln(x)) + 38*A*a*b*m*x^2*e^(m*ln(x)) + 16*B*a*b*x^3*e^(m*ln(x)) + 8*A*b^2*x^3
*e^(m*ln(x)) + 26*A*a^2*m*x*e^(m*ln(x)) + 12*B*a^2*x^2*e^(m*ln(x)) + 24*A*a*b*x^
2*e^(m*ln(x)) + 24*A*a^2*x*e^(m*ln(x)))/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)