3.140 \(\int (a+b x+c x^2)^4 (A+C x^2) \, dx\)

Optimal. Leaf size=254 \[ \frac{1}{7} x^7 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+\frac{1}{5} x^5 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+a b x^4 \left (a^2 C+A \left (3 a c+b^2\right )\right )+\frac{1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+2 a^3 A b x^2+a^4 A x+\frac{1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac{1}{2} b c x^8 \left (C \left (3 a c+b^2\right )+A c^2\right )+\frac{2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac{2}{5} b c^3 C x^{10}+\frac{1}{11} c^4 C x^{11} \]

[Out]

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

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Rubi [A]  time = 0.325565, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1657} \[ \frac{1}{7} x^7 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+\frac{1}{5} x^5 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+a b x^4 \left (a^2 C+A \left (3 a c+b^2\right )\right )+\frac{1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+2 a^3 A b x^2+a^4 A x+\frac{1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac{1}{2} b c x^8 \left (C \left (3 a c+b^2\right )+A c^2\right )+\frac{2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac{2}{5} b c^3 C x^{10}+\frac{1}{11} c^4 C x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4*(A + C*x^2),x]

[Out]

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

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

Mathematica [A]  time = 0.0970406, size = 256, normalized size = 1.01 \[ \frac{1}{7} x^7 \left (6 a^2 c^2 C+4 a A c^3+12 a b^2 c C+6 A b^2 c^2+b^4 C\right )+\frac{1}{5} x^5 \left (6 a^2 A c^2+6 a^2 b^2 C+4 a^3 c C+12 a A b^2 c+A b^4\right )+a b x^4 \left (a^2 C+3 a A c+A b^2\right )+\frac{1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+2 a^3 A b x^2+a^4 A x+\frac{1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac{1}{2} b c x^8 \left (3 a c C+A c^2+b^2 C\right )+\frac{2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac{2}{5} b c^3 C x^{10}+\frac{1}{11} c^4 C x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 343, normalized size = 1.4 \begin{align*}{\frac{{c}^{4}C{x}^{11}}{11}}+{\frac{2\,b{c}^{3}C{x}^{10}}{5}}+{\frac{ \left ( \left ( 2\, \left ( 2\,ac+{b}^{2} \right ){c}^{2}+4\,{b}^{2}{c}^{2} \right ) C+{c}^{4}A \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,ba{c}^{2}+4\, \left ( 2\,ac+{b}^{2} \right ) bc \right ) C+4\,b{c}^{3}A \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 2\,{a}^{2}{c}^{2}+8\,ac{b}^{2}+ \left ( 2\,ac+{b}^{2} \right ) ^{2} \right ) C+ \left ( 2\, \left ( 2\,ac+{b}^{2} \right ){c}^{2}+4\,{b}^{2}{c}^{2} \right ) A \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,{a}^{2}bc+4\,ab \left ( 2\,ac+{b}^{2} \right ) \right ) C+ \left ( 4\,ba{c}^{2}+4\, \left ( 2\,ac+{b}^{2} \right ) bc \right ) A \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 2\,{a}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,{b}^{2}{a}^{2} \right ) C+ \left ( 2\,{a}^{2}{c}^{2}+8\,ac{b}^{2}+ \left ( 2\,ac+{b}^{2} \right ) ^{2} \right ) A \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{3}bC+ \left ( 4\,{a}^{2}bc+4\,ab \left ( 2\,ac+{b}^{2} \right ) \right ) A \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{4}C+ \left ( 2\,{a}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,{b}^{2}{a}^{2} \right ) A \right ){x}^{3}}{3}}+2\,{a}^{3}Ab{x}^{2}+{a}^{4}Ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^4*C*x^11+2/5*b*c^3*C*x^10+1/9*((2*(2*a*c+b^2)*c^2+4*b^2*c^2)*C+c^4*A)*x^9+1/8*((4*b*a*c^2+4*(2*a*c+b^2)
*b*c)*C+4*b*c^3*A)*x^8+1/7*((2*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)^2)*C+(2*(2*a*c+b^2)*c^2+4*b^2*c^2)*A)*x^7+1/6*((4
*a^2*b*c+4*a*b*(2*a*c+b^2))*C+(4*b*a*c^2+4*(2*a*c+b^2)*b*c)*A)*x^6+1/5*((2*a^2*(2*a*c+b^2)+4*b^2*a^2)*C+(2*a^2
*c^2+8*a*c*b^2+(2*a*c+b^2)^2)*A)*x^5+1/4*(4*a^3*b*C+(4*a^2*b*c+4*a*b*(2*a*c+b^2))*A)*x^4+1/3*(a^4*C+(2*a^2*(2*
a*c+b^2)+4*b^2*a^2)*A)*x^3+2*a^3*A*b*x^2+a^4*A*x

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Maxima [A]  time = 1.00883, size = 355, normalized size = 1.4 \begin{align*} \frac{1}{11} \, C c^{4} x^{11} + \frac{2}{5} \, C b c^{3} x^{10} + \frac{1}{9} \,{\left (6 \, C b^{2} c^{2} + 4 \, C a c^{3} + A c^{4}\right )} x^{9} + \frac{1}{2} \,{\left (C b^{3} c + 3 \, C a b c^{2} + A b c^{3}\right )} x^{8} + \frac{1}{7} \,{\left (C b^{4} + 12 \, C a b^{2} c + 4 \, A a c^{3} + 6 \,{\left (C a^{2} + A b^{2}\right )} c^{2}\right )} x^{7} + 2 \, A a^{3} b x^{2} + \frac{2}{3} \,{\left (C a b^{3} + 3 \, A a b c^{2} +{\left (3 \, C a^{2} b + A b^{3}\right )} c\right )} x^{6} + A a^{4} x + \frac{1}{5} \,{\left (6 \, C a^{2} b^{2} + A b^{4} + 6 \, A a^{2} c^{2} + 4 \,{\left (C a^{3} + 3 \, A a b^{2}\right )} c\right )} x^{5} +{\left (C a^{3} b + A a b^{3} + 3 \, A a^{2} b c\right )} x^{4} + \frac{1}{3} \,{\left (C a^{4} + 6 \, A a^{2} b^{2} + 4 \, A a^{3} c\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.22246, size = 718, normalized size = 2.83 \begin{align*} \frac{1}{11} x^{11} c^{4} C + \frac{2}{5} x^{10} c^{3} b C + \frac{2}{3} x^{9} c^{2} b^{2} C + \frac{4}{9} x^{9} c^{3} a C + \frac{1}{9} x^{9} c^{4} A + \frac{1}{2} x^{8} c b^{3} C + \frac{3}{2} x^{8} c^{2} b a C + \frac{1}{2} x^{8} c^{3} b A + \frac{1}{7} x^{7} b^{4} C + \frac{12}{7} x^{7} c b^{2} a C + \frac{6}{7} x^{7} c^{2} a^{2} C + \frac{6}{7} x^{7} c^{2} b^{2} A + \frac{4}{7} x^{7} c^{3} a A + \frac{2}{3} x^{6} b^{3} a C + 2 x^{6} c b a^{2} C + \frac{2}{3} x^{6} c b^{3} A + 2 x^{6} c^{2} b a A + \frac{6}{5} x^{5} b^{2} a^{2} C + \frac{4}{5} x^{5} c a^{3} C + \frac{1}{5} x^{5} b^{4} A + \frac{12}{5} x^{5} c b^{2} a A + \frac{6}{5} x^{5} c^{2} a^{2} A + x^{4} b a^{3} C + x^{4} b^{3} a A + 3 x^{4} c b a^{2} A + \frac{1}{3} x^{3} a^{4} C + 2 x^{3} b^{2} a^{2} A + \frac{4}{3} x^{3} c a^{3} A + 2 x^{2} b a^{3} A + x a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.112548, size = 320, normalized size = 1.26 \begin{align*} A a^{4} x + 2 A a^{3} b x^{2} + \frac{2 C b c^{3} x^{10}}{5} + \frac{C c^{4} x^{11}}{11} + x^{9} \left (\frac{A c^{4}}{9} + \frac{4 C a c^{3}}{9} + \frac{2 C b^{2} c^{2}}{3}\right ) + x^{8} \left (\frac{A b c^{3}}{2} + \frac{3 C a b c^{2}}{2} + \frac{C b^{3} c}{2}\right ) + x^{7} \left (\frac{4 A a c^{3}}{7} + \frac{6 A b^{2} c^{2}}{7} + \frac{6 C a^{2} c^{2}}{7} + \frac{12 C a b^{2} c}{7} + \frac{C b^{4}}{7}\right ) + x^{6} \left (2 A a b c^{2} + \frac{2 A b^{3} c}{3} + 2 C a^{2} b c + \frac{2 C a b^{3}}{3}\right ) + x^{5} \left (\frac{6 A a^{2} c^{2}}{5} + \frac{12 A a b^{2} c}{5} + \frac{A b^{4}}{5} + \frac{4 C a^{3} c}{5} + \frac{6 C a^{2} b^{2}}{5}\right ) + x^{4} \left (3 A a^{2} b c + A a b^{3} + C a^{3} b\right ) + x^{3} \left (\frac{4 A a^{3} c}{3} + 2 A a^{2} b^{2} + \frac{C a^{4}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1739, size = 416, normalized size = 1.64 \begin{align*} \frac{1}{11} \, C c^{4} x^{11} + \frac{2}{5} \, C b c^{3} x^{10} + \frac{2}{3} \, C b^{2} c^{2} x^{9} + \frac{4}{9} \, C a c^{3} x^{9} + \frac{1}{9} \, A c^{4} x^{9} + \frac{1}{2} \, C b^{3} c x^{8} + \frac{3}{2} \, C a b c^{2} x^{8} + \frac{1}{2} \, A b c^{3} x^{8} + \frac{1}{7} \, C b^{4} x^{7} + \frac{12}{7} \, C a b^{2} c x^{7} + \frac{6}{7} \, C a^{2} c^{2} x^{7} + \frac{6}{7} \, A b^{2} c^{2} x^{7} + \frac{4}{7} \, A a c^{3} x^{7} + \frac{2}{3} \, C a b^{3} x^{6} + 2 \, C a^{2} b c x^{6} + \frac{2}{3} \, A b^{3} c x^{6} + 2 \, A a b c^{2} x^{6} + \frac{6}{5} \, C a^{2} b^{2} x^{5} + \frac{1}{5} \, A b^{4} x^{5} + \frac{4}{5} \, C a^{3} c x^{5} + \frac{12}{5} \, A a b^{2} c x^{5} + \frac{6}{5} \, A a^{2} c^{2} x^{5} + C a^{3} b x^{4} + A a b^{3} x^{4} + 3 \, A a^{2} b c x^{4} + \frac{1}{3} \, C a^{4} x^{3} + 2 \, A a^{2} b^{2} x^{3} + \frac{4}{3} \, A a^{3} c x^{3} + 2 \, A a^{3} b x^{2} + A a^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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