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

Optimal. Leaf size=28 \[ \frac{1}{4} x^4 (a d+b c)+a c x+\frac{1}{7} b d x^7 \]

[Out]

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

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Rubi [A]  time = 0.0393509, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{1}{4} x^4 (a d+b c)+a c x+\frac{1}{7} b d x^7 \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)*(c + d*x^3),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00892945, size = 28, normalized size = 1. \[ \frac{1}{4} x^4 (a d+b c)+a c x+\frac{1}{7} b d x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 25, normalized size = 0.9 \[ acx+{\frac{ \left ( ad+bc \right ){x}^{4}}{4}}+{\frac{bd{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(d*x^3+c),x)

[Out]

a*c*x+1/4*(a*d+b*c)*x^4+1/7*b*d*x^7

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Maxima [A]  time = 1.35272, size = 32, normalized size = 1.14 \[ \frac{1}{7} \, b d x^{7} + \frac{1}{4} \,{\left (b c + a d\right )} x^{4} + a c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b*d*x^7 + 1/4*(b*c + a*d)*x^4 + a*c*x

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Fricas [A]  time = 0.18203, size = 1, normalized size = 0.04 \[ \frac{1}{7} x^{7} d b + \frac{1}{4} x^{4} c b + \frac{1}{4} x^{4} d a + x c a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.074719, size = 26, normalized size = 0.93 \[ a c x + \frac{b d x^{7}}{7} + x^{4} \left (\frac{a d}{4} + \frac{b c}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c*x + b*d*x**7/7 + x**4*(a*d/4 + b*c/4)

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GIAC/XCAS [A]  time = 0.211328, size = 35, normalized size = 1.25 \[ \frac{1}{7} \, b d x^{7} + \frac{1}{4} \, b c x^{4} + \frac{1}{4} \, a d x^{4} + a c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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