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

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

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1657} \begin {gather*} 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} \end {gather*}

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 = 0.03, size = 154, normalized size = 1.00 \begin {gather*} 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} \end {gather*}

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.98, size = 151, normalized size = 0.98 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

giac [A]  time = 0.30, size = 151, normalized size = 0.98 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 139, normalized size = 0.90 \begin {gather*} \frac {C \,c^{2} x^{11}}{11}+\frac {B \,c^{2} x^{10}}{10}+\frac {B b c \,x^{8}}{4}+\frac {\left (A \,c^{2}+2 C b c \right ) x^{9}}{9}+\frac {B a b \,x^{4}}{2}+\frac {\left (2 a c +b^{2}\right ) B \,x^{6}}{6}+\frac {\left (2 A b c +\left (2 a c +b^{2}\right ) C \right ) x^{7}}{7}+\frac {B \,a^{2} x^{2}}{2}+\frac {\left (2 C a b +\left (2 a c +b^{2}\right ) A \right ) x^{5}}{5}+A \,a^{2} x +\frac {\left (2 A a b +C \,a^{2}\right ) x^{3}}{3} \end {gather*}

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*(2*C*a*b+(2*a*c+b^2)*A)*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

________________________________________________________________________________________

maxima [A]  time = 0.63, size = 140, normalized size = 0.91 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 138, normalized size = 0.90 \begin {gather*} x^3\,\left (\frac {C\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )+x^9\,\left (\frac {A\,c^2}{9}+\frac {2\,C\,b\,c}{9}\right )+x^5\,\left (\frac {A\,b^2}{5}+\frac {2\,C\,a\,b}{5}+\frac {2\,A\,a\,c}{5}\right )+x^7\,\left (\frac {C\,b^2}{7}+\frac {2\,A\,c\,b}{7}+\frac {2\,C\,a\,c}{7}\right )+\frac {B\,a^2\,x^2}{2}+\frac {B\,c^2\,x^{10}}{10}+\frac {C\,c^2\,x^{11}}{11}+\frac {B\,x^6\,\left (b^2+2\,a\,c\right )}{6}+A\,a^2\,x+\frac {B\,a\,b\,x^4}{2}+\frac {B\,b\,c\,x^8}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 165, normalized size = 1.07 \begin {gather*} 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 {gather*}

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)

________________________________________________________________________________________