3.273 \(\int x^2 (A+B x) (a+c x^2)^4 \, dx\)

Optimal. Leaf size=121 \[ \frac{6}{7} a^2 A c^2 x^7+\frac{4}{5} a^3 A c x^5+\frac{1}{3} a^4 A x^3+\frac{3}{4} a^2 B c^2 x^8+\frac{2}{3} a^3 B c x^6+\frac{1}{4} a^4 B x^4+\frac{4}{9} a A c^3 x^9+\frac{2}{5} a B c^3 x^{10}+\frac{1}{11} A c^4 x^{11}+\frac{1}{12} B c^4 x^{12} \]

[Out]

(a^4*A*x^3)/3 + (a^4*B*x^4)/4 + (4*a^3*A*c*x^5)/5 + (2*a^3*B*c*x^6)/3 + (6*a^2*A*c^2*x^7)/7 + (3*a^2*B*c^2*x^8
)/4 + (4*a*A*c^3*x^9)/9 + (2*a*B*c^3*x^10)/5 + (A*c^4*x^11)/11 + (B*c^4*x^12)/12

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Rubi [A]  time = 0.123798, antiderivative size = 121, 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} \[ \frac{6}{7} a^2 A c^2 x^7+\frac{4}{5} a^3 A c x^5+\frac{1}{3} a^4 A x^3+\frac{3}{4} a^2 B c^2 x^8+\frac{2}{3} a^3 B c x^6+\frac{1}{4} a^4 B x^4+\frac{4}{9} a A c^3 x^9+\frac{2}{5} a B c^3 x^{10}+\frac{1}{11} A c^4 x^{11}+\frac{1}{12} B c^4 x^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(A + B*x)*(a + c*x^2)^4,x]

[Out]

(a^4*A*x^3)/3 + (a^4*B*x^4)/4 + (4*a^3*A*c*x^5)/5 + (2*a^3*B*c*x^6)/3 + (6*a^2*A*c^2*x^7)/7 + (3*a^2*B*c^2*x^8
)/4 + (4*a*A*c^3*x^9)/9 + (2*a*B*c^3*x^10)/5 + (A*c^4*x^11)/11 + (B*c^4*x^12)/12

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 )^4 \, dx &=\int \left (a^4 A x^2+a^4 B x^3+4 a^3 A c x^4+4 a^3 B c x^5+6 a^2 A c^2 x^6+6 a^2 B c^2 x^7+4 a A c^3 x^8+4 a B c^3 x^9+A c^4 x^{10}+B c^4 x^{11}\right ) \, dx\\ &=\frac{1}{3} a^4 A x^3+\frac{1}{4} a^4 B x^4+\frac{4}{5} a^3 A c x^5+\frac{2}{3} a^3 B c x^6+\frac{6}{7} a^2 A c^2 x^7+\frac{3}{4} a^2 B c^2 x^8+\frac{4}{9} a A c^3 x^9+\frac{2}{5} a B c^3 x^{10}+\frac{1}{11} A c^4 x^{11}+\frac{1}{12} B c^4 x^{12}\\ \end{align*}

Mathematica [A]  time = 0.0033255, size = 121, normalized size = 1. \[ \frac{6}{7} a^2 A c^2 x^7+\frac{4}{5} a^3 A c x^5+\frac{1}{3} a^4 A x^3+\frac{3}{4} a^2 B c^2 x^8+\frac{2}{3} a^3 B c x^6+\frac{1}{4} a^4 B x^4+\frac{4}{9} a A c^3 x^9+\frac{2}{5} a B c^3 x^{10}+\frac{1}{11} A c^4 x^{11}+\frac{1}{12} B c^4 x^{12} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(A + B*x)*(a + c*x^2)^4,x]

[Out]

(a^4*A*x^3)/3 + (a^4*B*x^4)/4 + (4*a^3*A*c*x^5)/5 + (2*a^3*B*c*x^6)/3 + (6*a^2*A*c^2*x^7)/7 + (3*a^2*B*c^2*x^8
)/4 + (4*a*A*c^3*x^9)/9 + (2*a*B*c^3*x^10)/5 + (A*c^4*x^11)/11 + (B*c^4*x^12)/12

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Maple [A]  time = 0.002, size = 102, normalized size = 0.8 \begin{align*}{\frac{{a}^{4}A{x}^{3}}{3}}+{\frac{{a}^{4}B{x}^{4}}{4}}+{\frac{4\,{a}^{3}Ac{x}^{5}}{5}}+{\frac{2\,{a}^{3}Bc{x}^{6}}{3}}+{\frac{6\,{a}^{2}A{c}^{2}{x}^{7}}{7}}+{\frac{3\,{a}^{2}B{c}^{2}{x}^{8}}{4}}+{\frac{4\,aA{c}^{3}{x}^{9}}{9}}+{\frac{2\,aB{c}^{3}{x}^{10}}{5}}+{\frac{A{c}^{4}{x}^{11}}{11}}+{\frac{B{c}^{4}{x}^{12}}{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(B*x+A)*(c*x^2+a)^4,x)

[Out]

1/3*a^4*A*x^3+1/4*a^4*B*x^4+4/5*a^3*A*c*x^5+2/3*a^3*B*c*x^6+6/7*a^2*A*c^2*x^7+3/4*a^2*B*c^2*x^8+4/9*a*A*c^3*x^
9+2/5*a*B*c^3*x^10+1/11*A*c^4*x^11+1/12*B*c^4*x^12

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Maxima [A]  time = 1.07576, size = 136, normalized size = 1.12 \begin{align*} \frac{1}{12} \, B c^{4} x^{12} + \frac{1}{11} \, A c^{4} x^{11} + \frac{2}{5} \, B a c^{3} x^{10} + \frac{4}{9} \, A a c^{3} x^{9} + \frac{3}{4} \, B a^{2} c^{2} x^{8} + \frac{6}{7} \, A a^{2} c^{2} x^{7} + \frac{2}{3} \, B a^{3} c x^{6} + \frac{4}{5} \, A a^{3} c x^{5} + \frac{1}{4} \, B a^{4} x^{4} + \frac{1}{3} \, A a^{4} 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)^4,x, algorithm="maxima")

[Out]

1/12*B*c^4*x^12 + 1/11*A*c^4*x^11 + 2/5*B*a*c^3*x^10 + 4/9*A*a*c^3*x^9 + 3/4*B*a^2*c^2*x^8 + 6/7*A*a^2*c^2*x^7
+ 2/3*B*a^3*c*x^6 + 4/5*A*a^3*c*x^5 + 1/4*B*a^4*x^4 + 1/3*A*a^4*x^3

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Fricas [A]  time = 1.37526, size = 243, normalized size = 2.01 \begin{align*} \frac{1}{12} x^{12} c^{4} B + \frac{1}{11} x^{11} c^{4} A + \frac{2}{5} x^{10} c^{3} a B + \frac{4}{9} x^{9} c^{3} a A + \frac{3}{4} x^{8} c^{2} a^{2} B + \frac{6}{7} x^{7} c^{2} a^{2} A + \frac{2}{3} x^{6} c a^{3} B + \frac{4}{5} x^{5} c a^{3} A + \frac{1}{4} x^{4} a^{4} B + \frac{1}{3} x^{3} a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(B*x+A)*(c*x^2+a)^4,x, algorithm="fricas")

[Out]

1/12*x^12*c^4*B + 1/11*x^11*c^4*A + 2/5*x^10*c^3*a*B + 4/9*x^9*c^3*a*A + 3/4*x^8*c^2*a^2*B + 6/7*x^7*c^2*a^2*A
+ 2/3*x^6*c*a^3*B + 4/5*x^5*c*a^3*A + 1/4*x^4*a^4*B + 1/3*x^3*a^4*A

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Sympy [A]  time = 0.080918, size = 124, normalized size = 1.02 \begin{align*} \frac{A a^{4} x^{3}}{3} + \frac{4 A a^{3} c x^{5}}{5} + \frac{6 A a^{2} c^{2} x^{7}}{7} + \frac{4 A a c^{3} x^{9}}{9} + \frac{A c^{4} x^{11}}{11} + \frac{B a^{4} x^{4}}{4} + \frac{2 B a^{3} c x^{6}}{3} + \frac{3 B a^{2} c^{2} x^{8}}{4} + \frac{2 B a c^{3} x^{10}}{5} + \frac{B c^{4} x^{12}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(B*x+A)*(c*x**2+a)**4,x)

[Out]

A*a**4*x**3/3 + 4*A*a**3*c*x**5/5 + 6*A*a**2*c**2*x**7/7 + 4*A*a*c**3*x**9/9 + A*c**4*x**11/11 + B*a**4*x**4/4
+ 2*B*a**3*c*x**6/3 + 3*B*a**2*c**2*x**8/4 + 2*B*a*c**3*x**10/5 + B*c**4*x**12/12

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Giac [A]  time = 1.15824, size = 136, normalized size = 1.12 \begin{align*} \frac{1}{12} \, B c^{4} x^{12} + \frac{1}{11} \, A c^{4} x^{11} + \frac{2}{5} \, B a c^{3} x^{10} + \frac{4}{9} \, A a c^{3} x^{9} + \frac{3}{4} \, B a^{2} c^{2} x^{8} + \frac{6}{7} \, A a^{2} c^{2} x^{7} + \frac{2}{3} \, B a^{3} c x^{6} + \frac{4}{5} \, A a^{3} c x^{5} + \frac{1}{4} \, B a^{4} x^{4} + \frac{1}{3} \, A a^{4} 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)^4,x, algorithm="giac")

[Out]

1/12*B*c^4*x^12 + 1/11*A*c^4*x^11 + 2/5*B*a*c^3*x^10 + 4/9*A*a*c^3*x^9 + 3/4*B*a^2*c^2*x^8 + 6/7*A*a^2*c^2*x^7
+ 2/3*B*a^3*c*x^6 + 4/5*A*a^3*c*x^5 + 1/4*B*a^4*x^4 + 1/3*A*a^4*x^3