3.104 \(\int x^4 (a+b x)^3 (A+B x) \, dx\)

Optimal. Leaf size=75 \[ \frac{1}{6} a^2 x^6 (a B+3 A b)+\frac{1}{5} a^3 A x^5+\frac{1}{8} b^2 x^8 (3 a B+A b)+\frac{3}{7} a b x^7 (a B+A b)+\frac{1}{9} b^3 B x^9 \]

[Out]

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

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Rubi [A]  time = 0.0541689, 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} \[ \frac{1}{6} a^2 x^6 (a B+3 A b)+\frac{1}{5} a^3 A x^5+\frac{1}{8} b^2 x^8 (3 a B+A b)+\frac{3}{7} a b x^7 (a B+A b)+\frac{1}{9} b^3 B x^9 \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x)^3*(A + B*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0101744, size = 75, normalized size = 1. \[ \frac{1}{6} a^2 x^6 (a B+3 A b)+\frac{1}{5} a^3 A x^5+\frac{1}{8} b^2 x^8 (3 a B+A b)+\frac{3}{7} a b x^7 (a B+A b)+\frac{1}{9} b^3 B x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x)^3*(A + B*x),x]

[Out]

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

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Maple [A]  time = 0., size = 76, normalized size = 1. \begin{align*}{\frac{{b}^{3}B{x}^{9}}{9}}+{\frac{ \left ({b}^{3}A+3\,a{b}^{2}B \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2}bA+{a}^{3}B \right ){x}^{6}}{6}}+{\frac{{a}^{3}A{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x+a)^3*(B*x+A),x)

[Out]

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

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Maxima [A]  time = 1.05251, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{9} \, B b^{3} x^{9} + \frac{1}{5} \, A a^{3} x^{5} + \frac{1}{8} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{8} + \frac{3}{7} \,{\left (B a^{2} b + A a b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^3*(B*x+A),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.62268, size = 182, normalized size = 2.43 \begin{align*} \frac{1}{9} x^{9} b^{3} B + \frac{3}{8} x^{8} b^{2} a B + \frac{1}{8} x^{8} b^{3} A + \frac{3}{7} x^{7} b a^{2} B + \frac{3}{7} x^{7} b^{2} a A + \frac{1}{6} x^{6} a^{3} B + \frac{1}{2} x^{6} b a^{2} A + \frac{1}{5} x^{5} a^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^3*(B*x+A),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.085205, size = 82, normalized size = 1.09 \begin{align*} \frac{A a^{3} x^{5}}{5} + \frac{B b^{3} x^{9}}{9} + x^{8} \left (\frac{A b^{3}}{8} + \frac{3 B a b^{2}}{8}\right ) + x^{7} \left (\frac{3 A a b^{2}}{7} + \frac{3 B a^{2} b}{7}\right ) + x^{6} \left (\frac{A a^{2} b}{2} + \frac{B a^{3}}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x+a)**3*(B*x+A),x)

[Out]

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

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Giac [A]  time = 1.2033, size = 104, normalized size = 1.39 \begin{align*} \frac{1}{9} \, B b^{3} x^{9} + \frac{3}{8} \, B a b^{2} x^{8} + \frac{1}{8} \, A b^{3} x^{8} + \frac{3}{7} \, B a^{2} b x^{7} + \frac{3}{7} \, A a b^{2} x^{7} + \frac{1}{6} \, B a^{3} x^{6} + \frac{1}{2} \, A a^{2} b x^{6} + \frac{1}{5} \, A a^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^3*(B*x+A),x, algorithm="giac")

[Out]

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