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

Optimal. Leaf size=94 \[ \frac{1}{4} c^3 x^4 (4 a d+b c)+\frac{2}{7} c^2 d x^7 (3 a d+2 b c)+\frac{1}{13} d^3 x^{13} (a d+4 b c)+\frac{1}{5} c d^2 x^{10} (2 a d+3 b c)+a c^4 x+\frac{1}{16} b d^4 x^{16} \]

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^4)/4 + (2*c^2*d*(2*b*c + 3*a*d)*x^7)/7 + (c*d^2*(
3*b*c + 2*a*d)*x^10)/5 + (d^3*(4*b*c + a*d)*x^13)/13 + (b*d^4*x^16)/16

_______________________________________________________________________________________

Rubi [A]  time = 0.14945, antiderivative size = 94, 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 \[ \frac{1}{4} c^3 x^4 (4 a d+b c)+\frac{2}{7} c^2 d x^7 (3 a d+2 b c)+\frac{1}{13} d^3 x^{13} (a d+4 b c)+\frac{1}{5} c d^2 x^{10} (2 a d+3 b c)+a c^4 x+\frac{1}{16} b d^4 x^{16} \]

Antiderivative was successfully verified.

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

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^4)/4 + (2*c^2*d*(2*b*c + 3*a*d)*x^7)/7 + (c*d^2*(
3*b*c + 2*a*d)*x^10)/5 + (d^3*(4*b*c + a*d)*x^13)/13 + (b*d^4*x^16)/16

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b d^{4} x^{16}}{16} + c^{4} \int a\, dx + \frac{c^{3} x^{4} \left (4 a d + b c\right )}{4} + \frac{2 c^{2} d x^{7} \left (3 a d + 2 b c\right )}{7} + \frac{c d^{2} x^{10} \left (2 a d + 3 b c\right )}{5} + \frac{d^{3} x^{13} \left (a d + 4 b c\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*d**4*x**16/16 + c**4*Integral(a, x) + c**3*x**4*(4*a*d + b*c)/4 + 2*c**2*d*x**
7*(3*a*d + 2*b*c)/7 + c*d**2*x**10*(2*a*d + 3*b*c)/5 + d**3*x**13*(a*d + 4*b*c)/
13

_______________________________________________________________________________________

Mathematica [A]  time = 0.0345473, size = 94, normalized size = 1. \[ \frac{1}{4} c^3 x^4 (4 a d+b c)+\frac{2}{7} c^2 d x^7 (3 a d+2 b c)+\frac{1}{13} d^3 x^{13} (a d+4 b c)+\frac{1}{5} c d^2 x^{10} (2 a d+3 b c)+a c^4 x+\frac{1}{16} b d^4 x^{16} \]

Antiderivative was successfully verified.

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

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^4)/4 + (2*c^2*d*(2*b*c + 3*a*d)*x^7)/7 + (c*d^2*(
3*b*c + 2*a*d)*x^10)/5 + (d^3*(4*b*c + a*d)*x^13)/13 + (b*d^4*x^16)/16

_______________________________________________________________________________________

Maple [A]  time = 0., size = 97, normalized size = 1. \[{\frac{b{d}^{4}{x}^{16}}{16}}+{\frac{ \left ( a{d}^{4}+4\,bc{d}^{3} \right ){x}^{13}}{13}}+{\frac{ \left ( 4\,ac{d}^{3}+6\,{c}^{2}{d}^{2}b \right ){x}^{10}}{10}}+{\frac{ \left ( 6\,a{c}^{2}{d}^{2}+4\,{c}^{3}db \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,a{c}^{3}d+b{c}^{4} \right ){x}^{4}}{4}}+a{c}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(d*x^3+c)^4,x)

[Out]

1/16*b*d^4*x^16+1/13*(a*d^4+4*b*c*d^3)*x^13+1/10*(4*a*c*d^3+6*b*c^2*d^2)*x^10+1/
7*(6*a*c^2*d^2+4*b*c^3*d)*x^7+1/4*(4*a*c^3*d+b*c^4)*x^4+a*c^4*x

_______________________________________________________________________________________

Maxima [A]  time = 1.40995, size = 130, normalized size = 1.38 \[ \frac{1}{16} \, b d^{4} x^{16} + \frac{1}{13} \,{\left (4 \, b c d^{3} + a d^{4}\right )} x^{13} + \frac{1}{5} \,{\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{10} + \frac{2}{7} \,{\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{7} + a c^{4} x + \frac{1}{4} \,{\left (b c^{4} + 4 \, a c^{3} d\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*b*d^4*x^16 + 1/13*(4*b*c*d^3 + a*d^4)*x^13 + 1/5*(3*b*c^2*d^2 + 2*a*c*d^3)*
x^10 + 2/7*(2*b*c^3*d + 3*a*c^2*d^2)*x^7 + a*c^4*x + 1/4*(b*c^4 + 4*a*c^3*d)*x^4

_______________________________________________________________________________________

Fricas [A]  time = 0.179713, size = 1, normalized size = 0.01 \[ \frac{1}{16} x^{16} d^{4} b + \frac{4}{13} x^{13} d^{3} c b + \frac{1}{13} x^{13} d^{4} a + \frac{3}{5} x^{10} d^{2} c^{2} b + \frac{2}{5} x^{10} d^{3} c a + \frac{4}{7} x^{7} d c^{3} b + \frac{6}{7} x^{7} d^{2} c^{2} a + \frac{1}{4} x^{4} c^{4} b + x^{4} d c^{3} a + x c^{4} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*x^16*d^4*b + 4/13*x^13*d^3*c*b + 1/13*x^13*d^4*a + 3/5*x^10*d^2*c^2*b + 2/5
*x^10*d^3*c*a + 4/7*x^7*d*c^3*b + 6/7*x^7*d^2*c^2*a + 1/4*x^4*c^4*b + x^4*d*c^3*
a + x*c^4*a

_______________________________________________________________________________________

Sympy [A]  time = 0.144622, size = 104, normalized size = 1.11 \[ a c^{4} x + \frac{b d^{4} x^{16}}{16} + x^{13} \left (\frac{a d^{4}}{13} + \frac{4 b c d^{3}}{13}\right ) + x^{10} \left (\frac{2 a c d^{3}}{5} + \frac{3 b c^{2} d^{2}}{5}\right ) + x^{7} \left (\frac{6 a c^{2} d^{2}}{7} + \frac{4 b c^{3} d}{7}\right ) + x^{4} \left (a c^{3} d + \frac{b c^{4}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**4*x + b*d**4*x**16/16 + x**13*(a*d**4/13 + 4*b*c*d**3/13) + x**10*(2*a*c*d*
*3/5 + 3*b*c**2*d**2/5) + x**7*(6*a*c**2*d**2/7 + 4*b*c**3*d/7) + x**4*(a*c**3*d
+ b*c**4/4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217729, size = 131, normalized size = 1.39 \[ \frac{1}{16} \, b d^{4} x^{16} + \frac{4}{13} \, b c d^{3} x^{13} + \frac{1}{13} \, a d^{4} x^{13} + \frac{3}{5} \, b c^{2} d^{2} x^{10} + \frac{2}{5} \, a c d^{3} x^{10} + \frac{4}{7} \, b c^{3} d x^{7} + \frac{6}{7} \, a c^{2} d^{2} x^{7} + \frac{1}{4} \, b c^{4} x^{4} + a c^{3} d x^{4} + a c^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*b*d^4*x^16 + 4/13*b*c*d^3*x^13 + 1/13*a*d^4*x^13 + 3/5*b*c^2*d^2*x^10 + 2/5
*a*c*d^3*x^10 + 4/7*b*c^3*d*x^7 + 6/7*a*c^2*d^2*x^7 + 1/4*b*c^4*x^4 + a*c^3*d*x^
4 + a*c^4*x