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

Optimal. Leaf size=55 \[ \frac{1}{5} a^2 A x^5+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{6} a x^6 (a B+2 A b)+\frac{1}{8} b^2 B x^8 \]

[Out]

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

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Rubi [A]  time = 0.0483102, antiderivative size = 55, 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} \[ \frac{1}{5} a^2 A x^5+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{6} a x^6 (a B+2 A b)+\frac{1}{8} b^2 B x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0079606, size = 55, normalized size = 1. \[ \frac{1}{5} a^2 A x^5+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{6} a x^6 (a B+2 A b)+\frac{1}{8} b^2 B x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{8}}{8}}+{\frac{ \left ( A{b}^{2}+2\,abB \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,Aab+B{a}^{2} \right ){x}^{6}}{6}}+{\frac{{a}^{2}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),x)

[Out]

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

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Maxima [A]  time = 1.01765, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{8} \, B b^{2} x^{8} + \frac{1}{5} \, A a^{2} x^{5} + \frac{1}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B a^{2} + 2 \, A a 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.07104, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{8} x^{8} b^{2} B + \frac{2}{7} x^{7} b a B + \frac{1}{7} x^{7} b^{2} A + \frac{1}{6} x^{6} a^{2} B + \frac{1}{3} x^{6} b a A + \frac{1}{5} x^{5} a^{2} 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.085482, size = 54, normalized size = 0.98 \begin{align*} \frac{A a^{2} x^{5}}{5} + \frac{B b^{2} x^{8}}{8} + x^{7} \left (\frac{A b^{2}}{7} + \frac{2 B a b}{7}\right ) + x^{6} \left (\frac{A a b}{3} + \frac{B a^{2}}{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),x)

[Out]

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

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Giac [A]  time = 1.1157, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{8} \, B b^{2} x^{8} + \frac{2}{7} \, B a b x^{7} + \frac{1}{7} \, A b^{2} x^{7} + \frac{1}{6} \, B a^{2} x^{6} + \frac{1}{3} \, A a b x^{6} + \frac{1}{5} \, A a^{2} 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),x, algorithm="giac")

[Out]

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