3.64 \(\int x^4 (a+b x)^3 \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^5}{5}+\frac{3}{7} a b^2 x^7+\frac{b^3 x^8}{8} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0181752, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^5}{5}+\frac{3}{7} a b^2 x^7+\frac{b^3 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x)^3,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0021594, size = 43, normalized size = 1. \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^5}{5}+\frac{3}{7} a b^2 x^7+\frac{b^3 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x)^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0., size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{5}}{5}}+{\frac{{a}^{2}b{x}^{6}}{2}}+{\frac{3\,a{b}^{2}{x}^{7}}{7}}+{\frac{{b}^{3}{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.01717, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{8} \, b^{3} x^{8} + \frac{3}{7} \, a b^{2} x^{7} + \frac{1}{2} \, a^{2} b x^{6} + \frac{1}{5} \, a^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.29981, size = 80, normalized size = 1.86 \begin{align*} \frac{1}{8} x^{8} b^{3} + \frac{3}{7} x^{7} b^{2} a + \frac{1}{2} x^{6} b a^{2} + \frac{1}{5} x^{5} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.072973, size = 37, normalized size = 0.86 \begin{align*} \frac{a^{3} x^{5}}{5} + \frac{a^{2} b x^{6}}{2} + \frac{3 a b^{2} x^{7}}{7} + \frac{b^{3} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**5/5 + a**2*b*x**6/2 + 3*a*b**2*x**7/7 + b**3*x**8/8

________________________________________________________________________________________

Giac [A]  time = 1.17695, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{8} \, b^{3} x^{8} + \frac{3}{7} \, a b^{2} x^{7} + \frac{1}{2} \, a^{2} b x^{6} + \frac{1}{5} \, a^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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