3.65 \(\int (a+b x^2) (A+B x+C x^2+D x^3) \, dx\)

Optimal. Leaf size=60 \[ \frac {1}{3} x^3 (a C+A b)+a A x+\frac {1}{4} x^4 (a D+b B)+\frac {1}{2} a B x^2+\frac {1}{5} b C x^5+\frac {1}{6} b D x^6 \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 60, 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 = {1810} \[ \frac {1}{3} x^3 (a C+A b)+a A x+\frac {1}{4} x^4 (a D+b B)+\frac {1}{2} a B x^2+\frac {1}{5} b C x^5+\frac {1}{6} b D x^6 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]

[Out]

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

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 60, normalized size = 1.00 \[ \frac {1}{3} x^3 (a C+A b)+a A x+\frac {1}{4} x^4 (a D+b B)+\frac {1}{2} a B x^2+\frac {1}{5} b C x^5+\frac {1}{6} b D x^6 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")

[Out]

Exception raised: TypeError >> keys do not match self's parent

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giac [A]  time = 0.36, size = 54, normalized size = 0.90 \[ \frac {1}{6} \, D b x^{6} + \frac {1}{5} \, C b x^{5} + \frac {1}{4} \, D a x^{4} + \frac {1}{4} \, B b x^{4} + \frac {1}{3} \, C a x^{3} + \frac {1}{3} \, A b x^{3} + \frac {1}{2} \, B a x^{2} + A a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")

[Out]

1/6*D*b*x^6 + 1/5*C*b*x^5 + 1/4*D*a*x^4 + 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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maple [A]  time = 0.00, size = 51, normalized size = 0.85 \[ \frac {D b \,x^{6}}{6}+\frac {C b \,x^{5}}{5}+\frac {B a \,x^{2}}{2}+\frac {\left (b B +a D\right ) x^{4}}{4}+A a x +\frac {\left (A b +a C \right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)*(D*x^3+C*x^2+B*x+A),x)

[Out]

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

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maxima [A]  time = 1.32, size = 50, normalized size = 0.83 \[ \frac {1}{6} \, D b x^{6} + \frac {1}{5} \, C b x^{5} + \frac {1}{4} \, {\left (D a + B b\right )} x^{4} + \frac {1}{2} \, B a x^{2} + \frac {1}{3} \, {\left (C a + A b\right )} x^{3} + A a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.16, size = 54, normalized size = 0.90 \[ \frac {a\,x^4\,D}{4}+\frac {b\,x^6\,D}{6}+A\,a\,x+\frac {B\,a\,x^2}{2}+\frac {A\,b\,x^3}{3}+\frac {C\,a\,x^3}{3}+\frac {B\,b\,x^4}{4}+\frac {C\,b\,x^5}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)*(A + B*x + C*x^2 + x^3*D),x)

[Out]

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

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sympy [A]  time = 0.12, size = 56, normalized size = 0.93 \[ A a x + \frac {B a x^{2}}{2} + \frac {C b x^{5}}{5} + \frac {D b x^{6}}{6} + x^{4} \left (\frac {B b}{4} + \frac {D a}{4}\right ) + x^{3} \left (\frac {A b}{3} + \frac {C a}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*(D*x**3+C*x**2+B*x+A),x)

[Out]

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

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