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

Optimal. Leaf size=67 \[ \frac{\left (a+b x^3\right )^7 (A b-2 a B)}{21 b^3}-\frac{a \left (a+b x^3\right )^6 (A b-a B)}{18 b^3}+\frac{B \left (a+b x^3\right )^8}{24 b^3} \]

[Out]

-(a*(A*b - a*B)*(a + b*x^3)^6)/(18*b^3) + ((A*b - 2*a*B)*(a + b*x^3)^7)/(21*b^3)
+ (B*(a + b*x^3)^8)/(24*b^3)

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Rubi [A]  time = 0.447123, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\left (a+b x^3\right )^7 (A b-2 a B)}{21 b^3}-\frac{a \left (a+b x^3\right )^6 (A b-a B)}{18 b^3}+\frac{B \left (a+b x^3\right )^8}{24 b^3} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(A*b - a*B)*(a + b*x^3)^6)/(18*b^3) + ((A*b - 2*a*B)*(a + b*x^3)^7)/(21*b^3)
+ (B*(a + b*x^3)^8)/(24*b^3)

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Rubi in Sympy [A]  time = 24.8959, size = 58, normalized size = 0.87 \[ \frac{B \left (a + b x^{3}\right )^{8}}{24 b^{3}} - \frac{a \left (a + b x^{3}\right )^{6} \left (A b - B a\right )}{18 b^{3}} + \frac{\left (a + b x^{3}\right )^{7} \left (A b - 2 B a\right )}{21 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x**3)**8/(24*b**3) - a*(a + b*x**3)**6*(A*b - B*a)/(18*b**3) + (a + b*x
**3)**7*(A*b - 2*B*a)/(21*b**3)

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Mathematica [A]  time = 0.0491097, size = 107, normalized size = 1.6 \[ \frac{1}{504} x^6 \left (84 a^5 A+56 a^4 x^3 (a B+5 A b)+210 a^3 b x^6 (a B+2 A b)+336 a^2 b^2 x^9 (a B+A b)+24 b^4 x^{15} (5 a B+A b)+140 a b^3 x^{12} (2 a B+A b)+21 b^5 B x^{18}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^6*(84*a^5*A + 56*a^4*(5*A*b + a*B)*x^3 + 210*a^3*b*(2*A*b + a*B)*x^6 + 336*a^
2*b^2*(A*b + a*B)*x^9 + 140*a*b^3*(A*b + 2*a*B)*x^12 + 24*b^4*(A*b + 5*a*B)*x^15
+ 21*b^5*B*x^18))/504

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Maple [B]  time = 0.002, size = 124, normalized size = 1.9 \[{\frac{{b}^{5}B{x}^{24}}{24}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{21}}{21}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{18}}{18}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{15}}{15}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{12}}{12}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{9}}{9}}+{\frac{{a}^{5}A{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*b^5*B*x^24+1/21*(A*b^5+5*B*a*b^4)*x^21+1/18*(5*A*a*b^4+10*B*a^2*b^3)*x^18+1
/15*(10*A*a^2*b^3+10*B*a^3*b^2)*x^15+1/12*(10*A*a^3*b^2+5*B*a^4*b)*x^12+1/9*(5*A
*a^4*b+B*a^5)*x^9+1/6*a^5*A*x^6

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Maxima [A]  time = 1.46211, size = 161, normalized size = 2.4 \[ \frac{1}{24} \, B b^{5} x^{24} + \frac{1}{21} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{21} + \frac{5}{18} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{18} + \frac{2}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{15} + \frac{5}{12} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{12} + \frac{1}{6} \, A a^{5} x^{6} + \frac{1}{9} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*B*b^5*x^24 + 1/21*(5*B*a*b^4 + A*b^5)*x^21 + 5/18*(2*B*a^2*b^3 + A*a*b^4)*x
^18 + 2/3*(B*a^3*b^2 + A*a^2*b^3)*x^15 + 5/12*(B*a^4*b + 2*A*a^3*b^2)*x^12 + 1/6
*A*a^5*x^6 + 1/9*(B*a^5 + 5*A*a^4*b)*x^9

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Fricas [A]  time = 0.202244, size = 1, normalized size = 0.01 \[ \frac{1}{24} x^{24} b^{5} B + \frac{5}{21} x^{21} b^{4} a B + \frac{1}{21} x^{21} b^{5} A + \frac{5}{9} x^{18} b^{3} a^{2} B + \frac{5}{18} x^{18} b^{4} a A + \frac{2}{3} x^{15} b^{2} a^{3} B + \frac{2}{3} x^{15} b^{3} a^{2} A + \frac{5}{12} x^{12} b a^{4} B + \frac{5}{6} x^{12} b^{2} a^{3} A + \frac{1}{9} x^{9} a^{5} B + \frac{5}{9} x^{9} b a^{4} A + \frac{1}{6} x^{6} a^{5} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*x^24*b^5*B + 5/21*x^21*b^4*a*B + 1/21*x^21*b^5*A + 5/9*x^18*b^3*a^2*B + 5/1
8*x^18*b^4*a*A + 2/3*x^15*b^2*a^3*B + 2/3*x^15*b^3*a^2*A + 5/12*x^12*b*a^4*B + 5
/6*x^12*b^2*a^3*A + 1/9*x^9*a^5*B + 5/9*x^9*b*a^4*A + 1/6*x^6*a^5*A

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Sympy [A]  time = 0.166453, size = 138, normalized size = 2.06 \[ \frac{A a^{5} x^{6}}{6} + \frac{B b^{5} x^{24}}{24} + x^{21} \left (\frac{A b^{5}}{21} + \frac{5 B a b^{4}}{21}\right ) + x^{18} \left (\frac{5 A a b^{4}}{18} + \frac{5 B a^{2} b^{3}}{9}\right ) + x^{15} \left (\frac{2 A a^{2} b^{3}}{3} + \frac{2 B a^{3} b^{2}}{3}\right ) + x^{12} \left (\frac{5 A a^{3} b^{2}}{6} + \frac{5 B a^{4} b}{12}\right ) + x^{9} \left (\frac{5 A a^{4} b}{9} + \frac{B a^{5}}{9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**5*x**6/6 + B*b**5*x**24/24 + x**21*(A*b**5/21 + 5*B*a*b**4/21) + x**18*(5*A
*a*b**4/18 + 5*B*a**2*b**3/9) + x**15*(2*A*a**2*b**3/3 + 2*B*a**3*b**2/3) + x**1
2*(5*A*a**3*b**2/6 + 5*B*a**4*b/12) + x**9*(5*A*a**4*b/9 + B*a**5/9)

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GIAC/XCAS [A]  time = 0.215473, size = 169, normalized size = 2.52 \[ \frac{1}{24} \, B b^{5} x^{24} + \frac{5}{21} \, B a b^{4} x^{21} + \frac{1}{21} \, A b^{5} x^{21} + \frac{5}{9} \, B a^{2} b^{3} x^{18} + \frac{5}{18} \, A a b^{4} x^{18} + \frac{2}{3} \, B a^{3} b^{2} x^{15} + \frac{2}{3} \, A a^{2} b^{3} x^{15} + \frac{5}{12} \, B a^{4} b x^{12} + \frac{5}{6} \, A a^{3} b^{2} x^{12} + \frac{1}{9} \, B a^{5} x^{9} + \frac{5}{9} \, A a^{4} b x^{9} + \frac{1}{6} \, A a^{5} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*B*b^5*x^24 + 5/21*B*a*b^4*x^21 + 1/21*A*b^5*x^21 + 5/9*B*a^2*b^3*x^18 + 5/1
8*A*a*b^4*x^18 + 2/3*B*a^3*b^2*x^15 + 2/3*A*a^2*b^3*x^15 + 5/12*B*a^4*b*x^12 + 5
/6*A*a^3*b^2*x^12 + 1/9*B*a^5*x^9 + 5/9*A*a^4*b*x^9 + 1/6*A*a^5*x^6