3.1.1 \(\int x^2 (A+B x+C x^2) (a+b x^2+c x^4) \, dx\) [1]

Optimal. Leaf size=74 \[ \frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{5} (A b+a C) x^5+\frac {1}{6} b B x^6+\frac {1}{7} (A c+b C) x^7+\frac {1}{8} B c x^8+\frac {1}{9} c C x^9 \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 74, 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 = {1642} \begin {gather*} \frac {1}{5} x^5 (a C+A b)+\frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{7} x^7 (A c+b C)+\frac {1}{6} b B x^6+\frac {1}{8} B c x^8+\frac {1}{9} c C x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1642

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

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Mathematica [A]
time = 0.01, size = 74, normalized size = 1.00 \begin {gather*} \frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{5} (A b+a C) x^5+\frac {1}{6} b B x^6+\frac {1}{7} (A c+b C) x^7+\frac {1}{8} B c x^8+\frac {1}{9} c C x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.07, size = 61, normalized size = 0.82

method	result	size
default	\(\frac {a A \,x^{3}}{3}+\frac {a B \,x^{4}}{4}+\frac {\left (A b +a C \right ) x^{5}}{5}+\frac {b B \,x^{6}}{6}+\frac {\left (A c +b C \right ) x^{7}}{7}+\frac {B c \,x^{8}}{8}+\frac {c C \,x^{9}}{9}\)	\(61\)
norman	\(\frac {c C \,x^{9}}{9}+\frac {B c \,x^{8}}{8}+\left (\frac {A c}{7}+\frac {b C}{7}\right ) x^{7}+\frac {b B \,x^{6}}{6}+\left (\frac {A b}{5}+\frac {a C}{5}\right ) x^{5}+\frac {a B \,x^{4}}{4}+\frac {a A \,x^{3}}{3}\)	\(63\)
gosper	\(\frac {1}{9} c C \,x^{9}+\frac {1}{8} B c \,x^{8}+\frac {1}{7} x^{7} A c +\frac {1}{7} x^{7} b C +\frac {1}{6} b B \,x^{6}+\frac {1}{5} x^{5} A b +\frac {1}{5} x^{5} a C +\frac {1}{4} a B \,x^{4}+\frac {1}{3} a A \,x^{3}\)	\(65\)
risch	\(\frac {1}{9} c C \,x^{9}+\frac {1}{8} B c \,x^{8}+\frac {1}{7} x^{7} A c +\frac {1}{7} x^{7} b C +\frac {1}{6} b B \,x^{6}+\frac {1}{5} x^{5} A b +\frac {1}{5} x^{5} a C +\frac {1}{4} a B \,x^{4}+\frac {1}{3} a A \,x^{3}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 60, normalized size = 0.81 \begin {gather*} \frac {1}{9} \, C c x^{9} + \frac {1}{8} \, B c x^{8} + \frac {1}{6} \, B b x^{6} + \frac {1}{7} \, {\left (C b + A c\right )} x^{7} + \frac {1}{4} \, B a x^{4} + \frac {1}{5} \, {\left (C a + A b\right )} x^{5} + \frac {1}{3} \, A a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 60, normalized size = 0.81 \begin {gather*} \frac {1}{9} \, C c x^{9} + \frac {1}{8} \, B c x^{8} + \frac {1}{6} \, B b x^{6} + \frac {1}{7} \, {\left (C b + A c\right )} x^{7} + \frac {1}{4} \, B a x^{4} + \frac {1}{5} \, {\left (C a + A b\right )} x^{5} + \frac {1}{3} \, A a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 68, normalized size = 0.92 \begin {gather*} \frac {A a x^{3}}{3} + \frac {B a x^{4}}{4} + \frac {B b x^{6}}{6} + \frac {B c x^{8}}{8} + \frac {C c x^{9}}{9} + x^{7} \left (\frac {A c}{7} + \frac {C b}{7}\right ) + x^{5} \left (\frac {A b}{5} + \frac {C a}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.79, size = 64, normalized size = 0.86 \begin {gather*} \frac {1}{9} \, C c x^{9} + \frac {1}{8} \, B c x^{8} + \frac {1}{7} \, C b x^{7} + \frac {1}{7} \, A c x^{7} + \frac {1}{6} \, B b x^{6} + \frac {1}{5} \, C a x^{5} + \frac {1}{5} \, A b x^{5} + \frac {1}{4} \, B a x^{4} + \frac {1}{3} \, A a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 62, normalized size = 0.84 \begin {gather*} \frac {C\,c\,x^9}{9}+\frac {B\,c\,x^8}{8}+\left (\frac {A\,c}{7}+\frac {C\,b}{7}\right )\,x^7+\frac {B\,b\,x^6}{6}+\left (\frac {A\,b}{5}+\frac {C\,a}{5}\right )\,x^5+\frac {B\,a\,x^4}{4}+\frac {A\,a\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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