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

Optimal. Leaf size=87 \[ \frac{a^2 (a+b x)^5 (A b-a B)}{5 b^4}+\frac{(a+b x)^7 (A b-3 a B)}{7 b^4}-\frac{a (a+b x)^6 (2 A b-3 a B)}{6 b^4}+\frac{B (a+b x)^8}{8 b^4} \]

[Out]

(a^2*(A*b - a*B)*(a + b*x)^5)/(5*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x)^6)/(6*b^4) + ((A*b - 3*a*B)*(a + b*x)^7)/
(7*b^4) + (B*(a + b*x)^8)/(8*b^4)

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Rubi [A]  time = 0.0529759, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ \frac{a^2 (a+b x)^5 (A b-a B)}{5 b^4}+\frac{(a+b x)^7 (A b-3 a B)}{7 b^4}-\frac{a (a+b x)^6 (2 A b-3 a B)}{6 b^4}+\frac{B (a+b x)^8}{8 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(A*b - a*B)*(a + b*x)^5)/(5*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x)^6)/(6*b^4) + ((A*b - 3*a*B)*(a + b*x)^7)/
(7*b^4) + (B*(a + b*x)^8)/(8*b^4)

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^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int x^2 (a+b x)^4 (A+B x) \, dx\\ &=\int \left (-\frac{a^2 (-A b+a B) (a+b x)^4}{b^3}+\frac{a (-2 A b+3 a B) (a+b x)^5}{b^3}+\frac{(A b-3 a B) (a+b x)^6}{b^3}+\frac{B (a+b x)^7}{b^3}\right ) \, dx\\ &=\frac{a^2 (A b-a B) (a+b x)^5}{5 b^4}-\frac{a (2 A b-3 a B) (a+b x)^6}{6 b^4}+\frac{(A b-3 a B) (a+b x)^7}{7 b^4}+\frac{B (a+b x)^8}{8 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0257239, size = 88, normalized size = 1.01 \[ \frac{1}{840} x^3 \left (168 a^2 b^2 x^2 (6 A+5 B x)+168 a^3 b x (5 A+4 B x)+70 a^4 (4 A+3 B x)+80 a b^3 x^3 (7 A+6 B x)+15 b^4 x^4 (8 A+7 B x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(70*a^4*(4*A + 3*B*x) + 168*a^3*b*x*(5*A + 4*B*x) + 168*a^2*b^2*x^2*(6*A + 5*B*x) + 80*a*b^3*x^3*(7*A + 6
*B*x) + 15*b^4*x^4*(8*A + 7*B*x)))/840

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Maple [A]  time = 0., size = 100, normalized size = 1.2 \begin{align*}{\frac{{b}^{4}B{x}^{8}}{8}}+{\frac{ \left ( A{b}^{4}+4\,Ba{b}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,Aa{b}^{3}+6\,B{a}^{2}{b}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,A{a}^{2}{b}^{2}+4\,B{a}^{3}b \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,A{a}^{3}b+B{a}^{4} \right ){x}^{4}}{4}}+{\frac{{a}^{4}A{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^4*B*x^8+1/7*(A*b^4+4*B*a*b^3)*x^7+1/6*(4*A*a*b^3+6*B*a^2*b^2)*x^6+1/5*(6*A*a^2*b^2+4*B*a^3*b)*x^5+1/4*(4
*A*a^3*b+B*a^4)*x^4+1/3*a^4*A*x^3

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Maxima [A]  time = 1.02124, size = 134, normalized size = 1.54 \begin{align*} \frac{1}{8} \, B b^{4} x^{8} + \frac{1}{3} \, A a^{4} x^{3} + \frac{1}{7} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{7} + \frac{1}{3} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{6} + \frac{2}{5} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*B*b^4*x^8 + 1/3*A*a^4*x^3 + 1/7*(4*B*a*b^3 + A*b^4)*x^7 + 1/3*(3*B*a^2*b^2 + 2*A*a*b^3)*x^6 + 2/5*(2*B*a^3
*b + 3*A*a^2*b^2)*x^5 + 1/4*(B*a^4 + 4*A*a^3*b)*x^4

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Fricas [A]  time = 1.11358, size = 225, normalized size = 2.59 \begin{align*} \frac{1}{8} x^{8} b^{4} B + \frac{4}{7} x^{7} b^{3} a B + \frac{1}{7} x^{7} b^{4} A + x^{6} b^{2} a^{2} B + \frac{2}{3} x^{6} b^{3} a A + \frac{4}{5} x^{5} b a^{3} B + \frac{6}{5} x^{5} b^{2} a^{2} A + \frac{1}{4} x^{4} a^{4} B + x^{4} b a^{3} A + \frac{1}{3} x^{3} a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*b^4*B + 4/7*x^7*b^3*a*B + 1/7*x^7*b^4*A + x^6*b^2*a^2*B + 2/3*x^6*b^3*a*A + 4/5*x^5*b*a^3*B + 6/5*x^5*
b^2*a^2*A + 1/4*x^4*a^4*B + x^4*b*a^3*A + 1/3*x^3*a^4*A

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Sympy [A]  time = 0.081985, size = 104, normalized size = 1.2 \begin{align*} \frac{A a^{4} x^{3}}{3} + \frac{B b^{4} x^{8}}{8} + x^{7} \left (\frac{A b^{4}}{7} + \frac{4 B a b^{3}}{7}\right ) + x^{6} \left (\frac{2 A a b^{3}}{3} + B a^{2} b^{2}\right ) + x^{5} \left (\frac{6 A a^{2} b^{2}}{5} + \frac{4 B a^{3} b}{5}\right ) + x^{4} \left (A a^{3} b + \frac{B a^{4}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**4*x**3/3 + B*b**4*x**8/8 + x**7*(A*b**4/7 + 4*B*a*b**3/7) + x**6*(2*A*a*b**3/3 + B*a**2*b**2) + x**5*(6*A
*a**2*b**2/5 + 4*B*a**3*b/5) + x**4*(A*a**3*b + B*a**4/4)

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Giac [A]  time = 1.13566, size = 134, normalized size = 1.54 \begin{align*} \frac{1}{8} \, B b^{4} x^{8} + \frac{4}{7} \, B a b^{3} x^{7} + \frac{1}{7} \, A b^{4} x^{7} + B a^{2} b^{2} x^{6} + \frac{2}{3} \, A a b^{3} x^{6} + \frac{4}{5} \, B a^{3} b x^{5} + \frac{6}{5} \, A a^{2} b^{2} x^{5} + \frac{1}{4} \, B a^{4} x^{4} + A a^{3} b x^{4} + \frac{1}{3} \, A a^{4} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*B*b^4*x^8 + 4/7*B*a*b^3*x^7 + 1/7*A*b^4*x^7 + B*a^2*b^2*x^6 + 2/3*A*a*b^3*x^6 + 4/5*B*a^3*b*x^5 + 6/5*A*a^
2*b^2*x^5 + 1/4*B*a^4*x^4 + A*a^3*b*x^4 + 1/3*A*a^4*x^3