3.55 \(\int \left (a+b x^3\right ) \left (a c+a d x+b c x^3+b d x^4\right ) \, dx\)

Optimal. Leaf size=60 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8 \]

[Out]

a^2*c*x + (a^2*d*x^2)/2 + (a*b*c*x^4)/2 + (2*a*b*d*x^5)/5 + (b^2*c*x^7)/7 + (b^2
*d*x^8)/8

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Rubi [A]  time = 0.0713776, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8 \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]

[Out]

a^2*c*x + (a^2*d*x^2)/2 + (a*b*c*x^4)/2 + (2*a*b*d*x^5)/5 + (b^2*c*x^7)/7 + (b^2
*d*x^8)/8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} d \int x\, dx + a^{2} \int c\, dx + \frac{a b c x^{4}}{2} + \frac{2 a b d x^{5}}{5} + \frac{b^{2} c x^{7}}{7} + \frac{b^{2} d x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)*(b*d*x**4+b*c*x**3+a*d*x+a*c),x)

[Out]

a**2*d*Integral(x, x) + a**2*Integral(c, x) + a*b*c*x**4/2 + 2*a*b*d*x**5/5 + b*
*2*c*x**7/7 + b**2*d*x**8/8

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Mathematica [A]  time = 0.00360621, size = 60, normalized size = 1. \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8 \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]

[Out]

a^2*c*x + (a^2*d*x^2)/2 + (a*b*c*x^4)/2 + (2*a*b*d*x^5)/5 + (b^2*c*x^7)/7 + (b^2
*d*x^8)/8

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Maple [A]  time = 0.003, size = 51, normalized size = 0.9 \[{a}^{2}cx+{\frac{{a}^{2}d{x}^{2}}{2}}+{\frac{abc{x}^{4}}{2}}+{\frac{2\,abd{x}^{5}}{5}}+{\frac{{b}^{2}c{x}^{7}}{7}}+{\frac{{b}^{2}d{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(b*d*x^4+b*c*x^3+a*d*x+a*c),x)

[Out]

a^2*c*x+1/2*a^2*d*x^2+1/2*a*b*c*x^4+2/5*a*b*d*x^5+1/7*b^2*c*x^7+1/8*b^2*d*x^8

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Maxima [A]  time = 1.44795, size = 68, normalized size = 1.13 \[ \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{7} \, b^{2} c x^{7} + \frac{2}{5} \, a b d x^{5} + \frac{1}{2} \, a b c x^{4} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^4 + b*c*x^3 + a*d*x + a*c)*(b*x^3 + a),x, algorithm="maxima")

[Out]

1/8*b^2*d*x^8 + 1/7*b^2*c*x^7 + 2/5*a*b*d*x^5 + 1/2*a*b*c*x^4 + 1/2*a^2*d*x^2 +
a^2*c*x

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Fricas [A]  time = 0.192193, size = 1, normalized size = 0.02 \[ \frac{1}{8} x^{8} d b^{2} + \frac{1}{7} x^{7} c b^{2} + \frac{2}{5} x^{5} d b a + \frac{1}{2} x^{4} c b a + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^4 + b*c*x^3 + a*d*x + a*c)*(b*x^3 + a),x, algorithm="fricas")

[Out]

1/8*x^8*d*b^2 + 1/7*x^7*c*b^2 + 2/5*x^5*d*b*a + 1/2*x^4*c*b*a + 1/2*x^2*d*a^2 +
x*c*a^2

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Sympy [A]  time = 0.055978, size = 58, normalized size = 0.97 \[ a^{2} c x + \frac{a^{2} d x^{2}}{2} + \frac{a b c x^{4}}{2} + \frac{2 a b d x^{5}}{5} + \frac{b^{2} c x^{7}}{7} + \frac{b^{2} d x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)*(b*d*x**4+b*c*x**3+a*d*x+a*c),x)

[Out]

a**2*c*x + a**2*d*x**2/2 + a*b*c*x**4/2 + 2*a*b*d*x**5/5 + b**2*c*x**7/7 + b**2*
d*x**8/8

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GIAC/XCAS [A]  time = 0.21294, size = 68, normalized size = 1.13 \[ \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{7} \, b^{2} c x^{7} + \frac{2}{5} \, a b d x^{5} + \frac{1}{2} \, a b c x^{4} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^4 + b*c*x^3 + a*d*x + a*c)*(b*x^3 + a),x, algorithm="giac")

[Out]

1/8*b^2*d*x^8 + 1/7*b^2*c*x^7 + 2/5*a*b*d*x^5 + 1/2*a*b*c*x^4 + 1/2*a^2*d*x^2 +
a^2*c*x