3.44 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{12}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{a^5 A}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{5 a^2 b^2 (a B+A b)}{x^2}+\frac{1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{7} b^5 B x^7 \]

[Out]

-(a^5*A)/(11*x^11) - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x^5 - (
5*a^2*b^2*(A*b + a*B))/x^2 + 5*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5*a*B)*x^4)/4
+ (b^5*B*x^7)/7

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Rubi [A]  time = 0.19607, antiderivative size = 109, 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 \[ -\frac{a^5 A}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{5 a^2 b^2 (a B+A b)}{x^2}+\frac{1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{7} b^5 B x^7 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^12,x]

[Out]

-(a^5*A)/(11*x^11) - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x^5 - (
5*a^2*b^2*(A*b + a*B))/x^2 + 5*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5*a*B)*x^4)/4
+ (b^5*B*x^7)/7

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Rubi in Sympy [A]  time = 25.0005, size = 105, normalized size = 0.96 \[ - \frac{A a^{5}}{11 x^{11}} + \frac{B b^{5} x^{7}}{7} - \frac{a^{4} \left (5 A b + B a\right )}{8 x^{8}} - \frac{a^{3} b \left (2 A b + B a\right )}{x^{5}} - \frac{5 a^{2} b^{2} \left (A b + B a\right )}{x^{2}} + 5 a b^{3} x \left (A b + 2 B a\right ) + \frac{b^{4} x^{4} \left (A b + 5 B a\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**12,x)

[Out]

-A*a**5/(11*x**11) + B*b**5*x**7/7 - a**4*(5*A*b + B*a)/(8*x**8) - a**3*b*(2*A*b
+ B*a)/x**5 - 5*a**2*b**2*(A*b + B*a)/x**2 + 5*a*b**3*x*(A*b + 2*B*a) + b**4*x*
*4*(A*b + 5*B*a)/4

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Mathematica [A]  time = 0.0763636, size = 109, normalized size = 1. \[ -\frac{a^5 A}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{5 a^2 b^2 (a B+A b)}{x^2}+\frac{1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{7} b^5 B x^7 \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^12,x]

[Out]

-(a^5*A)/(11*x^11) - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x^5 - (
5*a^2*b^2*(A*b + a*B))/x^2 + 5*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5*a*B)*x^4)/4
+ (b^5*B*x^7)/7

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Maple [A]  time = 0.009, size = 108, normalized size = 1. \[{\frac{{b}^{5}B{x}^{7}}{7}}+{\frac{A{x}^{4}{b}^{5}}{4}}+{\frac{5\,B{x}^{4}a{b}^{4}}{4}}+5\,Axa{b}^{4}+10\,Bx{a}^{2}{b}^{3}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{8\,{x}^{8}}}-{\frac{A{a}^{5}}{11\,{x}^{11}}}-5\,{\frac{{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{{x}^{2}}}-{\frac{{a}^{3}b \left ( 2\,Ab+Ba \right ) }{{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^12,x)

[Out]

1/7*b^5*B*x^7+1/4*A*x^4*b^5+5/4*B*x^4*a*b^4+5*A*x*a*b^4+10*B*x*a^2*b^3-1/8*a^4*(
5*A*b+B*a)/x^8-1/11*a^5*A/x^11-5*a^2*b^2*(A*b+B*a)/x^2-a^3*b*(2*A*b+B*a)/x^5

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Maxima [A]  time = 1.4429, size = 162, normalized size = 1.49 \[ \frac{1}{7} \, B b^{5} x^{7} + \frac{1}{4} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{4} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x - \frac{440 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 88 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 8 \, A a^{5} + 11 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{88 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^12,x, algorithm="maxima")

[Out]

1/7*B*b^5*x^7 + 1/4*(5*B*a*b^4 + A*b^5)*x^4 + 5*(2*B*a^2*b^3 + A*a*b^4)*x - 1/88
*(440*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 88*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 8*A*a^5 + 1
1*(B*a^5 + 5*A*a^4*b)*x^3)/x^11

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Fricas [A]  time = 0.211696, size = 163, normalized size = 1.5 \[ \frac{88 \, B b^{5} x^{18} + 154 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 3080 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 3080 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 616 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 56 \, A a^{5} - 77 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^12,x, algorithm="fricas")

[Out]

1/616*(88*B*b^5*x^18 + 154*(5*B*a*b^4 + A*b^5)*x^15 + 3080*(2*B*a^2*b^3 + A*a*b^
4)*x^12 - 3080*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 616*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 5
6*A*a^5 - 77*(B*a^5 + 5*A*a^4*b)*x^3)/x^11

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Sympy [A]  time = 13.0614, size = 126, normalized size = 1.16 \[ \frac{B b^{5} x^{7}}{7} + x^{4} \left (\frac{A b^{5}}{4} + \frac{5 B a b^{4}}{4}\right ) + x \left (5 A a b^{4} + 10 B a^{2} b^{3}\right ) - \frac{8 A a^{5} + x^{9} \left (440 A a^{2} b^{3} + 440 B a^{3} b^{2}\right ) + x^{6} \left (176 A a^{3} b^{2} + 88 B a^{4} b\right ) + x^{3} \left (55 A a^{4} b + 11 B a^{5}\right )}{88 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5*(B*x**3+A)/x**12,x)

[Out]

B*b**5*x**7/7 + x**4*(A*b**5/4 + 5*B*a*b**4/4) + x*(5*A*a*b**4 + 10*B*a**2*b**3)
- (8*A*a**5 + x**9*(440*A*a**2*b**3 + 440*B*a**3*b**2) + x**6*(176*A*a**3*b**2
+ 88*B*a**4*b) + x**3*(55*A*a**4*b + 11*B*a**5))/(88*x**11)

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GIAC/XCAS [A]  time = 0.218003, size = 167, normalized size = 1.53 \[ \frac{1}{7} \, B b^{5} x^{7} + \frac{5}{4} \, B a b^{4} x^{4} + \frac{1}{4} \, A b^{5} x^{4} + 10 \, B a^{2} b^{3} x + 5 \, A a b^{4} x - \frac{440 \, B a^{3} b^{2} x^{9} + 440 \, A a^{2} b^{3} x^{9} + 88 \, B a^{4} b x^{6} + 176 \, A a^{3} b^{2} x^{6} + 11 \, B a^{5} x^{3} + 55 \, A a^{4} b x^{3} + 8 \, A a^{5}}{88 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^12,x, algorithm="giac")

[Out]

1/7*B*b^5*x^7 + 5/4*B*a*b^4*x^4 + 1/4*A*b^5*x^4 + 10*B*a^2*b^3*x + 5*A*a*b^4*x -
1/88*(440*B*a^3*b^2*x^9 + 440*A*a^2*b^3*x^9 + 88*B*a^4*b*x^6 + 176*A*a^3*b^2*x^
6 + 11*B*a^5*x^3 + 55*A*a^4*b*x^3 + 8*A*a^5)/x^11