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

Optimal. Leaf size=50 \[ \frac{1}{7} d x^7 (a d+2 b c)+\frac{1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac{1}{10} b d^2 x^{10} \]

[Out]

a*c^2*x + (c*(b*c + 2*a*d)*x^4)/4 + (d*(2*b*c + a*d)*x^7)/7 + (b*d^2*x^10)/10

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Rubi [A]  time = 0.0742223, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{1}{7} d x^7 (a d+2 b c)+\frac{1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac{1}{10} b d^2 x^{10} \]

Antiderivative was successfully verified.

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

[Out]

a*c^2*x + (c*(b*c + 2*a*d)*x^4)/4 + (d*(2*b*c + a*d)*x^7)/7 + (b*d^2*x^10)/10

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b d^{2} x^{10}}{10} + c^{2} \int a\, dx + \frac{c x^{4} \left (2 a d + b c\right )}{4} + \frac{d x^{7} \left (a d + 2 b c\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*d**2*x**10/10 + c**2*Integral(a, x) + c*x**4*(2*a*d + b*c)/4 + d*x**7*(a*d + 2
*b*c)/7

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Mathematica [A]  time = 0.0141192, size = 50, normalized size = 1. \[ \frac{1}{7} d x^7 (a d+2 b c)+\frac{1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac{1}{10} b d^2 x^{10} \]

Antiderivative was successfully verified.

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

[Out]

a*c^2*x + (c*(b*c + 2*a*d)*x^4)/4 + (d*(2*b*c + a*d)*x^7)/7 + (b*d^2*x^10)/10

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Maple [A]  time = 0.001, size = 49, normalized size = 1. \[{\frac{b{d}^{2}{x}^{10}}{10}}+{\frac{ \left ( a{d}^{2}+2\,bcd \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,acd+b{c}^{2} \right ){x}^{4}}{4}}+a{c}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(d*x^3+c)^2,x)

[Out]

1/10*b*d^2*x^10+1/7*(a*d^2+2*b*c*d)*x^7+1/4*(2*a*c*d+b*c^2)*x^4+a*c^2*x

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Maxima [A]  time = 1.36066, size = 65, normalized size = 1.3 \[ \frac{1}{10} \, b d^{2} x^{10} + \frac{1}{7} \,{\left (2 \, b c d + a d^{2}\right )} x^{7} + \frac{1}{4} \,{\left (b c^{2} + 2 \, a c d\right )} x^{4} + a c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b*d^2*x^10 + 1/7*(2*b*c*d + a*d^2)*x^7 + 1/4*(b*c^2 + 2*a*c*d)*x^4 + a*c^2*
x

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Fricas [A]  time = 0.180838, size = 1, normalized size = 0.02 \[ \frac{1}{10} x^{10} d^{2} b + \frac{2}{7} x^{7} d c b + \frac{1}{7} x^{7} d^{2} a + \frac{1}{4} x^{4} c^{2} b + \frac{1}{2} x^{4} d c a + x c^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*x^10*d^2*b + 2/7*x^7*d*c*b + 1/7*x^7*d^2*a + 1/4*x^4*c^2*b + 1/2*x^4*d*c*a
+ x*c^2*a

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Sympy [A]  time = 0.108516, size = 51, normalized size = 1.02 \[ a c^{2} x + \frac{b d^{2} x^{10}}{10} + x^{7} \left (\frac{a d^{2}}{7} + \frac{2 b c d}{7}\right ) + x^{4} \left (\frac{a c d}{2} + \frac{b c^{2}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**2*x + b*d**2*x**10/10 + x**7*(a*d**2/7 + 2*b*c*d/7) + x**4*(a*c*d/2 + b*c**
2/4)

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GIAC/XCAS [A]  time = 0.213265, size = 68, normalized size = 1.36 \[ \frac{1}{10} \, b d^{2} x^{10} + \frac{2}{7} \, b c d x^{7} + \frac{1}{7} \, a d^{2} x^{7} + \frac{1}{4} \, b c^{2} x^{4} + \frac{1}{2} \, a c d x^{4} + a c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b*d^2*x^10 + 2/7*b*c*d*x^7 + 1/7*a*d^2*x^7 + 1/4*b*c^2*x^4 + 1/2*a*c*d*x^4
+ a*c^2*x