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

Optimal. Leaf size=159 \[ \frac {1}{2} a^2 A x^2+\frac {1}{3} a^2 B x^3+\frac {1}{8} x^8 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac {1}{6} x^6 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac {1}{4} a x^4 (a C+2 A b)+\frac {1}{7} B x^7 \left (2 a c+b^2\right )+\frac {2}{5} a b B x^5+\frac {1}{10} c x^{10} (A c+2 b C)+\frac {2}{9} b B c x^9+\frac {1}{11} B c^2 x^{11}+\frac {1}{12} c^2 C x^{12} \]

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Rubi [A]  time = 0.14, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1628} \begin {gather*} \frac {1}{2} a^2 A x^2+\frac {1}{3} a^2 B x^3+\frac {1}{8} x^8 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac {1}{6} x^6 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac {1}{4} a x^4 (a C+2 A b)+\frac {1}{7} B x^7 \left (2 a c+b^2\right )+\frac {2}{5} a b B x^5+\frac {1}{10} c x^{10} (A c+2 b C)+\frac {2}{9} b B c x^9+\frac {1}{11} B c^2 x^{11}+\frac {1}{12} c^2 C x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 159, normalized size = 1.00 \begin {gather*} \frac {1}{2} a^2 A x^2+\frac {1}{3} a^2 B x^3+\frac {1}{8} x^8 \left (2 a c C+2 A b c+b^2 C\right )+\frac {1}{6} x^6 \left (2 a A c+2 a b C+A b^2\right )+\frac {1}{4} a x^4 (a C+2 A b)+\frac {1}{7} B x^7 \left (2 a c+b^2\right )+\frac {2}{5} a b B x^5+\frac {1}{10} c x^{10} (A c+2 b C)+\frac {2}{9} b B c x^9+\frac {1}{11} B c^2 x^{11}+\frac {1}{12} c^2 C x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

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fricas [A]  time = 1.09, size = 154, normalized size = 0.97 \begin {gather*} \frac {1}{12} x^{12} c^{2} C + \frac {1}{11} x^{11} c^{2} B + \frac {1}{5} x^{10} c b C + \frac {1}{10} x^{10} c^{2} A + \frac {2}{9} x^{9} c b B + \frac {1}{8} x^{8} b^{2} C + \frac {1}{4} x^{8} c a C + \frac {1}{4} x^{8} c b A + \frac {1}{7} x^{7} b^{2} B + \frac {2}{7} x^{7} c a B + \frac {1}{3} x^{6} b a C + \frac {1}{6} x^{6} b^{2} A + \frac {1}{3} x^{6} c a A + \frac {2}{5} x^{5} b a B + \frac {1}{4} x^{4} a^{2} C + \frac {1}{2} x^{4} b a A + \frac {1}{3} x^{3} a^{2} B + \frac {1}{2} x^{2} a^{2} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.31, size = 154, normalized size = 0.97 \begin {gather*} \frac {1}{12} \, C c^{2} x^{12} + \frac {1}{11} \, B c^{2} x^{11} + \frac {1}{5} \, C b c x^{10} + \frac {1}{10} \, A c^{2} x^{10} + \frac {2}{9} \, B b c x^{9} + \frac {1}{8} \, C b^{2} x^{8} + \frac {1}{4} \, C a c x^{8} + \frac {1}{4} \, A b c x^{8} + \frac {1}{7} \, B b^{2} x^{7} + \frac {2}{7} \, B a c x^{7} + \frac {1}{3} \, C a b x^{6} + \frac {1}{6} \, A b^{2} x^{6} + \frac {1}{3} \, A a c x^{6} + \frac {2}{5} \, B a b x^{5} + \frac {1}{4} \, C a^{2} x^{4} + \frac {1}{2} \, A a b x^{4} + \frac {1}{3} \, B a^{2} x^{3} + \frac {1}{2} \, A a^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 142, normalized size = 0.89 \begin {gather*} \frac {C \,c^{2} x^{12}}{12}+\frac {B \,c^{2} x^{11}}{11}+\frac {2 B b c \,x^{9}}{9}+\frac {\left (A \,c^{2}+2 C b c \right ) x^{10}}{10}+\frac {2 B a b \,x^{5}}{5}+\frac {\left (2 a c +b^{2}\right ) B \,x^{7}}{7}+\frac {\left (2 A b c +\left (2 a c +b^{2}\right ) C \right ) x^{8}}{8}+\frac {B \,a^{2} x^{3}}{3}+\frac {\left (2 C a b +\left (2 a c +b^{2}\right ) A \right ) x^{6}}{6}+\frac {A \,a^{2} x^{2}}{2}+\frac {\left (2 A a b +C \,a^{2}\right ) x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.18, size = 143, normalized size = 0.90 \begin {gather*} \frac {1}{12} \, C c^{2} x^{12} + \frac {1}{11} \, B c^{2} x^{11} + \frac {2}{9} \, B b c x^{9} + \frac {1}{10} \, {\left (2 \, C b c + A c^{2}\right )} x^{10} + \frac {1}{8} \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{8} + \frac {2}{5} \, B a b x^{5} + \frac {1}{7} \, {\left (B b^{2} + 2 \, B a c\right )} x^{7} + \frac {1}{6} \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{6} + \frac {1}{3} \, B a^{2} x^{3} + \frac {1}{2} \, A a^{2} x^{2} + \frac {1}{4} \, {\left (C a^{2} + 2 \, A a b\right )} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 141, normalized size = 0.89 \begin {gather*} x^4\,\left (\frac {C\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^{10}\,\left (\frac {A\,c^2}{10}+\frac {C\,b\,c}{5}\right )+x^6\,\left (\frac {A\,b^2}{6}+\frac {C\,a\,b}{3}+\frac {A\,a\,c}{3}\right )+x^8\,\left (\frac {C\,b^2}{8}+\frac {A\,c\,b}{4}+\frac {C\,a\,c}{4}\right )+\frac {A\,a^2\,x^2}{2}+\frac {B\,a^2\,x^3}{3}+\frac {B\,c^2\,x^{11}}{11}+\frac {C\,c^2\,x^{12}}{12}+\frac {B\,x^7\,\left (b^2+2\,a\,c\right )}{7}+\frac {2\,B\,a\,b\,x^5}{5}+\frac {2\,B\,b\,c\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.09, size = 163, normalized size = 1.03 \begin {gather*} \frac {A a^{2} x^{2}}{2} + \frac {B a^{2} x^{3}}{3} + \frac {2 B a b x^{5}}{5} + \frac {2 B b c x^{9}}{9} + \frac {B c^{2} x^{11}}{11} + \frac {C c^{2} x^{12}}{12} + x^{10} \left (\frac {A c^{2}}{10} + \frac {C b c}{5}\right ) + x^{8} \left (\frac {A b c}{4} + \frac {C a c}{4} + \frac {C b^{2}}{8}\right ) + x^{7} \left (\frac {2 B a c}{7} + \frac {B b^{2}}{7}\right ) + x^{6} \left (\frac {A a c}{3} + \frac {A b^{2}}{6} + \frac {C a b}{3}\right ) + x^{4} \left (\frac {A a b}{2} + \frac {C a^{2}}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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