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

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

[Out]

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

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Rubi [A]  time = 0.109861, antiderivative size = 65, 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 \[ \frac{1}{2} a^2 A x^2+\frac{1}{3} a^2 B x^3+\frac{1}{2} a A c x^4+\frac{2}{5} a B c x^5+\frac{1}{6} A c^2 x^6+\frac{1}{7} B c^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a^{2} \int x\, dx + \frac{A a c x^{4}}{2} + \frac{A c^{2} x^{6}}{6} + \frac{B a^{2} x^{3}}{3} + \frac{2 B a c x^{5}}{5} + \frac{B c^{2} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*Integral(x, x) + A*a*c*x**4/2 + A*c**2*x**6/6 + B*a**2*x**3/3 + 2*B*a*c*x
**5/5 + B*c**2*x**7/7

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Mathematica [A]  time = 0.00385836, size = 65, normalized size = 1. \[ \frac{1}{2} a^2 A x^2+\frac{1}{3} a^2 B x^3+\frac{1}{2} a A c x^4+\frac{2}{5} a B c x^5+\frac{1}{6} A c^2 x^6+\frac{1}{7} B c^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 54, normalized size = 0.8 \[{\frac{{a}^{2}A{x}^{2}}{2}}+{\frac{{a}^{2}B{x}^{3}}{3}}+{\frac{aAc{x}^{4}}{2}}+{\frac{2\,aBc{x}^{5}}{5}}+{\frac{A{c}^{2}{x}^{6}}{6}}+{\frac{B{c}^{2}{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)*(c*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.680825, size = 72, normalized size = 1.11 \[ \frac{1}{7} \, B c^{2} x^{7} + \frac{1}{6} \, A c^{2} x^{6} + \frac{2}{5} \, B a c x^{5} + \frac{1}{2} \, A a c x^{4} + \frac{1}{3} \, B a^{2} x^{3} + \frac{1}{2} \, A a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.26559, size = 1, normalized size = 0.02 \[ \frac{1}{7} x^{7} c^{2} B + \frac{1}{6} x^{6} c^{2} A + \frac{2}{5} x^{5} c a B + \frac{1}{2} x^{4} c a A + \frac{1}{3} x^{3} a^{2} B + \frac{1}{2} x^{2} a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.106856, size = 61, normalized size = 0.94 \[ \frac{A a^{2} x^{2}}{2} + \frac{A a c x^{4}}{2} + \frac{A c^{2} x^{6}}{6} + \frac{B a^{2} x^{3}}{3} + \frac{2 B a c x^{5}}{5} + \frac{B c^{2} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*x**2/2 + A*a*c*x**4/2 + A*c**2*x**6/6 + B*a**2*x**3/3 + 2*B*a*c*x**5/5 +
B*c**2*x**7/7

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GIAC/XCAS [A]  time = 0.266846, size = 72, normalized size = 1.11 \[ \frac{1}{7} \, B c^{2} x^{7} + \frac{1}{6} \, A c^{2} x^{6} + \frac{2}{5} \, B a c x^{5} + \frac{1}{2} \, A a c x^{4} + \frac{1}{3} \, B a^{2} x^{3} + \frac{1}{2} \, A a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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