3.267 \(\int x (A+B x) (a+c x^2)^3 \, dx\)

Optimal. Leaf size=93 \[ \frac{3}{4} a^2 A c x^4+\frac{1}{2} a^3 A x^2+\frac{3}{5} a^2 B c x^5+\frac{1}{3} a^3 B x^3+\frac{1}{2} a A c^2 x^6+\frac{3}{7} a B c^2 x^7+\frac{1}{8} A c^3 x^8+\frac{1}{9} B c^3 x^9 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0792587, antiderivative size = 93, 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 = {766} \[ \frac{3}{4} a^2 A c x^4+\frac{1}{2} a^3 A x^2+\frac{3}{5} a^2 B c x^5+\frac{1}{3} a^3 B x^3+\frac{1}{2} a A c^2 x^6+\frac{3}{7} a B c^2 x^7+\frac{1}{8} A c^3 x^8+\frac{1}{9} B c^3 x^9 \]

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(a + c*x^2)^3,x]

[Out]

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

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0027795, size = 93, normalized size = 1. \[ \frac{3}{4} a^2 A c x^4+\frac{1}{2} a^3 A x^2+\frac{3}{5} a^2 B c x^5+\frac{1}{3} a^3 B x^3+\frac{1}{2} a A c^2 x^6+\frac{3}{7} a B c^2 x^7+\frac{1}{8} A c^3 x^8+\frac{1}{9} B c^3 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 78, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}A{x}^{2}}{2}}+{\frac{{a}^{3}B{x}^{3}}{3}}+{\frac{3\,{a}^{2}Ac{x}^{4}}{4}}+{\frac{3\,{a}^{2}Bc{x}^{5}}{5}}+{\frac{aA{c}^{2}{x}^{6}}{2}}+{\frac{3\,aB{c}^{2}{x}^{7}}{7}}+{\frac{A{c}^{3}{x}^{8}}{8}}+{\frac{B{c}^{3}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02881, size = 104, normalized size = 1.12 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} + \frac{1}{8} \, A c^{3} x^{8} + \frac{3}{7} \, B a c^{2} x^{7} + \frac{1}{2} \, A a c^{2} x^{6} + \frac{3}{5} \, B a^{2} c x^{5} + \frac{3}{4} \, A a^{2} c x^{4} + \frac{1}{3} \, B a^{3} x^{3} + \frac{1}{2} \, A a^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.58022, size = 182, normalized size = 1.96 \begin{align*} \frac{1}{9} x^{9} c^{3} B + \frac{1}{8} x^{8} c^{3} A + \frac{3}{7} x^{7} c^{2} a B + \frac{1}{2} x^{6} c^{2} a A + \frac{3}{5} x^{5} c a^{2} B + \frac{3}{4} x^{4} c a^{2} A + \frac{1}{3} x^{3} a^{3} B + \frac{1}{2} x^{2} a^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.083398, size = 92, normalized size = 0.99 \begin{align*} \frac{A a^{3} x^{2}}{2} + \frac{3 A a^{2} c x^{4}}{4} + \frac{A a c^{2} x^{6}}{2} + \frac{A c^{3} x^{8}}{8} + \frac{B a^{3} x^{3}}{3} + \frac{3 B a^{2} c x^{5}}{5} + \frac{3 B a c^{2} x^{7}}{7} + \frac{B c^{3} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**3*x**2/2 + 3*A*a**2*c*x**4/4 + A*a*c**2*x**6/2 + A*c**3*x**8/8 + B*a**3*x**3/3 + 3*B*a**2*c*x**5/5 + 3*B*
a*c**2*x**7/7 + B*c**3*x**9/9

________________________________________________________________________________________

Giac [A]  time = 1.11881, size = 104, normalized size = 1.12 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} + \frac{1}{8} \, A c^{3} x^{8} + \frac{3}{7} \, B a c^{2} x^{7} + \frac{1}{2} \, A a c^{2} x^{6} + \frac{3}{5} \, B a^{2} c x^{5} + \frac{3}{4} \, A a^{2} c x^{4} + \frac{1}{3} \, B a^{3} x^{3} + \frac{1}{2} \, A a^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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