3.24 \(\int x^8 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=95 \[ \frac{a^2 \left (a+b x^3\right )^6 (A b-a B)}{18 b^4}+\frac{\left (a+b x^3\right )^8 (A b-3 a B)}{24 b^4}-\frac{a \left (a+b x^3\right )^7 (2 A b-3 a B)}{21 b^4}+\frac{B \left (a+b x^3\right )^9}{27 b^4} \]

[Out]

(a^2*(A*b - a*B)*(a + b*x^3)^6)/(18*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x^3)^7)/(21
*b^4) + ((A*b - 3*a*B)*(a + b*x^3)^8)/(24*b^4) + (B*(a + b*x^3)^9)/(27*b^4)

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Rubi [A]  time = 0.607123, antiderivative size = 95, 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 \[ \frac{a^2 \left (a+b x^3\right )^6 (A b-a B)}{18 b^4}+\frac{\left (a+b x^3\right )^8 (A b-3 a B)}{24 b^4}-\frac{a \left (a+b x^3\right )^7 (2 A b-3 a B)}{21 b^4}+\frac{B \left (a+b x^3\right )^9}{27 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(a^2*(A*b - a*B)*(a + b*x^3)^6)/(18*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x^3)^7)/(21
*b^4) + ((A*b - 3*a*B)*(a + b*x^3)^8)/(24*b^4) + (B*(a + b*x^3)^9)/(27*b^4)

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Rubi in Sympy [A]  time = 30.6162, size = 85, normalized size = 0.89 \[ \frac{B \left (a + b x^{3}\right )^{9}}{27 b^{4}} + \frac{a^{2} \left (a + b x^{3}\right )^{6} \left (A b - B a\right )}{18 b^{4}} - \frac{a \left (a + b x^{3}\right )^{7} \left (2 A b - 3 B a\right )}{21 b^{4}} + \frac{\left (a + b x^{3}\right )^{8} \left (A b - 3 B a\right )}{24 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(b*x**3+a)**5*(B*x**3+A),x)

[Out]

B*(a + b*x**3)**9/(27*b**4) + a**2*(a + b*x**3)**6*(A*b - B*a)/(18*b**4) - a*(a
+ b*x**3)**7*(2*A*b - 3*B*a)/(21*b**4) + (a + b*x**3)**8*(A*b - 3*B*a)/(24*b**4)

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Mathematica [A]  time = 0.0551935, size = 107, normalized size = 1.13 \[ \frac{x^9 \left (168 a^5 A+126 a^4 x^3 (a B+5 A b)+504 a^3 b x^6 (a B+2 A b)+840 a^2 b^2 x^9 (a B+A b)+63 b^4 x^{15} (5 a B+A b)+360 a b^3 x^{12} (2 a B+A b)+56 b^5 B x^{18}\right )}{1512} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(x^9*(168*a^5*A + 126*a^4*(5*A*b + a*B)*x^3 + 504*a^3*b*(2*A*b + a*B)*x^6 + 840*
a^2*b^2*(A*b + a*B)*x^9 + 360*a*b^3*(A*b + 2*a*B)*x^12 + 63*b^4*(A*b + 5*a*B)*x^
15 + 56*b^5*B*x^18))/1512

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Maple [A]  time = 0.002, size = 124, normalized size = 1.3 \[{\frac{{b}^{5}B{x}^{27}}{27}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{24}}{24}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{21}}{21}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{18}}{18}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{15}}{15}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{12}}{12}}+{\frac{{a}^{5}A{x}^{9}}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b*x^3+a)^5*(B*x^3+A),x)

[Out]

1/27*b^5*B*x^27+1/24*(A*b^5+5*B*a*b^4)*x^24+1/21*(5*A*a*b^4+10*B*a^2*b^3)*x^21+1
/18*(10*A*a^2*b^3+10*B*a^3*b^2)*x^18+1/15*(10*A*a^3*b^2+5*B*a^4*b)*x^15+1/12*(5*
A*a^4*b+B*a^5)*x^12+1/9*a^5*A*x^9

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Maxima [A]  time = 1.41794, size = 161, normalized size = 1.69 \[ \frac{1}{27} \, B b^{5} x^{27} + \frac{1}{24} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{24} + \frac{5}{21} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{21} + \frac{5}{9} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{18} + \frac{1}{3} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{15} + \frac{1}{9} \, A a^{5} x^{9} + \frac{1}{12} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5*x^8,x, algorithm="maxima")

[Out]

1/27*B*b^5*x^27 + 1/24*(5*B*a*b^4 + A*b^5)*x^24 + 5/21*(2*B*a^2*b^3 + A*a*b^4)*x
^21 + 5/9*(B*a^3*b^2 + A*a^2*b^3)*x^18 + 1/3*(B*a^4*b + 2*A*a^3*b^2)*x^15 + 1/9*
A*a^5*x^9 + 1/12*(B*a^5 + 5*A*a^4*b)*x^12

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Fricas [A]  time = 0.193193, size = 1, normalized size = 0.01 \[ \frac{1}{27} x^{27} b^{5} B + \frac{5}{24} x^{24} b^{4} a B + \frac{1}{24} x^{24} b^{5} A + \frac{10}{21} x^{21} b^{3} a^{2} B + \frac{5}{21} x^{21} b^{4} a A + \frac{5}{9} x^{18} b^{2} a^{3} B + \frac{5}{9} x^{18} b^{3} a^{2} A + \frac{1}{3} x^{15} b a^{4} B + \frac{2}{3} x^{15} b^{2} a^{3} A + \frac{1}{12} x^{12} a^{5} B + \frac{5}{12} x^{12} b a^{4} A + \frac{1}{9} x^{9} a^{5} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5*x^8,x, algorithm="fricas")

[Out]

1/27*x^27*b^5*B + 5/24*x^24*b^4*a*B + 1/24*x^24*b^5*A + 10/21*x^21*b^3*a^2*B + 5
/21*x^21*b^4*a*A + 5/9*x^18*b^2*a^3*B + 5/9*x^18*b^3*a^2*A + 1/3*x^15*b*a^4*B +
2/3*x^15*b^2*a^3*A + 1/12*x^12*a^5*B + 5/12*x^12*b*a^4*A + 1/9*x^9*a^5*A

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Sympy [A]  time = 0.168798, size = 136, normalized size = 1.43 \[ \frac{A a^{5} x^{9}}{9} + \frac{B b^{5} x^{27}}{27} + x^{24} \left (\frac{A b^{5}}{24} + \frac{5 B a b^{4}}{24}\right ) + x^{21} \left (\frac{5 A a b^{4}}{21} + \frac{10 B a^{2} b^{3}}{21}\right ) + x^{18} \left (\frac{5 A a^{2} b^{3}}{9} + \frac{5 B a^{3} b^{2}}{9}\right ) + x^{15} \left (\frac{2 A a^{3} b^{2}}{3} + \frac{B a^{4} b}{3}\right ) + x^{12} \left (\frac{5 A a^{4} b}{12} + \frac{B a^{5}}{12}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(b*x**3+a)**5*(B*x**3+A),x)

[Out]

A*a**5*x**9/9 + B*b**5*x**27/27 + x**24*(A*b**5/24 + 5*B*a*b**4/24) + x**21*(5*A
*a*b**4/21 + 10*B*a**2*b**3/21) + x**18*(5*A*a**2*b**3/9 + 5*B*a**3*b**2/9) + x*
*15*(2*A*a**3*b**2/3 + B*a**4*b/3) + x**12*(5*A*a**4*b/12 + B*a**5/12)

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GIAC/XCAS [A]  time = 0.215987, size = 169, normalized size = 1.78 \[ \frac{1}{27} \, B b^{5} x^{27} + \frac{5}{24} \, B a b^{4} x^{24} + \frac{1}{24} \, A b^{5} x^{24} + \frac{10}{21} \, B a^{2} b^{3} x^{21} + \frac{5}{21} \, A a b^{4} x^{21} + \frac{5}{9} \, B a^{3} b^{2} x^{18} + \frac{5}{9} \, A a^{2} b^{3} x^{18} + \frac{1}{3} \, B a^{4} b x^{15} + \frac{2}{3} \, A a^{3} b^{2} x^{15} + \frac{1}{12} \, B a^{5} x^{12} + \frac{5}{12} \, A a^{4} b x^{12} + \frac{1}{9} \, A a^{5} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5*x^8,x, algorithm="giac")

[Out]

1/27*B*b^5*x^27 + 5/24*B*a*b^4*x^24 + 1/24*A*b^5*x^24 + 10/21*B*a^2*b^3*x^21 + 5
/21*A*a*b^4*x^21 + 5/9*B*a^3*b^2*x^18 + 5/9*A*a^2*b^3*x^18 + 1/3*B*a^4*b*x^15 +
2/3*A*a^3*b^2*x^15 + 1/12*B*a^5*x^12 + 5/12*A*a^4*b*x^12 + 1/9*A*a^5*x^9