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

Optimal. Leaf size=55 \[ \frac{1}{4} a^2 A x^4+\frac{1}{6} b x^6 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{7} b^2 B x^7 \]

[Out]

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

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Rubi [A]  time = 0.113341, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{1}{4} a^2 A x^4+\frac{1}{6} b x^6 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{7} b^2 B x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.2049, size = 49, normalized size = 0.89 \[ \frac{A a^{2} x^{4}}{4} + \frac{B b^{2} x^{7}}{7} + \frac{a x^{5} \left (2 A b + B a\right )}{5} + \frac{b x^{6} \left (A b + 2 B a\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0126515, size = 55, normalized size = 1. \[ \frac{1}{4} a^2 A x^4+\frac{1}{6} b x^6 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{7} b^2 B x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 52, normalized size = 1. \[{\frac{{b}^{2}B{x}^{7}}{7}}+{\frac{ \left ({b}^{2}A+2\,abB \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,abA+{a}^{2}B \right ){x}^{5}}{5}}+{\frac{{a}^{2}A{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.67877, size = 69, normalized size = 1.25 \[ \frac{1}{7} \, B b^{2} x^{7} + \frac{1}{4} \, A a^{2} x^{4} + \frac{1}{6} \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B a^{2} + 2 \, A a b\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.249938, size = 1, normalized size = 0.02 \[ \frac{1}{7} x^{7} b^{2} B + \frac{1}{3} x^{6} b a B + \frac{1}{6} x^{6} b^{2} A + \frac{1}{5} x^{5} a^{2} B + \frac{2}{5} x^{5} b a A + \frac{1}{4} x^{4} a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.117123, size = 54, normalized size = 0.98 \[ \frac{A a^{2} x^{4}}{4} + \frac{B b^{2} x^{7}}{7} + x^{6} \left (\frac{A b^{2}}{6} + \frac{B a b}{3}\right ) + x^{5} \left (\frac{2 A a b}{5} + \frac{B a^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.267175, size = 72, normalized size = 1.31 \[ \frac{1}{7} \, B b^{2} x^{7} + \frac{1}{3} \, B a b x^{6} + \frac{1}{6} \, A b^{2} x^{6} + \frac{1}{5} \, B a^{2} x^{5} + \frac{2}{5} \, A a b x^{5} + \frac{1}{4} \, A a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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