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

Optimal. Leaf size=93 \[ \frac{3}{5} a^2 A c x^5+\frac{1}{3} a^3 A x^3+\frac{1}{2} a^2 B c x^6+\frac{1}{4} a^3 B x^4+\frac{3}{7} a A c^2 x^7+\frac{3}{8} a B c^2 x^8+\frac{1}{9} A c^3 x^9+\frac{1}{10} B c^3 x^{10} \]

[Out]

(a^3*A*x^3)/3 + (a^3*B*x^4)/4 + (3*a^2*A*c*x^5)/5 + (a^2*B*c*x^6)/2 + (3*a*A*c^2*x^7)/7 + (3*a*B*c^2*x^8)/8 +
(A*c^3*x^9)/9 + (B*c^3*x^10)/10

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Rubi [A]  time = 0.088014, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ \frac{3}{5} a^2 A c x^5+\frac{1}{3} a^3 A x^3+\frac{1}{2} a^2 B c x^6+\frac{1}{4} a^3 B x^4+\frac{3}{7} a A c^2 x^7+\frac{3}{8} a B c^2 x^8+\frac{1}{9} A c^3 x^9+\frac{1}{10} B c^3 x^{10} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(A + B*x)*(a + c*x^2)^3,x]

[Out]

(a^3*A*x^3)/3 + (a^3*B*x^4)/4 + (3*a^2*A*c*x^5)/5 + (a^2*B*c*x^6)/2 + (3*a*A*c^2*x^7)/7 + (3*a*B*c^2*x^8)/8 +
(A*c^3*x^9)/9 + (B*c^3*x^10)/10

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^2 (A+B x) \left (a+c x^2\right )^3 \, dx &=\int \left (a^3 A x^2+a^3 B x^3+3 a^2 A c x^4+3 a^2 B c x^5+3 a A c^2 x^6+3 a B c^2 x^7+A c^3 x^8+B c^3 x^9\right ) \, dx\\ &=\frac{1}{3} a^3 A x^3+\frac{1}{4} a^3 B x^4+\frac{3}{5} a^2 A c x^5+\frac{1}{2} a^2 B c x^6+\frac{3}{7} a A c^2 x^7+\frac{3}{8} a B c^2 x^8+\frac{1}{9} A c^3 x^9+\frac{1}{10} B c^3 x^{10}\\ \end{align*}

Mathematica [A]  time = 0.003031, size = 93, normalized size = 1. \[ \frac{3}{5} a^2 A c x^5+\frac{1}{3} a^3 A x^3+\frac{1}{2} a^2 B c x^6+\frac{1}{4} a^3 B x^4+\frac{3}{7} a A c^2 x^7+\frac{3}{8} a B c^2 x^8+\frac{1}{9} A c^3 x^9+\frac{1}{10} B c^3 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*A*x^3)/3 + (a^3*B*x^4)/4 + (3*a^2*A*c*x^5)/5 + (a^2*B*c*x^6)/2 + (3*a*A*c^2*x^7)/7 + (3*a*B*c^2*x^8)/8 +
(A*c^3*x^9)/9 + (B*c^3*x^10)/10

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Maple [A]  time = 0.001, size = 78, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}A{x}^{3}}{3}}+{\frac{{a}^{3}B{x}^{4}}{4}}+{\frac{3\,{a}^{2}Ac{x}^{5}}{5}}+{\frac{{a}^{2}Bc{x}^{6}}{2}}+{\frac{3\,aA{c}^{2}{x}^{7}}{7}}+{\frac{3\,aB{c}^{2}{x}^{8}}{8}}+{\frac{A{c}^{3}{x}^{9}}{9}}+{\frac{B{c}^{3}{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*A*x^3+1/4*a^3*B*x^4+3/5*a^2*A*c*x^5+1/2*a^2*B*c*x^6+3/7*a*A*c^2*x^7+3/8*a*B*c^2*x^8+1/9*A*c^3*x^9+1/10
*B*c^3*x^10

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Maxima [A]  time = 1.04009, size = 104, normalized size = 1.12 \begin{align*} \frac{1}{10} \, B c^{3} x^{10} + \frac{1}{9} \, A c^{3} x^{9} + \frac{3}{8} \, B a c^{2} x^{8} + \frac{3}{7} \, A a c^{2} x^{7} + \frac{1}{2} \, B a^{2} c x^{6} + \frac{3}{5} \, A a^{2} c x^{5} + \frac{1}{4} \, B a^{3} 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)*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/10*B*c^3*x^10 + 1/9*A*c^3*x^9 + 3/8*B*a*c^2*x^8 + 3/7*A*a*c^2*x^7 + 1/2*B*a^2*c*x^6 + 3/5*A*a^2*c*x^5 + 1/4*
B*a^3*x^4 + 1/3*A*a^3*x^3

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Fricas [A]  time = 1.54742, size = 185, normalized size = 1.99 \begin{align*} \frac{1}{10} x^{10} c^{3} B + \frac{1}{9} x^{9} c^{3} A + \frac{3}{8} x^{8} c^{2} a B + \frac{3}{7} x^{7} c^{2} a A + \frac{1}{2} x^{6} c a^{2} B + \frac{3}{5} x^{5} c a^{2} A + \frac{1}{4} x^{4} a^{3} B + \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)*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/10*x^10*c^3*B + 1/9*x^9*c^3*A + 3/8*x^8*c^2*a*B + 3/7*x^7*c^2*a*A + 1/2*x^6*c*a^2*B + 3/5*x^5*c*a^2*A + 1/4*
x^4*a^3*B + 1/3*x^3*a^3*A

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Sympy [A]  time = 0.083413, size = 92, normalized size = 0.99 \begin{align*} \frac{A a^{3} x^{3}}{3} + \frac{3 A a^{2} c x^{5}}{5} + \frac{3 A a c^{2} x^{7}}{7} + \frac{A c^{3} x^{9}}{9} + \frac{B a^{3} x^{4}}{4} + \frac{B a^{2} c x^{6}}{2} + \frac{3 B a c^{2} x^{8}}{8} + \frac{B c^{3} x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**3*x**3/3 + 3*A*a**2*c*x**5/5 + 3*A*a*c**2*x**7/7 + A*c**3*x**9/9 + B*a**3*x**4/4 + B*a**2*c*x**6/2 + 3*B*
a*c**2*x**8/8 + B*c**3*x**10/10

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Giac [A]  time = 1.13507, size = 104, normalized size = 1.12 \begin{align*} \frac{1}{10} \, B c^{3} x^{10} + \frac{1}{9} \, A c^{3} x^{9} + \frac{3}{8} \, B a c^{2} x^{8} + \frac{3}{7} \, A a c^{2} x^{7} + \frac{1}{2} \, B a^{2} c x^{6} + \frac{3}{5} \, A a^{2} c x^{5} + \frac{1}{4} \, B a^{3} 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)*(c*x^2+a)^3,x, algorithm="giac")

[Out]

1/10*B*c^3*x^10 + 1/9*A*c^3*x^9 + 3/8*B*a*c^2*x^8 + 3/7*A*a*c^2*x^7 + 1/2*B*a^2*c*x^6 + 3/5*A*a^2*c*x^5 + 1/4*
B*a^3*x^4 + 1/3*A*a^3*x^3