3.1.3 \(\int (A+B x+C x^2) (a+b x^2+c x^4) \, dx\) [3]

Optimal. Leaf size=69 \[ a A x+\frac {1}{2} a B x^2+\frac {1}{3} (A b+a C) x^3+\frac {1}{4} b B x^4+\frac {1}{5} (A c+b C) x^5+\frac {1}{6} B c x^6+\frac {1}{7} c C x^7 \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1671} \begin {gather*} \frac {1}{3} x^3 (a C+A b)+a A x+\frac {1}{2} a B x^2+\frac {1}{5} x^5 (A c+b C)+\frac {1}{4} b B x^4+\frac {1}{6} B c x^6+\frac {1}{7} c C x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1671

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

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Mathematica [A]
time = 0.01, size = 69, normalized size = 1.00 \begin {gather*} a A x+\frac {1}{2} a B x^2+\frac {1}{3} (A b+a C) x^3+\frac {1}{4} b B x^4+\frac {1}{5} (A c+b C) x^5+\frac {1}{6} B c x^6+\frac {1}{7} c C x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.04, size = 58, normalized size = 0.84

method	result	size
default	\(a A x +\frac {a B \,x^{2}}{2}+\frac {\left (A b +a C \right ) x^{3}}{3}+\frac {b B \,x^{4}}{4}+\frac {\left (A c +b C \right ) x^{5}}{5}+\frac {B c \,x^{6}}{6}+\frac {c C \,x^{7}}{7}\)	\(58\)
norman	\(\frac {c C \,x^{7}}{7}+\frac {B c \,x^{6}}{6}+\left (\frac {A c}{5}+\frac {b C}{5}\right ) x^{5}+\frac {b B \,x^{4}}{4}+\left (\frac {A b}{3}+\frac {a C}{3}\right ) x^{3}+\frac {a B \,x^{2}}{2}+a A x\)	\(60\)
gosper	\(\frac {1}{7} c C \,x^{7}+\frac {1}{6} B c \,x^{6}+\frac {1}{5} x^{5} A c +\frac {1}{5} x^{5} b C +\frac {1}{4} b B \,x^{4}+\frac {1}{3} x^{3} A b +\frac {1}{3} x^{3} a C +\frac {1}{2} a B \,x^{2}+a A x\)	\(62\)
risch	\(\frac {1}{7} c C \,x^{7}+\frac {1}{6} B c \,x^{6}+\frac {1}{5} x^{5} A c +\frac {1}{5} x^{5} b C +\frac {1}{4} b B \,x^{4}+\frac {1}{3} x^{3} A b +\frac {1}{3} x^{3} a C +\frac {1}{2} a B \,x^{2}+a A x\)	\(62\)

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),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{7} \, C c x^{7} + \frac {1}{6} \, B c x^{6} + \frac {1}{4} \, B b x^{4} + \frac {1}{5} \, {\left (C b + A c\right )} x^{5} + \frac {1}{2} \, B a x^{2} + \frac {1}{3} \, {\left (C a + A b\right )} x^{3} + A a 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.38, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{7} \, C c x^{7} + \frac {1}{6} \, B c x^{6} + \frac {1}{4} \, B b x^{4} + \frac {1}{5} \, {\left (C b + A c\right )} x^{5} + \frac {1}{2} \, B a x^{2} + \frac {1}{3} \, {\left (C a + A b\right )} x^{3} + A a 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.01, size = 65, normalized size = 0.94 \begin {gather*} A a x + \frac {B a x^{2}}{2} + \frac {B b x^{4}}{4} + \frac {B c x^{6}}{6} + \frac {C c x^{7}}{7} + x^{5} \left (\frac {A c}{5} + \frac {C b}{5}\right ) + x^{3} \left (\frac {A b}{3} + \frac {C a}{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),x)

[Out]

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

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Giac [A]
time = 6.32, size = 61, normalized size = 0.88 \begin {gather*} \frac {1}{7} \, C c x^{7} + \frac {1}{6} \, B c x^{6} + \frac {1}{5} \, C b x^{5} + \frac {1}{5} \, A c x^{5} + \frac {1}{4} \, B b x^{4} + \frac {1}{3} \, C a x^{3} + \frac {1}{3} \, A b x^{3} + \frac {1}{2} \, B a x^{2} + A a 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),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.03, size = 59, normalized size = 0.86 \begin {gather*} \frac {C\,c\,x^7}{7}+\frac {B\,c\,x^6}{6}+\left (\frac {A\,c}{5}+\frac {C\,b}{5}\right )\,x^5+\frac {B\,b\,x^4}{4}+\left (\frac {A\,b}{3}+\frac {C\,a}{3}\right )\,x^3+\frac {B\,a\,x^2}{2}+A\,a\,x \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),x)

[Out]

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

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