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

Optimal. Leaf size=65 \[ \frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{6} x^6 (a D+b B)+\frac{1}{4} a B x^4+\frac{1}{7} b C x^7+\frac{1}{8} b D x^8 \]

[Out]

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

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Rubi [A]  time = 0.0764878, antiderivative size = 65, normalized size of antiderivative = 1., 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 = {1802} \[ \frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{6} x^6 (a D+b B)+\frac{1}{4} a B x^4+\frac{1}{7} b C x^7+\frac{1}{8} b D x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1802

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

Rubi steps

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

Mathematica [A]  time = 0.0155026, size = 65, normalized size = 1. \[ \frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{6} x^6 (a D+b B)+\frac{1}{4} a B x^4+\frac{1}{7} b C x^7+\frac{1}{8} b D x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 54, normalized size = 0.8 \begin{align*}{\frac{aA{x}^{3}}{3}}+{\frac{aB{x}^{4}}{4}}+{\frac{ \left ( Ab+aC \right ){x}^{5}}{5}}+{\frac{ \left ( Bb+aD \right ){x}^{6}}{6}}+{\frac{bC{x}^{7}}{7}}+{\frac{bD{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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+D*a)*x^6+1/7*b*C*x^7+1/8*b*D*x^8

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Maxima [A]  time = 1.05344, size = 72, normalized size = 1.11 \begin{align*} \frac{1}{8} \, D b x^{8} + \frac{1}{7} \, C b x^{7} + \frac{1}{6} \,{\left (D a + B b\right )} x^{6} + \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*D*b*x^8 + 1/7*C*b*x^7 + 1/6*(D*a + B*b)*x^6 + 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 = 1.23049, size = 150, normalized size = 2.31 \begin{align*} \frac{1}{8} x^{8} b D + \frac{1}{7} x^{7} b C + \frac{1}{6} x^{6} a D + \frac{1}{6} x^{6} b B + \frac{1}{5} x^{5} a C + \frac{1}{5} x^{5} b A + \frac{1}{4} x^{4} a B + \frac{1}{3} x^{3} a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.06503, size = 60, normalized size = 0.92 \begin{align*} \frac{A a x^{3}}{3} + \frac{B a x^{4}}{4} + \frac{C b x^{7}}{7} + \frac{D b x^{8}}{8} + x^{6} \left (\frac{B b}{6} + \frac{D a}{6}\right ) + x^{5} \left (\frac{A b}{5} + \frac{C a}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.53007, size = 77, normalized size = 1.18 \begin{align*} \frac{1}{8} \, D b x^{8} + \frac{1}{7} \, C b x^{7} + \frac{1}{6} \, D a x^{6} + \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*D*b*x^8 + 1/7*C*b*x^7 + 1/6*D*a*x^6 + 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