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

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

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Rubi [A]  time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {27, 76} \[ \frac {a^2 A x^{m+1}}{m+1}+\frac {a x^{m+2} (a B+2 A b)}{m+2}+\frac {b x^{m+3} (2 a B+A b)}{m+3}+\frac {b^2 B x^{m+4}}{m+4} \]

Antiderivative was successfully verified.

[In]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin {align*} \int x^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int x^m (a+b x)^2 (A+B x) \, dx\\ &=\int \left (a^2 A x^m+a (2 A b+a B) x^{1+m}+b (A b+2 a B) x^{2+m}+b^2 B x^{3+m}\right ) \, dx\\ &=\frac {a^2 A x^{1+m}}{1+m}+\frac {a (2 A b+a B) x^{2+m}}{2+m}+\frac {b (A b+2 a B) x^{3+m}}{3+m}+\frac {b^2 B x^{4+m}}{4+m}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 71, normalized size = 1.00 \[ \frac {x^{m+1} \left (\left (\frac {a^2}{m+1}+\frac {2 a b x}{m+2}+\frac {b^2 x^2}{m+3}\right ) (A b (m+4)-a B (m+1))+B (a+b x)^3\right )}{b (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.17, size = 215, normalized size = 3.03 \[ \frac {{\left ({\left (B b^{2} m^{3} + 6 \, B b^{2} m^{2} + 11 \, B b^{2} m + 6 \, B b^{2}\right )} x^{4} + {\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 16 \, B a b + 8 \, A b^{2} + 7 \, {\left (2 \, B a b + A b^{2}\right )} m^{2} + 14 \, {\left (2 \, B a b + A b^{2}\right )} m\right )} x^{3} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 12 \, B a^{2} + 24 \, A a b + 8 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 19 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{2} + {\left (A a^{2} m^{3} + 9 \, A a^{2} m^{2} + 26 \, A a^{2} m + 24 \, A a^{2}\right )} x\right )} x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2),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)

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giac [B]  time = 0.18, size = 332, normalized size = 4.68 \[ \frac {B b^{2} m^{3} x^{4} x^{m} + 2 \, B a b m^{3} x^{3} x^{m} + A b^{2} m^{3} x^{3} x^{m} + 6 \, B b^{2} m^{2} x^{4} x^{m} + B a^{2} m^{3} x^{2} x^{m} + 2 \, A a b m^{3} x^{2} x^{m} + 14 \, B a b m^{2} x^{3} x^{m} + 7 \, A b^{2} m^{2} x^{3} x^{m} + 11 \, B b^{2} m x^{4} x^{m} + A a^{2} m^{3} x x^{m} + 8 \, B a^{2} m^{2} x^{2} x^{m} + 16 \, A a b m^{2} x^{2} x^{m} + 28 \, B a b m x^{3} x^{m} + 14 \, A b^{2} m x^{3} x^{m} + 6 \, B b^{2} x^{4} x^{m} + 9 \, A a^{2} m^{2} x x^{m} + 19 \, B a^{2} m x^{2} x^{m} + 38 \, A a b m x^{2} x^{m} + 16 \, B a b x^{3} x^{m} + 8 \, A b^{2} x^{3} x^{m} + 26 \, A a^{2} m x x^{m} + 12 \, B a^{2} x^{2} x^{m} + 24 \, A a b x^{2} x^{m} + 24 \, A a^{2} x x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.06, size = 246, normalized size = 3.46 \[ \frac {\left (B \,b^{2} m^{3} x^{3}+A \,b^{2} m^{3} x^{2}+2 B a b \,m^{3} x^{2}+6 B \,b^{2} m^{2} x^{3}+2 A a b \,m^{3} x +7 A \,b^{2} m^{2} x^{2}+B \,a^{2} m^{3} x +14 B a b \,m^{2} x^{2}+11 B \,b^{2} m \,x^{3}+A \,a^{2} m^{3}+16 A a b \,m^{2} x +14 A \,b^{2} m \,x^{2}+8 B \,a^{2} m^{2} x +28 B a b m \,x^{2}+6 B \,b^{2} x^{3}+9 A \,a^{2} m^{2}+38 A a b m x +8 A \,b^{2} x^{2}+19 B \,a^{2} m x +16 B a b \,x^{2}+26 A \,a^{2} m +24 A a b x +12 B \,a^{2} x +24 A \,a^{2}\right ) x^{m +1}}{\left (m +4\right ) \left (m +3\right ) \left (m +2\right ) \left (m +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

x^(m+1)*(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)/
(m+4)/(m+3)/(m+2)/(m+1)

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maxima [A]  time = 0.46, size = 91, normalized size = 1.28 \[ \frac {B b^{2} x^{m + 4}}{m + 4} + \frac {2 \, B a b x^{m + 3}}{m + 3} + \frac {A b^{2} x^{m + 3}}{m + 3} + \frac {B a^{2} x^{m + 2}}{m + 2} + \frac {2 \, A a b x^{m + 2}}{m + 2} + \frac {A a^{2} x^{m + 1}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.29, size = 177, normalized size = 2.49 \[ x^m\,\left (\frac {B\,b^2\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {A\,a^2\,x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {a\,x^2\,\left (2\,A\,b+B\,a\right )\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {b\,x^3\,\left (A\,b+2\,B\,a\right )\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

x^m*((B*b^2*x^4*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (A*a^2*x*(26*m + 9*m^2 + m^3 +
24))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (a*x^2*(2*A*b + B*a)*(19*m + 8*m^2 + m^3 + 12))/(50*m + 35*m^2 + 1
0*m^3 + m^4 + 24) + (b*x^3*(A*b + 2*B*a)*(14*m + 7*m^2 + m^3 + 8))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24))

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sympy [A]  time = 1.13, size = 1020, normalized size = 14.37 \[ \begin {cases} - \frac {A a^{2}}{3 x^{3}} - \frac {A a b}{x^{2}} - \frac {A b^{2}}{x} - \frac {B a^{2}}{2 x^{2}} - \frac {2 B a b}{x} + B b^{2} \log {\relax (x )} & \text {for}\: m = -4 \\- \frac {A a^{2}}{2 x^{2}} - \frac {2 A a b}{x} + A b^{2} \log {\relax (x )} - \frac {B a^{2}}{x} + 2 B a b \log {\relax (x )} + B b^{2} x & \text {for}\: m = -3 \\- \frac {A a^{2}}{x} + 2 A a b \log {\relax (x )} + A b^{2} x + B a^{2} \log {\relax (x )} + 2 B a b x + \frac {B b^{2} x^{2}}{2} & \text {for}\: m = -2 \\A a^{2} \log {\relax (x )} + 2 A a b x + \frac {A b^{2} x^{2}}{2} + B a^{2} x + B a b x^{2} + \frac {B b^{2} x^{3}}{3} & \text {for}\: m = -1 \\\frac {A a^{2} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {9 A a^{2} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {26 A a^{2} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 A a^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {2 A a b m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {16 A a b m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {38 A a b m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 A a b x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {A b^{2} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {7 A b^{2} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {14 A b^{2} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 A b^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {B a^{2} m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 B a^{2} m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {19 B a^{2} m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {12 B a^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {2 B a b m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {14 B a b m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {28 B a b m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {16 B a b x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {B b^{2} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 B b^{2} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {11 B b^{2} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 B b^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(B*x+A)*(b**2*x**2+2*a*b*x+a**2),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 + 3
5*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 + 24) + 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))

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