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 + 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)

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Rubi [A]  time = 0.0371273, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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*}

Mathematica [A]  time = 0.0890492, size = 71, normalized size = 1. \[ \frac{x^{m+1} \left (\left (\frac{a^2}{m+1}+\frac{2 a b x}{m+2}+\frac{b^2 x^2}{m+3}\right ) (A b (m+4)-a B (m+1))+B (a+b x)^3\right )}{b (m+4)} \]

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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Maple [B]  time = 0.007, size = 246, normalized size = 3.5 \begin{align*}{\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}^{2}B{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{a}^{2} \right ) }{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}

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^(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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 1.33942, size = 479, normalized size = 6.75 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 1.32814, size = 1020, normalized size = 14.37 \begin{align*} \text{result too large to display} \end{align*}

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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Giac [B]  time = 1.15887, size = 448, normalized size = 6.31 \begin{align*} \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} \end{align*}

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)