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

Optimal. Leaf size=115 \[ a^2 A c^2 x^6+a^3 A c x^4+\frac{1}{2} a^4 A x^2+\frac{6}{7} a^2 B c^2 x^7+\frac{4}{5} a^3 B c x^5+\frac{1}{3} a^4 B x^3+\frac{1}{2} a A c^3 x^8+\frac{4}{9} a B c^3 x^9+\frac{1}{10} A c^4 x^{10}+\frac{1}{11} B c^4 x^{11} \]

[Out]

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

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Rubi [A]  time = 0.115734, antiderivative size = 115, 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} \[ a^2 A c^2 x^6+a^3 A c x^4+\frac{1}{2} a^4 A x^2+\frac{6}{7} a^2 B c^2 x^7+\frac{4}{5} a^3 B c x^5+\frac{1}{3} a^4 B x^3+\frac{1}{2} a A c^3 x^8+\frac{4}{9} a B c^3 x^9+\frac{1}{10} A c^4 x^{10}+\frac{1}{11} B c^4 x^{11} \]

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(a + c*x^2)^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0031218, size = 115, normalized size = 1. \[ a^2 A c^2 x^6+a^3 A c x^4+\frac{1}{2} a^4 A x^2+\frac{6}{7} a^2 B c^2 x^7+\frac{4}{5} a^3 B c x^5+\frac{1}{3} a^4 B x^3+\frac{1}{2} a A c^3 x^8+\frac{4}{9} a B c^3 x^9+\frac{1}{10} A c^4 x^{10}+\frac{1}{11} B c^4 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 100, normalized size = 0.9 \begin{align*}{\frac{{a}^{4}A{x}^{2}}{2}}+{\frac{{a}^{4}B{x}^{3}}{3}}+{a}^{3}Ac{x}^{4}+{\frac{4\,{a}^{3}Bc{x}^{5}}{5}}+{a}^{2}A{c}^{2}{x}^{6}+{\frac{6\,{a}^{2}B{c}^{2}{x}^{7}}{7}}+{\frac{aA{c}^{3}{x}^{8}}{2}}+{\frac{4\,aB{c}^{3}{x}^{9}}{9}}+{\frac{A{c}^{4}{x}^{10}}{10}}+{\frac{B{c}^{4}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02806, size = 134, normalized size = 1.17 \begin{align*} \frac{1}{11} \, B c^{4} x^{11} + \frac{1}{10} \, A c^{4} x^{10} + \frac{4}{9} \, B a c^{3} x^{9} + \frac{1}{2} \, A a c^{3} x^{8} + \frac{6}{7} \, B a^{2} c^{2} x^{7} + A a^{2} c^{2} x^{6} + \frac{4}{5} \, B a^{3} c x^{5} + A a^{3} c x^{4} + \frac{1}{3} \, B a^{4} x^{3} + \frac{1}{2} \, A a^{4} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.35765, size = 231, normalized size = 2.01 \begin{align*} \frac{1}{11} x^{11} c^{4} B + \frac{1}{10} x^{10} c^{4} A + \frac{4}{9} x^{9} c^{3} a B + \frac{1}{2} x^{8} c^{3} a A + \frac{6}{7} x^{7} c^{2} a^{2} B + x^{6} c^{2} a^{2} A + \frac{4}{5} x^{5} c a^{3} B + x^{4} c a^{3} A + \frac{1}{3} x^{3} a^{4} B + \frac{1}{2} x^{2} a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.083757, size = 116, normalized size = 1.01 \begin{align*} \frac{A a^{4} x^{2}}{2} + A a^{3} c x^{4} + A a^{2} c^{2} x^{6} + \frac{A a c^{3} x^{8}}{2} + \frac{A c^{4} x^{10}}{10} + \frac{B a^{4} x^{3}}{3} + \frac{4 B a^{3} c x^{5}}{5} + \frac{6 B a^{2} c^{2} x^{7}}{7} + \frac{4 B a c^{3} x^{9}}{9} + \frac{B c^{4} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15724, size = 134, normalized size = 1.17 \begin{align*} \frac{1}{11} \, B c^{4} x^{11} + \frac{1}{10} \, A c^{4} x^{10} + \frac{4}{9} \, B a c^{3} x^{9} + \frac{1}{2} \, A a c^{3} x^{8} + \frac{6}{7} \, B a^{2} c^{2} x^{7} + A a^{2} c^{2} x^{6} + \frac{4}{5} \, B a^{3} c x^{5} + A a^{3} c x^{4} + \frac{1}{3} \, B a^{4} x^{3} + \frac{1}{2} \, A a^{4} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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