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

Optimal. Leaf size=113 \[ a^4 c x+\frac{1}{2} a^4 d x^2+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14} \]

[Out]

a^4*c*x + (a^4*d*x^2)/2 + a^3*b*c*x^4 + (4*a^3*b*d*x^5)/5 + (6*a^2*b^2*c*x^7)/7
+ (3*a^2*b^2*d*x^8)/4 + (2*a*b^3*c*x^10)/5 + (4*a*b^3*d*x^11)/11 + (b^4*c*x^13)/
13 + (b^4*d*x^14)/14

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Rubi [A]  time = 0.174886, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ a^4 c x+\frac{1}{2} a^4 d x^2+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14} \]

Antiderivative was successfully verified.

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

[Out]

a^4*c*x + (a^4*d*x^2)/2 + a^3*b*c*x^4 + (4*a^3*b*d*x^5)/5 + (6*a^2*b^2*c*x^7)/7
+ (3*a^2*b^2*d*x^8)/4 + (2*a*b^3*c*x^10)/5 + (4*a*b^3*d*x^11)/11 + (b^4*c*x^13)/
13 + (b^4*d*x^14)/14

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{4} d \int x\, dx + a^{4} \int c\, dx + a^{3} b c x^{4} + \frac{4 a^{3} b d x^{5}}{5} + \frac{6 a^{2} b^{2} c x^{7}}{7} + \frac{3 a^{2} b^{2} d x^{8}}{4} + \frac{2 a b^{3} c x^{10}}{5} + \frac{4 a b^{3} d x^{11}}{11} + \frac{b^{4} c x^{13}}{13} + \frac{b^{4} d x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**4*d*Integral(x, x) + a**4*Integral(c, x) + a**3*b*c*x**4 + 4*a**3*b*d*x**5/5
+ 6*a**2*b**2*c*x**7/7 + 3*a**2*b**2*d*x**8/4 + 2*a*b**3*c*x**10/5 + 4*a*b**3*d*
x**11/11 + b**4*c*x**13/13 + b**4*d*x**14/14

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Mathematica [A]  time = 0.00787382, size = 113, normalized size = 1. \[ a^4 c x+\frac{1}{2} a^4 d x^2+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14} \]

Antiderivative was successfully verified.

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

[Out]

a^4*c*x + (a^4*d*x^2)/2 + a^3*b*c*x^4 + (4*a^3*b*d*x^5)/5 + (6*a^2*b^2*c*x^7)/7
+ (3*a^2*b^2*d*x^8)/4 + (2*a*b^3*c*x^10)/5 + (4*a*b^3*d*x^11)/11 + (b^4*c*x^13)/
13 + (b^4*d*x^14)/14

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Maple [A]  time = 0.003, size = 98, normalized size = 0.9 \[{a}^{4}cx+{\frac{{a}^{4}d{x}^{2}}{2}}+{a}^{3}bc{x}^{4}+{\frac{4\,{a}^{3}bd{x}^{5}}{5}}+{\frac{6\,{a}^{2}{b}^{2}c{x}^{7}}{7}}+{\frac{3\,{a}^{2}{b}^{2}d{x}^{8}}{4}}+{\frac{2\,a{b}^{3}c{x}^{10}}{5}}+{\frac{4\,a{b}^{3}d{x}^{11}}{11}}+{\frac{{b}^{4}c{x}^{13}}{13}}+{\frac{{b}^{4}d{x}^{14}}{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^4*c*x+1/2*a^4*d*x^2+a^3*b*c*x^4+4/5*a^3*b*d*x^5+6/7*a^2*b^2*c*x^7+3/4*a^2*b^2*
d*x^8+2/5*a*b^3*c*x^10+4/11*a*b^3*d*x^11+1/13*b^4*c*x^13+1/14*b^4*d*x^14

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Maxima [A]  time = 1.42368, size = 131, normalized size = 1.16 \[ \frac{1}{14} \, b^{4} d x^{14} + \frac{1}{13} \, b^{4} c x^{13} + \frac{4}{11} \, a b^{3} d x^{11} + \frac{2}{5} \, a b^{3} c x^{10} + \frac{3}{4} \, a^{2} b^{2} d x^{8} + \frac{6}{7} \, a^{2} b^{2} c x^{7} + \frac{4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \]

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)^3,x, algorithm="maxima")

[Out]

1/14*b^4*d*x^14 + 1/13*b^4*c*x^13 + 4/11*a*b^3*d*x^11 + 2/5*a*b^3*c*x^10 + 3/4*a
^2*b^2*d*x^8 + 6/7*a^2*b^2*c*x^7 + 4/5*a^3*b*d*x^5 + a^3*b*c*x^4 + 1/2*a^4*d*x^2
+ a^4*c*x

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Fricas [A]  time = 0.206957, size = 1, normalized size = 0.01 \[ \frac{1}{14} x^{14} d b^{4} + \frac{1}{13} x^{13} c b^{4} + \frac{4}{11} x^{11} d b^{3} a + \frac{2}{5} x^{10} c b^{3} a + \frac{3}{4} x^{8} d b^{2} a^{2} + \frac{6}{7} x^{7} c b^{2} a^{2} + \frac{4}{5} x^{5} d b a^{3} + x^{4} c b a^{3} + \frac{1}{2} x^{2} d a^{4} + x c a^{4} \]

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)^3,x, algorithm="fricas")

[Out]

1/14*x^14*d*b^4 + 1/13*x^13*c*b^4 + 4/11*x^11*d*b^3*a + 2/5*x^10*c*b^3*a + 3/4*x
^8*d*b^2*a^2 + 6/7*x^7*c*b^2*a^2 + 4/5*x^5*d*b*a^3 + x^4*c*b*a^3 + 1/2*x^2*d*a^4
+ x*c*a^4

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Sympy [A]  time = 0.082327, size = 117, normalized size = 1.04 \[ a^{4} c x + \frac{a^{4} d x^{2}}{2} + a^{3} b c x^{4} + \frac{4 a^{3} b d x^{5}}{5} + \frac{6 a^{2} b^{2} c x^{7}}{7} + \frac{3 a^{2} b^{2} d x^{8}}{4} + \frac{2 a b^{3} c x^{10}}{5} + \frac{4 a b^{3} d x^{11}}{11} + \frac{b^{4} c x^{13}}{13} + \frac{b^{4} d x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**4*c*x + a**4*d*x**2/2 + a**3*b*c*x**4 + 4*a**3*b*d*x**5/5 + 6*a**2*b**2*c*x**
7/7 + 3*a**2*b**2*d*x**8/4 + 2*a*b**3*c*x**10/5 + 4*a*b**3*d*x**11/11 + b**4*c*x
**13/13 + b**4*d*x**14/14

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GIAC/XCAS [A]  time = 0.214945, size = 131, normalized size = 1.16 \[ \frac{1}{14} \, b^{4} d x^{14} + \frac{1}{13} \, b^{4} c x^{13} + \frac{4}{11} \, a b^{3} d x^{11} + \frac{2}{5} \, a b^{3} c x^{10} + \frac{3}{4} \, a^{2} b^{2} d x^{8} + \frac{6}{7} \, a^{2} b^{2} c x^{7} + \frac{4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \]

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)^3,x, algorithm="giac")

[Out]

1/14*b^4*d*x^14 + 1/13*b^4*c*x^13 + 4/11*a*b^3*d*x^11 + 2/5*a*b^3*c*x^10 + 3/4*a
^2*b^2*d*x^8 + 6/7*a^2*b^2*c*x^7 + 4/5*a^3*b*d*x^5 + a^3*b*c*x^4 + 1/2*a^4*d*x^2
+ a^4*c*x