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

Optimal. Leaf size=65 \[ \frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{5} x^5 (a D+b B)+\frac{1}{3} a B x^3+\frac{1}{6} b C x^6+\frac{1}{7} b D x^7 \]

[Out]

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

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Rubi [A]  time = 0.123128, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{5} x^5 (a D+b B)+\frac{1}{3} a B x^3+\frac{1}{6} b C x^6+\frac{1}{7} b D x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a \int x\, dx + \frac{B a x^{3}}{3} + \frac{C b x^{6}}{6} + \frac{D b x^{7}}{7} + x^{5} \left (\frac{B b}{5} + \frac{D a}{5}\right ) + x^{4} \left (\frac{A b}{4} + \frac{C a}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0193494, size = 65, normalized size = 1. \[ \frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{5} x^5 (a D+b B)+\frac{1}{3} a B x^3+\frac{1}{6} b C x^6+\frac{1}{7} b D x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 54, normalized size = 0.8 \[{\frac{aA{x}^{2}}{2}}+{\frac{aB{x}^{3}}{3}}+{\frac{ \left ( Ab+aC \right ){x}^{4}}{4}}+{\frac{ \left ( Bb+aD \right ){x}^{5}}{5}}+{\frac{bC{x}^{6}}{6}}+{\frac{bD{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35071, size = 72, normalized size = 1.11 \[ \frac{1}{7} \, D b x^{7} + \frac{1}{6} \, C b x^{6} + \frac{1}{5} \,{\left (D a + B b\right )} x^{5} + \frac{1}{3} \, B a x^{3} + \frac{1}{4} \,{\left (C a + A b\right )} x^{4} + \frac{1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22847, size = 1, normalized size = 0.02 \[ \frac{1}{7} x^{7} b D + \frac{1}{6} x^{6} b C + \frac{1}{5} x^{5} a D + \frac{1}{5} x^{5} b B + \frac{1}{4} x^{4} a C + \frac{1}{4} x^{4} b A + \frac{1}{3} x^{3} a B + \frac{1}{2} x^{2} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.052273, size = 60, normalized size = 0.92 \[ \frac{A a x^{2}}{2} + \frac{B a x^{3}}{3} + \frac{C b x^{6}}{6} + \frac{D b x^{7}}{7} + x^{5} \left (\frac{B b}{5} + \frac{D a}{5}\right ) + x^{4} \left (\frac{A b}{4} + \frac{C a}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211772, size = 77, normalized size = 1.18 \[ \frac{1}{7} \, D b x^{7} + \frac{1}{6} \, C b x^{6} + \frac{1}{5} \, D a x^{5} + \frac{1}{5} \, B b x^{5} + \frac{1}{4} \, C a x^{4} + \frac{1}{4} \, A b x^{4} + \frac{1}{3} \, B a x^{3} + \frac{1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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