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 + (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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Rubi [A]  time = 0.0411648, antiderivative size = 60, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0089797, size = 60, normalized size = 1. \[ \frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{4} x^4 (a D+b B)+\frac{1}{2} a B x^2+\frac{1}{5} b C x^5+\frac{1}{6} b D x^6 \]

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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Maple [A]  time = 0.001, size = 51, normalized size = 0.9 \begin{align*} aAx+{\frac{Ba{x}^{2}}{2}}+{\frac{ \left ( Ab+aC \right ){x}^{3}}{3}}+{\frac{ \left ( Bb+aD \right ){x}^{4}}{4}}+{\frac{bC{x}^{5}}{5}}+{\frac{bD{x}^{6}}{6}} \end{align*}

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 = 1.03893, size = 68, normalized size = 1.13 \begin{align*} \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 \end{align*}

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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Fricas [A]  time = 1.28853, size = 142, normalized size = 2.37 \begin{align*} \frac{1}{6} x^{6} b D + \frac{1}{5} x^{5} b C + \frac{1}{4} x^{4} a D + \frac{1}{4} x^{4} b B + \frac{1}{3} x^{3} a C + \frac{1}{3} x^{3} b A + \frac{1}{2} x^{2} a B + x a A \end{align*}

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]

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

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Sympy [A]  time = 0.062735, size = 56, normalized size = 0.93 \begin{align*} 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 ) \end{align*}

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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Giac [A]  time = 1.15165, size = 73, normalized size = 1.22 \begin{align*} \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 \end{align*}

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