3.313 \(\int \left (c+d x+e x^2\right ) \left (a+b x^3\right ) \, dx\)

Optimal. Leaf size=50 \[ a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{4} b c x^4+\frac{1}{5} b d x^5+\frac{1}{6} b e x^6 \]

[Out]

a*c*x + (a*d*x^2)/2 + (a*e*x^3)/3 + (b*c*x^4)/4 + (b*d*x^5)/5 + (b*e*x^6)/6

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Rubi [A]  time = 0.0585857, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{4} b c x^4+\frac{1}{5} b d x^5+\frac{1}{6} b e x^6 \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2)*(a + b*x^3),x]

[Out]

a*c*x + (a*d*x^2)/2 + (a*e*x^3)/3 + (b*c*x^4)/4 + (b*d*x^5)/5 + (b*e*x^6)/6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d*x+c)*(b*x**3+a),x)

[Out]

a*d*Integral(x, x) + a*e*x**3/3 + b*c*x**4/4 + b*d*x**5/5 + b*e*x**6/6 + c*Integ
ral(a, x)

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Mathematica [A]  time = 0.00581697, size = 50, normalized size = 1. \[ a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{4} b c x^4+\frac{1}{5} b d x^5+\frac{1}{6} b e x^6 \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2)*(a + b*x^3),x]

[Out]

a*c*x + (a*d*x^2)/2 + (a*e*x^3)/3 + (b*c*x^4)/4 + (b*d*x^5)/5 + (b*e*x^6)/6

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Maple [A]  time = 0.002, size = 41, normalized size = 0.8 \[ acx+{\frac{ad{x}^{2}}{2}}+{\frac{ae{x}^{3}}{3}}+{\frac{bc{x}^{4}}{4}}+{\frac{bd{x}^{5}}{5}}+{\frac{be{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d*x+c)*(b*x^3+a),x)

[Out]

a*c*x+1/2*a*d*x^2+1/3*a*e*x^3+1/4*b*c*x^4+1/5*b*d*x^5+1/6*b*e*x^6

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Maxima [A]  time = 1.37736, size = 54, normalized size = 1.08 \[ \frac{1}{6} \, b e x^{6} + \frac{1}{5} \, b d x^{5} + \frac{1}{4} \, b c x^{4} + \frac{1}{3} \, a e x^{3} + \frac{1}{2} \, a d x^{2} + a c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)*(e*x^2 + d*x + c),x, algorithm="maxima")

[Out]

1/6*b*e*x^6 + 1/5*b*d*x^5 + 1/4*b*c*x^4 + 1/3*a*e*x^3 + 1/2*a*d*x^2 + a*c*x

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Fricas [A]  time = 0.203941, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} e b + \frac{1}{5} x^{5} d b + \frac{1}{4} x^{4} c b + \frac{1}{3} x^{3} e a + \frac{1}{2} x^{2} d a + x c a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)*(e*x^2 + d*x + c),x, algorithm="fricas")

[Out]

1/6*x^6*e*b + 1/5*x^5*d*b + 1/4*x^4*c*b + 1/3*x^3*e*a + 1/2*x^2*d*a + x*c*a

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Sympy [A]  time = 0.047076, size = 46, normalized size = 0.92 \[ a c x + \frac{a d x^{2}}{2} + \frac{a e x^{3}}{3} + \frac{b c x^{4}}{4} + \frac{b d x^{5}}{5} + \frac{b e x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d*x+c)*(b*x**3+a),x)

[Out]

a*c*x + a*d*x**2/2 + a*e*x**3/3 + b*c*x**4/4 + b*d*x**5/5 + b*e*x**6/6

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GIAC/XCAS [A]  time = 0.206528, size = 57, normalized size = 1.14 \[ \frac{1}{6} \, b x^{6} e + \frac{1}{5} \, b d x^{5} + \frac{1}{4} \, b c x^{4} + \frac{1}{3} \, a x^{3} e + \frac{1}{2} \, a d x^{2} + a c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)*(e*x^2 + d*x + c),x, algorithm="giac")

[Out]

1/6*b*x^6*e + 1/5*b*d*x^5 + 1/4*b*c*x^4 + 1/3*a*x^3*e + 1/2*a*d*x^2 + a*c*x