3.106 \(\int x^2 (a+b x)^3 (A+B x) \, dx\)

Optimal. Leaf size=75 \[ \frac{1}{4} a^2 x^4 (a B+3 A b)+\frac{1}{3} a^3 A x^3+\frac{1}{6} b^2 x^6 (3 a B+A b)+\frac{3}{5} a b x^5 (a B+A b)+\frac{1}{7} b^3 B x^7 \]

[Out]

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

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Rubi [A]  time = 0.0420364, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ \frac{1}{4} a^2 x^4 (a B+3 A b)+\frac{1}{3} a^3 A x^3+\frac{1}{6} b^2 x^6 (3 a B+A b)+\frac{3}{5} a b x^5 (a B+A b)+\frac{1}{7} b^3 B x^7 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^3*(A + B*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0090785, size = 75, normalized size = 1. \[ \frac{1}{4} a^2 x^4 (a B+3 A b)+\frac{1}{3} a^3 A x^3+\frac{1}{6} b^2 x^6 (3 a B+A b)+\frac{3}{5} a b x^5 (a B+A b)+\frac{1}{7} b^3 B x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^3*(A + B*x),x]

[Out]

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

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Maple [A]  time = 0.001, size = 76, normalized size = 1. \begin{align*}{\frac{{b}^{3}B{x}^{7}}{7}}+{\frac{ \left ({b}^{3}A+3\,a{b}^{2}B \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{a}^{2}bA+{a}^{3}B \right ){x}^{4}}{4}}+{\frac{{a}^{3}A{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)^3*(B*x+A),x)

[Out]

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

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Maxima [A]  time = 1.09362, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{7} \, B b^{3} x^{7} + \frac{1}{3} \, A a^{3} x^{3} + \frac{1}{6} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + \frac{3}{5} \,{\left (B a^{2} b + A a b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^3*(B*x+A),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.42089, size = 182, normalized size = 2.43 \begin{align*} \frac{1}{7} x^{7} b^{3} B + \frac{1}{2} x^{6} b^{2} a B + \frac{1}{6} x^{6} b^{3} A + \frac{3}{5} x^{5} b a^{2} B + \frac{3}{5} x^{5} b^{2} a A + \frac{1}{4} x^{4} a^{3} B + \frac{3}{4} x^{4} b a^{2} A + \frac{1}{3} x^{3} a^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^3*(B*x+A),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.07574, size = 82, normalized size = 1.09 \begin{align*} \frac{A a^{3} x^{3}}{3} + \frac{B b^{3} x^{7}}{7} + x^{6} \left (\frac{A b^{3}}{6} + \frac{B a b^{2}}{2}\right ) + x^{5} \left (\frac{3 A a b^{2}}{5} + \frac{3 B a^{2} b}{5}\right ) + x^{4} \left (\frac{3 A a^{2} b}{4} + \frac{B a^{3}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22999, size = 104, normalized size = 1.39 \begin{align*} \frac{1}{7} \, B b^{3} x^{7} + \frac{1}{2} \, B a b^{2} x^{6} + \frac{1}{6} \, A b^{3} x^{6} + \frac{3}{5} \, B a^{2} b x^{5} + \frac{3}{5} \, A a b^{2} x^{5} + \frac{1}{4} \, B a^{3} x^{4} + \frac{3}{4} \, A a^{2} b x^{4} + \frac{1}{3} \, A a^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^3*(B*x+A),x, algorithm="giac")

[Out]

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