3.272 \(\int x^3 (A+B x) (a+c x^2)^4 \, dx\)

Optimal. Leaf size=121 \[ \frac{3}{4} a^2 A c^2 x^8+\frac{2}{3} a^3 A c x^6+\frac{1}{4} a^4 A x^4+\frac{2}{3} a^2 B c^2 x^9+\frac{4}{7} a^3 B c x^7+\frac{1}{5} a^4 B x^5+\frac{2}{5} a A c^3 x^{10}+\frac{4}{11} a B c^3 x^{11}+\frac{1}{12} A c^4 x^{12}+\frac{1}{13} B c^4 x^{13} \]

[Out]

(a^4*A*x^4)/4 + (a^4*B*x^5)/5 + (2*a^3*A*c*x^6)/3 + (4*a^3*B*c*x^7)/7 + (3*a^2*A*c^2*x^8)/4 + (2*a^2*B*c^2*x^9
)/3 + (2*a*A*c^3*x^10)/5 + (4*a*B*c^3*x^11)/11 + (A*c^4*x^12)/12 + (B*c^4*x^13)/13

________________________________________________________________________________________

Rubi [A]  time = 0.141367, 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} \[ \frac{3}{4} a^2 A c^2 x^8+\frac{2}{3} a^3 A c x^6+\frac{1}{4} a^4 A x^4+\frac{2}{3} a^2 B c^2 x^9+\frac{4}{7} a^3 B c x^7+\frac{1}{5} a^4 B x^5+\frac{2}{5} a A c^3 x^{10}+\frac{4}{11} a B c^3 x^{11}+\frac{1}{12} A c^4 x^{12}+\frac{1}{13} B c^4 x^{13} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(A + B*x)*(a + c*x^2)^4,x]

[Out]

(a^4*A*x^4)/4 + (a^4*B*x^5)/5 + (2*a^3*A*c*x^6)/3 + (4*a^3*B*c*x^7)/7 + (3*a^2*A*c^2*x^8)/4 + (2*a^2*B*c^2*x^9
)/3 + (2*a*A*c^3*x^10)/5 + (4*a*B*c^3*x^11)/11 + (A*c^4*x^12)/12 + (B*c^4*x^13)/13

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^3 (A+B x) \left (a+c x^2\right )^4 \, dx &=\int \left (a^4 A x^3+a^4 B x^4+4 a^3 A c x^5+4 a^3 B c x^6+6 a^2 A c^2 x^7+6 a^2 B c^2 x^8+4 a A c^3 x^9+4 a B c^3 x^{10}+A c^4 x^{11}+B c^4 x^{12}\right ) \, dx\\ &=\frac{1}{4} a^4 A x^4+\frac{1}{5} a^4 B x^5+\frac{2}{3} a^3 A c x^6+\frac{4}{7} a^3 B c x^7+\frac{3}{4} a^2 A c^2 x^8+\frac{2}{3} a^2 B c^2 x^9+\frac{2}{5} a A c^3 x^{10}+\frac{4}{11} a B c^3 x^{11}+\frac{1}{12} A c^4 x^{12}+\frac{1}{13} B c^4 x^{13}\\ \end{align*}

Mathematica [A]  time = 0.0038559, size = 121, normalized size = 1. \[ \frac{3}{4} a^2 A c^2 x^8+\frac{2}{3} a^3 A c x^6+\frac{1}{4} a^4 A x^4+\frac{2}{3} a^2 B c^2 x^9+\frac{4}{7} a^3 B c x^7+\frac{1}{5} a^4 B x^5+\frac{2}{5} a A c^3 x^{10}+\frac{4}{11} a B c^3 x^{11}+\frac{1}{12} A c^4 x^{12}+\frac{1}{13} B c^4 x^{13} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(A + B*x)*(a + c*x^2)^4,x]

[Out]

(a^4*A*x^4)/4 + (a^4*B*x^5)/5 + (2*a^3*A*c*x^6)/3 + (4*a^3*B*c*x^7)/7 + (3*a^2*A*c^2*x^8)/4 + (2*a^2*B*c^2*x^9
)/3 + (2*a*A*c^3*x^10)/5 + (4*a*B*c^3*x^11)/11 + (A*c^4*x^12)/12 + (B*c^4*x^13)/13

________________________________________________________________________________________

Maple [A]  time = 0., size = 102, normalized size = 0.8 \begin{align*}{\frac{{a}^{4}A{x}^{4}}{4}}+{\frac{{a}^{4}B{x}^{5}}{5}}+{\frac{2\,{a}^{3}Ac{x}^{6}}{3}}+{\frac{4\,{a}^{3}Bc{x}^{7}}{7}}+{\frac{3\,{a}^{2}A{c}^{2}{x}^{8}}{4}}+{\frac{2\,{a}^{2}B{c}^{2}{x}^{9}}{3}}+{\frac{2\,aA{c}^{3}{x}^{10}}{5}}+{\frac{4\,aB{c}^{3}{x}^{11}}{11}}+{\frac{A{c}^{4}{x}^{12}}{12}}+{\frac{B{c}^{4}{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(B*x+A)*(c*x^2+a)^4,x)

[Out]

1/4*a^4*A*x^4+1/5*a^4*B*x^5+2/3*a^3*A*c*x^6+4/7*a^3*B*c*x^7+3/4*a^2*A*c^2*x^8+2/3*a^2*B*c^2*x^9+2/5*a*A*c^3*x^
10+4/11*a*B*c^3*x^11+1/12*A*c^4*x^12+1/13*B*c^4*x^13

________________________________________________________________________________________

Maxima [A]  time = 1.03292, size = 136, normalized size = 1.12 \begin{align*} \frac{1}{13} \, B c^{4} x^{13} + \frac{1}{12} \, A c^{4} x^{12} + \frac{4}{11} \, B a c^{3} x^{11} + \frac{2}{5} \, A a c^{3} x^{10} + \frac{2}{3} \, B a^{2} c^{2} x^{9} + \frac{3}{4} \, A a^{2} c^{2} x^{8} + \frac{4}{7} \, B a^{3} c x^{7} + \frac{2}{3} \, A a^{3} c x^{6} + \frac{1}{5} \, B a^{4} x^{5} + \frac{1}{4} \, A a^{4} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)*(c*x^2+a)^4,x, algorithm="maxima")

[Out]

1/13*B*c^4*x^13 + 1/12*A*c^4*x^12 + 4/11*B*a*c^3*x^11 + 2/5*A*a*c^3*x^10 + 2/3*B*a^2*c^2*x^9 + 3/4*A*a^2*c^2*x
^8 + 4/7*B*a^3*c*x^7 + 2/3*A*a^3*c*x^6 + 1/5*B*a^4*x^5 + 1/4*A*a^4*x^4

________________________________________________________________________________________

Fricas [A]  time = 1.26367, size = 246, normalized size = 2.03 \begin{align*} \frac{1}{13} x^{13} c^{4} B + \frac{1}{12} x^{12} c^{4} A + \frac{4}{11} x^{11} c^{3} a B + \frac{2}{5} x^{10} c^{3} a A + \frac{2}{3} x^{9} c^{2} a^{2} B + \frac{3}{4} x^{8} c^{2} a^{2} A + \frac{4}{7} x^{7} c a^{3} B + \frac{2}{3} x^{6} c a^{3} A + \frac{1}{5} x^{5} a^{4} B + \frac{1}{4} x^{4} a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)*(c*x^2+a)^4,x, algorithm="fricas")

[Out]

1/13*x^13*c^4*B + 1/12*x^12*c^4*A + 4/11*x^11*c^3*a*B + 2/5*x^10*c^3*a*A + 2/3*x^9*c^2*a^2*B + 3/4*x^8*c^2*a^2
*A + 4/7*x^7*c*a^3*B + 2/3*x^6*c*a^3*A + 1/5*x^5*a^4*B + 1/4*x^4*a^4*A

________________________________________________________________________________________

Sympy [A]  time = 0.082148, size = 124, normalized size = 1.02 \begin{align*} \frac{A a^{4} x^{4}}{4} + \frac{2 A a^{3} c x^{6}}{3} + \frac{3 A a^{2} c^{2} x^{8}}{4} + \frac{2 A a c^{3} x^{10}}{5} + \frac{A c^{4} x^{12}}{12} + \frac{B a^{4} x^{5}}{5} + \frac{4 B a^{3} c x^{7}}{7} + \frac{2 B a^{2} c^{2} x^{9}}{3} + \frac{4 B a c^{3} x^{11}}{11} + \frac{B c^{4} x^{13}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(B*x+A)*(c*x**2+a)**4,x)

[Out]

A*a**4*x**4/4 + 2*A*a**3*c*x**6/3 + 3*A*a**2*c**2*x**8/4 + 2*A*a*c**3*x**10/5 + A*c**4*x**12/12 + B*a**4*x**5/
5 + 4*B*a**3*c*x**7/7 + 2*B*a**2*c**2*x**9/3 + 4*B*a*c**3*x**11/11 + B*c**4*x**13/13

________________________________________________________________________________________

Giac [A]  time = 1.13318, size = 136, normalized size = 1.12 \begin{align*} \frac{1}{13} \, B c^{4} x^{13} + \frac{1}{12} \, A c^{4} x^{12} + \frac{4}{11} \, B a c^{3} x^{11} + \frac{2}{5} \, A a c^{3} x^{10} + \frac{2}{3} \, B a^{2} c^{2} x^{9} + \frac{3}{4} \, A a^{2} c^{2} x^{8} + \frac{4}{7} \, B a^{3} c x^{7} + \frac{2}{3} \, A a^{3} c x^{6} + \frac{1}{5} \, B a^{4} x^{5} + \frac{1}{4} \, A a^{4} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)*(c*x^2+a)^4,x, algorithm="giac")

[Out]

1/13*B*c^4*x^13 + 1/12*A*c^4*x^12 + 4/11*B*a*c^3*x^11 + 2/5*A*a*c^3*x^10 + 2/3*B*a^2*c^2*x^9 + 3/4*A*a^2*c^2*x
^8 + 4/7*B*a^3*c*x^7 + 2/3*A*a^3*c*x^6 + 1/5*B*a^4*x^5 + 1/4*A*a^4*x^4