3.523 \(\int x^4 (A+B x) (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=99 \[ \frac{2}{7} a^2 b x^7 (2 a B+3 A b)+\frac{1}{6} a^3 x^6 (a B+4 A b)+\frac{1}{5} a^4 A x^5+\frac{1}{9} b^3 x^9 (4 a B+A b)+\frac{1}{4} a b^2 x^8 (3 a B+2 A b)+\frac{1}{10} b^4 B x^{10} \]

[Out]

(a^4*A*x^5)/5 + (a^3*(4*A*b + a*B)*x^6)/6 + (2*a^2*b*(3*A*b + 2*a*B)*x^7)/7 + (a*b^2*(2*A*b + 3*a*B)*x^8)/4 +
(b^3*(A*b + 4*a*B)*x^9)/9 + (b^4*B*x^10)/10

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Rubi [A]  time = 0.0840065, antiderivative size = 99, 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} \[ \frac{2}{7} a^2 b x^7 (2 a B+3 A b)+\frac{1}{6} a^3 x^6 (a B+4 A b)+\frac{1}{5} a^4 A x^5+\frac{1}{9} b^3 x^9 (4 a B+A b)+\frac{1}{4} a b^2 x^8 (3 a B+2 A b)+\frac{1}{10} b^4 B x^{10} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(a^4*A*x^5)/5 + (a^3*(4*A*b + a*B)*x^6)/6 + (2*a^2*b*(3*A*b + 2*a*B)*x^7)/7 + (a*b^2*(2*A*b + 3*a*B)*x^8)/4 +
(b^3*(A*b + 4*a*B)*x^9)/9 + (b^4*B*x^10)/10

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^4 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int x^4 (a+b x)^4 (A+B x) \, dx\\ &=\int \left (a^4 A x^4+a^3 (4 A b+a B) x^5+2 a^2 b (3 A b+2 a B) x^6+2 a b^2 (2 A b+3 a B) x^7+b^3 (A b+4 a B) x^8+b^4 B x^9\right ) \, dx\\ &=\frac{1}{5} a^4 A x^5+\frac{1}{6} a^3 (4 A b+a B) x^6+\frac{2}{7} a^2 b (3 A b+2 a B) x^7+\frac{1}{4} a b^2 (2 A b+3 a B) x^8+\frac{1}{9} b^3 (A b+4 a B) x^9+\frac{1}{10} b^4 B x^{10}\\ \end{align*}

Mathematica [A]  time = 0.0141712, size = 99, normalized size = 1. \[ \frac{2}{7} a^2 b x^7 (2 a B+3 A b)+\frac{1}{6} a^3 x^6 (a B+4 A b)+\frac{1}{5} a^4 A x^5+\frac{1}{9} b^3 x^9 (4 a B+A b)+\frac{1}{4} a b^2 x^8 (3 a B+2 A b)+\frac{1}{10} b^4 B x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(a^4*A*x^5)/5 + (a^3*(4*A*b + a*B)*x^6)/6 + (2*a^2*b*(3*A*b + 2*a*B)*x^7)/7 + (a*b^2*(2*A*b + 3*a*B)*x^8)/4 +
(b^3*(A*b + 4*a*B)*x^9)/9 + (b^4*B*x^10)/10

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Maple [A]  time = 0.001, size = 100, normalized size = 1. \begin{align*}{\frac{{b}^{4}B{x}^{10}}{10}}+{\frac{ \left ( A{b}^{4}+4\,Ba{b}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,Aa{b}^{3}+6\,B{a}^{2}{b}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 6\,A{a}^{2}{b}^{2}+4\,B{a}^{3}b \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,A{a}^{3}b+B{a}^{4} \right ){x}^{6}}{6}}+{\frac{{a}^{4}A{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/10*b^4*B*x^10+1/9*(A*b^4+4*B*a*b^3)*x^9+1/8*(4*A*a*b^3+6*B*a^2*b^2)*x^8+1/7*(6*A*a^2*b^2+4*B*a^3*b)*x^7+1/6*
(4*A*a^3*b+B*a^4)*x^6+1/5*a^4*A*x^5

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Maxima [A]  time = 0.979778, size = 134, normalized size = 1.35 \begin{align*} \frac{1}{10} \, B b^{4} x^{10} + \frac{1}{5} \, A a^{4} x^{5} + \frac{1}{9} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{9} + \frac{1}{4} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{8} + \frac{2}{7} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/10*B*b^4*x^10 + 1/5*A*a^4*x^5 + 1/9*(4*B*a*b^3 + A*b^4)*x^9 + 1/4*(3*B*a^2*b^2 + 2*A*a*b^3)*x^8 + 2/7*(2*B*a
^3*b + 3*A*a^2*b^2)*x^7 + 1/6*(B*a^4 + 4*A*a^3*b)*x^6

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Fricas [A]  time = 1.19845, size = 239, normalized size = 2.41 \begin{align*} \frac{1}{10} x^{10} b^{4} B + \frac{4}{9} x^{9} b^{3} a B + \frac{1}{9} x^{9} b^{4} A + \frac{3}{4} x^{8} b^{2} a^{2} B + \frac{1}{2} x^{8} b^{3} a A + \frac{4}{7} x^{7} b a^{3} B + \frac{6}{7} x^{7} b^{2} a^{2} A + \frac{1}{6} x^{6} a^{4} B + \frac{2}{3} x^{6} b a^{3} A + \frac{1}{5} x^{5} a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/10*x^10*b^4*B + 4/9*x^9*b^3*a*B + 1/9*x^9*b^4*A + 3/4*x^8*b^2*a^2*B + 1/2*x^8*b^3*a*A + 4/7*x^7*b*a^3*B + 6/
7*x^7*b^2*a^2*A + 1/6*x^6*a^4*B + 2/3*x^6*b*a^3*A + 1/5*x^5*a^4*A

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Sympy [A]  time = 0.085009, size = 109, normalized size = 1.1 \begin{align*} \frac{A a^{4} x^{5}}{5} + \frac{B b^{4} x^{10}}{10} + x^{9} \left (\frac{A b^{4}}{9} + \frac{4 B a b^{3}}{9}\right ) + x^{8} \left (\frac{A a b^{3}}{2} + \frac{3 B a^{2} b^{2}}{4}\right ) + x^{7} \left (\frac{6 A a^{2} b^{2}}{7} + \frac{4 B a^{3} b}{7}\right ) + x^{6} \left (\frac{2 A a^{3} b}{3} + \frac{B a^{4}}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*x**5/5 + B*b**4*x**10/10 + x**9*(A*b**4/9 + 4*B*a*b**3/9) + x**8*(A*a*b**3/2 + 3*B*a**2*b**2/4) + x**7*
(6*A*a**2*b**2/7 + 4*B*a**3*b/7) + x**6*(2*A*a**3*b/3 + B*a**4/6)

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Giac [A]  time = 1.11999, size = 136, normalized size = 1.37 \begin{align*} \frac{1}{10} \, B b^{4} x^{10} + \frac{4}{9} \, B a b^{3} x^{9} + \frac{1}{9} \, A b^{4} x^{9} + \frac{3}{4} \, B a^{2} b^{2} x^{8} + \frac{1}{2} \, A a b^{3} x^{8} + \frac{4}{7} \, B a^{3} b x^{7} + \frac{6}{7} \, A a^{2} b^{2} x^{7} + \frac{1}{6} \, B a^{4} x^{6} + \frac{2}{3} \, A a^{3} b x^{6} + \frac{1}{5} \, A a^{4} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/10*B*b^4*x^10 + 4/9*B*a*b^3*x^9 + 1/9*A*b^4*x^9 + 3/4*B*a^2*b^2*x^8 + 1/2*A*a*b^3*x^8 + 4/7*B*a^3*b*x^7 + 6/
7*A*a^2*b^2*x^7 + 1/6*B*a^4*x^6 + 2/3*A*a^3*b*x^6 + 1/5*A*a^4*x^5