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

Optimal. Leaf size=117 \[ \frac{1}{4} a^5 A x^4+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{2} a^3 b x^{10} (a B+2 A b)+\frac{10}{13} a^2 b^2 x^{13} (a B+A b)+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{16} a b^3 x^{16} (2 a B+A b)+\frac{1}{22} b^5 B x^{22} \]

[Out]

(a^5*A*x^4)/4 + (a^4*(5*A*b + a*B)*x^7)/7 + (a^3*b*(2*A*b + a*B)*x^10)/2 + (10*a
^2*b^2*(A*b + a*B)*x^13)/13 + (5*a*b^3*(A*b + 2*a*B)*x^16)/16 + (b^4*(A*b + 5*a*
B)*x^19)/19 + (b^5*B*x^22)/22

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Rubi [A]  time = 0.242558, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{1}{4} a^5 A x^4+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{2} a^3 b x^{10} (a B+2 A b)+\frac{10}{13} a^2 b^2 x^{13} (a B+A b)+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{16} a b^3 x^{16} (2 a B+A b)+\frac{1}{22} b^5 B x^{22} \]

Antiderivative was successfully verified.

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

[Out]

(a^5*A*x^4)/4 + (a^4*(5*A*b + a*B)*x^7)/7 + (a^3*b*(2*A*b + a*B)*x^10)/2 + (10*a
^2*b^2*(A*b + a*B)*x^13)/13 + (5*a*b^3*(A*b + 2*a*B)*x^16)/16 + (b^4*(A*b + 5*a*
B)*x^19)/19 + (b^5*B*x^22)/22

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Rubi in Sympy [A]  time = 26.7946, size = 112, normalized size = 0.96 \[ \frac{A a^{5} x^{4}}{4} + \frac{B b^{5} x^{22}}{22} + \frac{a^{4} x^{7} \left (5 A b + B a\right )}{7} + \frac{a^{3} b x^{10} \left (2 A b + B a\right )}{2} + \frac{10 a^{2} b^{2} x^{13} \left (A b + B a\right )}{13} + \frac{5 a b^{3} x^{16} \left (A b + 2 B a\right )}{16} + \frac{b^{4} x^{19} \left (A b + 5 B a\right )}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**5*x**4/4 + B*b**5*x**22/22 + a**4*x**7*(5*A*b + B*a)/7 + a**3*b*x**10*(2*A*
b + B*a)/2 + 10*a**2*b**2*x**13*(A*b + B*a)/13 + 5*a*b**3*x**16*(A*b + 2*B*a)/16
+ b**4*x**19*(A*b + 5*B*a)/19

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Mathematica [A]  time = 0.0332382, size = 117, normalized size = 1. \[ \frac{1}{4} a^5 A x^4+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{2} a^3 b x^{10} (a B+2 A b)+\frac{10}{13} a^2 b^2 x^{13} (a B+A b)+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{16} a b^3 x^{16} (2 a B+A b)+\frac{1}{22} b^5 B x^{22} \]

Antiderivative was successfully verified.

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

[Out]

(a^5*A*x^4)/4 + (a^4*(5*A*b + a*B)*x^7)/7 + (a^3*b*(2*A*b + a*B)*x^10)/2 + (10*a
^2*b^2*(A*b + a*B)*x^13)/13 + (5*a*b^3*(A*b + 2*a*B)*x^16)/16 + (b^4*(A*b + 5*a*
B)*x^19)/19 + (b^5*B*x^22)/22

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Maple [A]  time = 0.002, size = 124, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{22}}{22}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{19}}{19}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{16}}{16}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{13}}{13}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{10}}{10}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{7}}{7}}+{\frac{{a}^{5}A{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/22*b^5*B*x^22+1/19*(A*b^5+5*B*a*b^4)*x^19+1/16*(5*A*a*b^4+10*B*a^2*b^3)*x^16+1
/13*(10*A*a^2*b^3+10*B*a^3*b^2)*x^13+1/10*(10*A*a^3*b^2+5*B*a^4*b)*x^10+1/7*(5*A
*a^4*b+B*a^5)*x^7+1/4*a^5*A*x^4

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Maxima [A]  time = 1.4285, size = 161, normalized size = 1.38 \[ \frac{1}{22} \, B b^{5} x^{22} + \frac{1}{19} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{19} + \frac{5}{16} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{16} + \frac{10}{13} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{13} + \frac{1}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{10} + \frac{1}{4} \, A a^{5} x^{4} + \frac{1}{7} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/22*B*b^5*x^22 + 1/19*(5*B*a*b^4 + A*b^5)*x^19 + 5/16*(2*B*a^2*b^3 + A*a*b^4)*x
^16 + 10/13*(B*a^3*b^2 + A*a^2*b^3)*x^13 + 1/2*(B*a^4*b + 2*A*a^3*b^2)*x^10 + 1/
4*A*a^5*x^4 + 1/7*(B*a^5 + 5*A*a^4*b)*x^7

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Fricas [A]  time = 0.208703, size = 1, normalized size = 0.01 \[ \frac{1}{22} x^{22} b^{5} B + \frac{5}{19} x^{19} b^{4} a B + \frac{1}{19} x^{19} b^{5} A + \frac{5}{8} x^{16} b^{3} a^{2} B + \frac{5}{16} x^{16} b^{4} a A + \frac{10}{13} x^{13} b^{2} a^{3} B + \frac{10}{13} x^{13} b^{3} a^{2} A + \frac{1}{2} x^{10} b a^{4} B + x^{10} b^{2} a^{3} A + \frac{1}{7} x^{7} a^{5} B + \frac{5}{7} x^{7} b a^{4} A + \frac{1}{4} x^{4} a^{5} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/22*x^22*b^5*B + 5/19*x^19*b^4*a*B + 1/19*x^19*b^5*A + 5/8*x^16*b^3*a^2*B + 5/1
6*x^16*b^4*a*A + 10/13*x^13*b^2*a^3*B + 10/13*x^13*b^3*a^2*A + 1/2*x^10*b*a^4*B
+ x^10*b^2*a^3*A + 1/7*x^7*a^5*B + 5/7*x^7*b*a^4*A + 1/4*x^4*a^5*A

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Sympy [A]  time = 0.163197, size = 133, normalized size = 1.14 \[ \frac{A a^{5} x^{4}}{4} + \frac{B b^{5} x^{22}}{22} + x^{19} \left (\frac{A b^{5}}{19} + \frac{5 B a b^{4}}{19}\right ) + x^{16} \left (\frac{5 A a b^{4}}{16} + \frac{5 B a^{2} b^{3}}{8}\right ) + x^{13} \left (\frac{10 A a^{2} b^{3}}{13} + \frac{10 B a^{3} b^{2}}{13}\right ) + x^{10} \left (A a^{3} b^{2} + \frac{B a^{4} b}{2}\right ) + x^{7} \left (\frac{5 A a^{4} b}{7} + \frac{B a^{5}}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**5*x**4/4 + B*b**5*x**22/22 + x**19*(A*b**5/19 + 5*B*a*b**4/19) + x**16*(5*A
*a*b**4/16 + 5*B*a**2*b**3/8) + x**13*(10*A*a**2*b**3/13 + 10*B*a**3*b**2/13) +
x**10*(A*a**3*b**2 + B*a**4*b/2) + x**7*(5*A*a**4*b/7 + B*a**5/7)

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GIAC/XCAS [A]  time = 0.215658, size = 167, normalized size = 1.43 \[ \frac{1}{22} \, B b^{5} x^{22} + \frac{5}{19} \, B a b^{4} x^{19} + \frac{1}{19} \, A b^{5} x^{19} + \frac{5}{8} \, B a^{2} b^{3} x^{16} + \frac{5}{16} \, A a b^{4} x^{16} + \frac{10}{13} \, B a^{3} b^{2} x^{13} + \frac{10}{13} \, A a^{2} b^{3} x^{13} + \frac{1}{2} \, B a^{4} b x^{10} + A a^{3} b^{2} x^{10} + \frac{1}{7} \, B a^{5} x^{7} + \frac{5}{7} \, A a^{4} b x^{7} + \frac{1}{4} \, A a^{5} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/22*B*b^5*x^22 + 5/19*B*a*b^4*x^19 + 1/19*A*b^5*x^19 + 5/8*B*a^2*b^3*x^16 + 5/1
6*A*a*b^4*x^16 + 10/13*B*a^3*b^2*x^13 + 10/13*A*a^2*b^3*x^13 + 1/2*B*a^4*b*x^10
+ A*a^3*b^2*x^10 + 1/7*B*a^5*x^7 + 5/7*A*a^4*b*x^7 + 1/4*A*a^5*x^4