3.32 \(\int \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=109 \[ a^5 A x+\frac{1}{4} a^4 x^4 (a B+5 A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+a^2 b^2 x^{10} (a B+A b)+\frac{1}{16} b^4 x^{16} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{19} b^5 B x^{19} \]

[Out]

a^5*A*x + (a^4*(5*A*b + a*B)*x^4)/4 + (5*a^3*b*(2*A*b + a*B)*x^7)/7 + a^2*b^2*(A
*b + a*B)*x^10 + (5*a*b^3*(A*b + 2*a*B)*x^13)/13 + (b^4*(A*b + 5*a*B)*x^16)/16 +
(b^5*B*x^19)/19

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Rubi [A]  time = 0.168, antiderivative size = 109, 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 \[ a^5 A x+\frac{1}{4} a^4 x^4 (a B+5 A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+a^2 b^2 x^{10} (a B+A b)+\frac{1}{16} b^4 x^{16} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{19} b^5 B x^{19} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^5*(A + B*x^3),x]

[Out]

a^5*A*x + (a^4*(5*A*b + a*B)*x^4)/4 + (5*a^3*b*(2*A*b + a*B)*x^7)/7 + a^2*b^2*(A
*b + a*B)*x^10 + (5*a*b^3*(A*b + 2*a*B)*x^13)/13 + (b^4*(A*b + 5*a*B)*x^16)/16 +
(b^5*B*x^19)/19

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B b^{5} x^{19}}{19} + a^{5} \int A\, dx + \frac{a^{4} x^{4} \left (5 A b + B a\right )}{4} + \frac{5 a^{3} b x^{7} \left (2 A b + B a\right )}{7} + a^{2} b^{2} x^{10} \left (A b + B a\right ) + \frac{5 a b^{3} x^{13} \left (A b + 2 B a\right )}{13} + \frac{b^{4} x^{16} \left (A b + 5 B a\right )}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A),x)

[Out]

B*b**5*x**19/19 + a**5*Integral(A, x) + a**4*x**4*(5*A*b + B*a)/4 + 5*a**3*b*x**
7*(2*A*b + B*a)/7 + a**2*b**2*x**10*(A*b + B*a) + 5*a*b**3*x**13*(A*b + 2*B*a)/1
3 + b**4*x**16*(A*b + 5*B*a)/16

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Mathematica [A]  time = 0.0326415, size = 109, normalized size = 1. \[ a^5 A x+\frac{1}{4} a^4 x^4 (a B+5 A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+a^2 b^2 x^{10} (a B+A b)+\frac{1}{16} b^4 x^{16} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{19} b^5 B x^{19} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^5*(A + B*x^3),x]

[Out]

a^5*A*x + (a^4*(5*A*b + a*B)*x^4)/4 + (5*a^3*b*(2*A*b + a*B)*x^7)/7 + a^2*b^2*(A
*b + a*B)*x^10 + (5*a*b^3*(A*b + 2*a*B)*x^13)/13 + (b^4*(A*b + 5*a*B)*x^16)/16 +
(b^5*B*x^19)/19

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Maple [A]  time = 0.002, size = 121, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{19}}{19}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{16}}{16}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{13}}{13}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{7}}{7}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{4}}{4}}+{a}^{5}Ax \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A),x)

[Out]

1/19*b^5*B*x^19+1/16*(A*b^5+5*B*a*b^4)*x^16+1/13*(5*A*a*b^4+10*B*a^2*b^3)*x^13+1
/10*(10*A*a^2*b^3+10*B*a^3*b^2)*x^10+1/7*(10*A*a^3*b^2+5*B*a^4*b)*x^7+1/4*(5*A*a
^4*b+B*a^5)*x^4+a^5*A*x

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Maxima [A]  time = 1.4104, size = 155, normalized size = 1.42 \[ \frac{1}{19} \, B b^{5} x^{19} + \frac{1}{16} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{16} + \frac{5}{13} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{13} +{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{10} + \frac{5}{7} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{7} + A a^{5} x + \frac{1}{4} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5,x, algorithm="maxima")

[Out]

1/19*B*b^5*x^19 + 1/16*(5*B*a*b^4 + A*b^5)*x^16 + 5/13*(2*B*a^2*b^3 + A*a*b^4)*x
^13 + (B*a^3*b^2 + A*a^2*b^3)*x^10 + 5/7*(B*a^4*b + 2*A*a^3*b^2)*x^7 + A*a^5*x +
1/4*(B*a^5 + 5*A*a^4*b)*x^4

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Fricas [A]  time = 0.199336, size = 1, normalized size = 0.01 \[ \frac{1}{19} x^{19} b^{5} B + \frac{5}{16} x^{16} b^{4} a B + \frac{1}{16} x^{16} b^{5} A + \frac{10}{13} x^{13} b^{3} a^{2} B + \frac{5}{13} x^{13} b^{4} a A + x^{10} b^{2} a^{3} B + x^{10} b^{3} a^{2} A + \frac{5}{7} x^{7} b a^{4} B + \frac{10}{7} x^{7} b^{2} a^{3} A + \frac{1}{4} x^{4} a^{5} B + \frac{5}{4} x^{4} b a^{4} A + x a^{5} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5,x, algorithm="fricas")

[Out]

1/19*x^19*b^5*B + 5/16*x^16*b^4*a*B + 1/16*x^16*b^5*A + 10/13*x^13*b^3*a^2*B + 5
/13*x^13*b^4*a*A + x^10*b^2*a^3*B + x^10*b^3*a^2*A + 5/7*x^7*b*a^4*B + 10/7*x^7*
b^2*a^3*A + 1/4*x^4*a^5*B + 5/4*x^4*b*a^4*A + x*a^5*A

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Sympy [A]  time = 0.16129, size = 128, normalized size = 1.17 \[ A a^{5} x + \frac{B b^{5} x^{19}}{19} + x^{16} \left (\frac{A b^{5}}{16} + \frac{5 B a b^{4}}{16}\right ) + x^{13} \left (\frac{5 A a b^{4}}{13} + \frac{10 B a^{2} b^{3}}{13}\right ) + x^{10} \left (A a^{2} b^{3} + B a^{3} b^{2}\right ) + x^{7} \left (\frac{10 A a^{3} b^{2}}{7} + \frac{5 B a^{4} b}{7}\right ) + x^{4} \left (\frac{5 A a^{4} b}{4} + \frac{B a^{5}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5*(B*x**3+A),x)

[Out]

A*a**5*x + B*b**5*x**19/19 + x**16*(A*b**5/16 + 5*B*a*b**4/16) + x**13*(5*A*a*b*
*4/13 + 10*B*a**2*b**3/13) + x**10*(A*a**2*b**3 + B*a**3*b**2) + x**7*(10*A*a**3
*b**2/7 + 5*B*a**4*b/7) + x**4*(5*A*a**4*b/4 + B*a**5/4)

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GIAC/XCAS [A]  time = 0.216203, size = 162, normalized size = 1.49 \[ \frac{1}{19} \, B b^{5} x^{19} + \frac{5}{16} \, B a b^{4} x^{16} + \frac{1}{16} \, A b^{5} x^{16} + \frac{10}{13} \, B a^{2} b^{3} x^{13} + \frac{5}{13} \, A a b^{4} x^{13} + B a^{3} b^{2} x^{10} + A a^{2} b^{3} x^{10} + \frac{5}{7} \, B a^{4} b x^{7} + \frac{10}{7} \, A a^{3} b^{2} x^{7} + \frac{1}{4} \, B a^{5} x^{4} + \frac{5}{4} \, A a^{4} b x^{4} + A a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5,x, algorithm="giac")

[Out]

1/19*B*b^5*x^19 + 5/16*B*a*b^4*x^16 + 1/16*A*b^5*x^16 + 10/13*B*a^2*b^3*x^13 + 5
/13*A*a*b^4*x^13 + B*a^3*b^2*x^10 + A*a^2*b^3*x^10 + 5/7*B*a^4*b*x^7 + 10/7*A*a^
3*b^2*x^7 + 1/4*B*a^5*x^4 + 5/4*A*a^4*b*x^4 + A*a^5*x