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

Optimal. Leaf size=37 \[ \frac{1}{4} a A x^4+\frac{1}{5} a B x^5+\frac{1}{6} A c x^6+\frac{1}{7} B c x^7 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0865103, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{1}{4} a A x^4+\frac{1}{5} a B x^5+\frac{1}{6} A c x^6+\frac{1}{7} B c x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.18964, size = 32, normalized size = 0.86 \[ \frac{A a x^{4}}{4} + \frac{A c x^{6}}{6} + \frac{B a x^{5}}{5} + \frac{B c x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**4/4 + A*c*x**6/6 + B*a*x**5/5 + B*c*x**7/7

_______________________________________________________________________________________

Mathematica [A]  time = 0.00395307, size = 37, normalized size = 1. \[ \frac{1}{4} a A x^4+\frac{1}{5} a B x^5+\frac{1}{6} A c x^6+\frac{1}{7} B c x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 30, normalized size = 0.8 \[{\frac{aA{x}^{4}}{4}}+{\frac{aB{x}^{5}}{5}}+{\frac{Ac{x}^{6}}{6}}+{\frac{Bc{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)*(c*x^2+a),x)

[Out]

1/4*a*A*x^4+1/5*a*B*x^5+1/6*A*c*x^6+1/7*B*c*x^7

_______________________________________________________________________________________

Maxima [A]  time = 0.685902, size = 39, normalized size = 1.05 \[ \frac{1}{7} \, B c x^{7} + \frac{1}{6} \, A c x^{6} + \frac{1}{5} \, B a x^{5} + \frac{1}{4} \, A a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*c*x^7 + 1/6*A*c*x^6 + 1/5*B*a*x^5 + 1/4*A*a*x^4

_______________________________________________________________________________________

Fricas [A]  time = 0.263153, size = 1, normalized size = 0.03 \[ \frac{1}{7} x^{7} c B + \frac{1}{6} x^{6} c A + \frac{1}{5} x^{5} a B + \frac{1}{4} x^{4} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*x^7*c*B + 1/6*x^6*c*A + 1/5*x^5*a*B + 1/4*x^4*a*A

_______________________________________________________________________________________

Sympy [A]  time = 0.087581, size = 32, normalized size = 0.86 \[ \frac{A a x^{4}}{4} + \frac{A c x^{6}}{6} + \frac{B a x^{5}}{5} + \frac{B c x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**4/4 + A*c*x**6/6 + B*a*x**5/5 + B*c*x**7/7

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.266438, size = 39, normalized size = 1.05 \[ \frac{1}{7} \, B c x^{7} + \frac{1}{6} \, A c x^{6} + \frac{1}{5} \, B a x^{5} + \frac{1}{4} \, A a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*c*x^7 + 1/6*A*c*x^6 + 1/5*B*a*x^5 + 1/4*A*a*x^4