### 3.13 $$\int (A+B x+C x^2) (a+b x^2+c x^4)^2 \, dx$$

Optimal. Leaf size=154 $a^2 A x+\frac{1}{2} a^2 B x^2+\frac{1}{7} x^7 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac{1}{3} a x^3 (a C+2 A b)+\frac{1}{6} B x^6 \left (2 a c+b^2\right )+\frac{1}{2} a b B x^4+\frac{1}{9} c x^9 (A c+2 b C)+\frac{1}{4} b B c x^8+\frac{1}{10} B c^2 x^{10}+\frac{1}{11} c^2 C x^{11}$

[Out]

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

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Rubi [A]  time = 0.110979, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.04, Rules used = {1657} $a^2 A x+\frac{1}{2} a^2 B x^2+\frac{1}{7} x^7 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac{1}{3} a x^3 (a C+2 A b)+\frac{1}{6} B x^6 \left (2 a c+b^2\right )+\frac{1}{2} a b B x^4+\frac{1}{9} c x^9 (A c+2 b C)+\frac{1}{4} b B c x^8+\frac{1}{10} B c^2 x^{10}+\frac{1}{11} c^2 C x^{11}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0305159, size = 154, normalized size = 1. $a^2 A x+\frac{1}{2} a^2 B x^2+\frac{1}{7} x^7 \left (2 a c C+2 A b c+b^2 C\right )+\frac{1}{5} x^5 \left (2 a A c+2 a b C+A b^2\right )+\frac{1}{3} a x^3 (a C+2 A b)+\frac{1}{6} B x^6 \left (2 a c+b^2\right )+\frac{1}{2} a b B x^4+\frac{1}{9} c x^9 (A c+2 b C)+\frac{1}{4} b B c x^8+\frac{1}{10} B c^2 x^{10}+\frac{1}{11} c^2 C x^{11}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 139, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}C{x}^{11}}{11}}+{\frac{B{c}^{2}{x}^{10}}{10}}+{\frac{ \left ( A{c}^{2}+2\,Cbc \right ){x}^{9}}{9}}+{\frac{bBc{x}^{8}}{4}}+{\frac{ \left ( 2\,Abc+ \left ( 2\,ac+{b}^{2} \right ) C \right ){x}^{7}}{7}}+{\frac{B \left ( 2\,ac+{b}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( A \left ( 2\,ac+{b}^{2} \right ) +2\,abC \right ){x}^{5}}{5}}+{\frac{abB{x}^{4}}{2}}+{\frac{ \left ( 2\,Aab+C{a}^{2} \right ){x}^{3}}{3}}+{\frac{{a}^{2}B{x}^{2}}{2}}+{a}^{2}Ax \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.972651, size = 189, normalized size = 1.23 \begin{align*} \frac{1}{11} \, C c^{2} x^{11} + \frac{1}{10} \, B c^{2} x^{10} + \frac{1}{4} \, B b c x^{8} + \frac{1}{9} \,{\left (2 \, C b c + A c^{2}\right )} x^{9} + \frac{1}{7} \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{7} + \frac{1}{2} \, B a b x^{4} + \frac{1}{6} \,{\left (B b^{2} + 2 \, B a c\right )} x^{6} + \frac{1}{5} \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{5} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x + \frac{1}{3} \,{\left (C a^{2} + 2 \, A a b\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.10337, size = 385, normalized size = 2.5 \begin{align*} \frac{1}{11} x^{11} c^{2} C + \frac{1}{10} x^{10} c^{2} B + \frac{2}{9} x^{9} c b C + \frac{1}{9} x^{9} c^{2} A + \frac{1}{4} x^{8} c b B + \frac{1}{7} x^{7} b^{2} C + \frac{2}{7} x^{7} c a C + \frac{2}{7} x^{7} c b A + \frac{1}{6} x^{6} b^{2} B + \frac{1}{3} x^{6} c a B + \frac{2}{5} x^{5} b a C + \frac{1}{5} x^{5} b^{2} A + \frac{2}{5} x^{5} c a A + \frac{1}{2} x^{4} b a B + \frac{1}{3} x^{3} a^{2} C + \frac{2}{3} x^{3} b a A + \frac{1}{2} x^{2} a^{2} B + x a^{2} A \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.095202, size = 165, normalized size = 1.07 \begin{align*} A a^{2} x + \frac{B a^{2} x^{2}}{2} + \frac{B a b x^{4}}{2} + \frac{B b c x^{8}}{4} + \frac{B c^{2} x^{10}}{10} + \frac{C c^{2} x^{11}}{11} + x^{9} \left (\frac{A c^{2}}{9} + \frac{2 C b c}{9}\right ) + x^{7} \left (\frac{2 A b c}{7} + \frac{2 C a c}{7} + \frac{C b^{2}}{7}\right ) + x^{6} \left (\frac{B a c}{3} + \frac{B b^{2}}{6}\right ) + x^{5} \left (\frac{2 A a c}{5} + \frac{A b^{2}}{5} + \frac{2 C a b}{5}\right ) + x^{3} \left (\frac{2 A a b}{3} + \frac{C a^{2}}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09106, size = 204, normalized size = 1.32 \begin{align*} \frac{1}{11} \, C c^{2} x^{11} + \frac{1}{10} \, B c^{2} x^{10} + \frac{2}{9} \, C b c x^{9} + \frac{1}{9} \, A c^{2} x^{9} + \frac{1}{4} \, B b c x^{8} + \frac{1}{7} \, C b^{2} x^{7} + \frac{2}{7} \, C a c x^{7} + \frac{2}{7} \, A b c x^{7} + \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{3} \, B a c x^{6} + \frac{2}{5} \, C a b x^{5} + \frac{1}{5} \, A b^{2} x^{5} + \frac{2}{5} \, A a c x^{5} + \frac{1}{2} \, B a b x^{4} + \frac{1}{3} \, C a^{2} x^{3} + \frac{2}{3} \, A a b x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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