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

Optimal. Leaf size=65 \[ \frac {1}{6} x^6 (a C+A b)+\frac {1}{4} a A x^4+\frac {1}{7} x^7 (a D+b B)+\frac {1}{5} a B x^5+\frac {1}{8} b C x^8+\frac {1}{9} b D x^9 \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 65, 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 = {1802} \[ \frac {1}{6} x^6 (a C+A b)+\frac {1}{4} a A x^4+\frac {1}{7} x^7 (a D+b B)+\frac {1}{5} a B x^5+\frac {1}{8} b C x^8+\frac {1}{9} b D x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 65, normalized size = 1.00 \[ \frac {1}{6} x^6 (a C+A b)+\frac {1}{4} a A x^4+\frac {1}{7} x^7 (a D+b B)+\frac {1}{5} a B x^5+\frac {1}{8} b C x^8+\frac {1}{9} b D x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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(x^3*(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 = 0.37, size = 57, normalized size = 0.88 \[ \frac {1}{9} \, D b x^{9} + \frac {1}{8} \, C b x^{8} + \frac {1}{7} \, D a x^{7} + \frac {1}{7} \, B b x^{7} + \frac {1}{6} \, C a x^{6} + \frac {1}{6} \, A b x^{6} + \frac {1}{5} \, B a x^{5} + \frac {1}{4} \, A a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 54, normalized size = 0.83 \[ \frac {D b \,x^{9}}{9}+\frac {C b \,x^{8}}{8}+\frac {B a \,x^{5}}{5}+\frac {\left (b B +a D\right ) x^{7}}{7}+\frac {A a \,x^{4}}{4}+\frac {\left (A b +a C \right ) x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 53, normalized size = 0.82 \[ \frac {1}{9} \, D b x^{9} + \frac {1}{8} \, C b x^{8} + \frac {1}{7} \, {\left (D a + B b\right )} x^{7} + \frac {1}{5} \, B a x^{5} + \frac {1}{6} \, {\left (C a + A b\right )} x^{6} + \frac {1}{4} \, A a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.20, size = 57, normalized size = 0.88 \[ \frac {a\,x^7\,D}{7}+\frac {b\,x^9\,D}{9}+\frac {A\,a\,x^4}{4}+\frac {B\,a\,x^5}{5}+\frac {A\,b\,x^6}{6}+\frac {C\,a\,x^6}{6}+\frac {B\,b\,x^7}{7}+\frac {C\,b\,x^8}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.10, size = 60, normalized size = 0.92 \[ \frac {A a x^{4}}{4} + \frac {B a x^{5}}{5} + \frac {C b x^{8}}{8} + \frac {D b x^{9}}{9} + x^{7} \left (\frac {B b}{7} + \frac {D a}{7}\right ) + x^{6} \left (\frac {A b}{6} + \frac {C a}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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