3.43 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{a^5 A}{10 x^{10}}-\frac{a^4 (a B+5 A b)}{7 x^7}-\frac{5 a^3 b (a B+2 A b)}{4 x^4}-\frac{10 a^2 b^2 (a B+A b)}{x}+\frac{1}{5} b^4 x^5 (5 a B+A b)+\frac{5}{2} a b^3 x^2 (2 a B+A b)+\frac{1}{8} b^5 B x^8 \]

[Out]

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

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Rubi [A]  time = 0.212146, antiderivative size = 115, 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 \[ -\frac{a^5 A}{10 x^{10}}-\frac{a^4 (a B+5 A b)}{7 x^7}-\frac{5 a^3 b (a B+2 A b)}{4 x^4}-\frac{10 a^2 b^2 (a B+A b)}{x}+\frac{1}{5} b^4 x^5 (5 a B+A b)+\frac{5}{2} a b^3 x^2 (2 a B+A b)+\frac{1}{8} b^5 B x^8 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^11,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{10 x^{10}} + \frac{B b^{5} x^{8}}{8} - \frac{a^{4} \left (5 A b + B a\right )}{7 x^{7}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{4 x^{4}} - \frac{10 a^{2} b^{2} \left (A b + B a\right )}{x} + 5 a b^{3} \left (A b + 2 B a\right ) \int x\, dx + \frac{b^{4} x^{5} \left (A b + 5 B a\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**11,x)

[Out]

-A*a**5/(10*x**10) + B*b**5*x**8/8 - a**4*(5*A*b + B*a)/(7*x**7) - 5*a**3*b*(2*A
*b + B*a)/(4*x**4) - 10*a**2*b**2*(A*b + B*a)/x + 5*a*b**3*(A*b + 2*B*a)*Integra
l(x, x) + b**4*x**5*(A*b + 5*B*a)/5

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Mathematica [A]  time = 0.0467742, size = 118, normalized size = 1.03 \[ \frac{-4 a^5 \left (7 A+10 B x^3\right )-50 a^4 b x^3 \left (4 A+7 B x^3\right )-700 a^3 b^2 x^6 \left (A+4 B x^3\right )+1400 a^2 b^3 x^9 \left (B x^3-2 A\right )+140 a b^4 x^{12} \left (5 A+2 B x^3\right )+7 b^5 x^{15} \left (8 A+5 B x^3\right )}{280 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^11,x]

[Out]

(1400*a^2*b^3*x^9*(-2*A + B*x^3) + 140*a*b^4*x^12*(5*A + 2*B*x^3) - 700*a^3*b^2*
x^6*(A + 4*B*x^3) + 7*b^5*x^15*(8*A + 5*B*x^3) - 50*a^4*b*x^3*(4*A + 7*B*x^3) -
4*a^5*(7*A + 10*B*x^3))/(280*x^10)

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Maple [A]  time = 0.008, size = 111, normalized size = 1. \[{\frac{{b}^{5}B{x}^{8}}{8}}+{\frac{A{x}^{5}{b}^{5}}{5}}+B{x}^{5}a{b}^{4}+{\frac{5\,A{x}^{2}a{b}^{4}}{2}}+5\,B{x}^{2}{a}^{2}{b}^{3}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{4\,{x}^{4}}}-{\frac{A{a}^{5}}{10\,{x}^{10}}}-10\,{\frac{{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{x}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^11,x)

[Out]

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

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Maxima [A]  time = 1.43299, size = 165, normalized size = 1.43 \[ \frac{1}{8} \, B b^{5} x^{8} + \frac{1}{5} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac{5}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{2} - \frac{1400 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 175 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 14 \, A a^{5} + 20 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{140 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^11,x, algorithm="maxima")

[Out]

1/8*B*b^5*x^8 + 1/5*(5*B*a*b^4 + A*b^5)*x^5 + 5/2*(2*B*a^2*b^3 + A*a*b^4)*x^2 -
1/140*(1400*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 175*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 14*A
*a^5 + 20*(B*a^5 + 5*A*a^4*b)*x^3)/x^10

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Fricas [A]  time = 0.216433, size = 163, normalized size = 1.42 \[ \frac{35 \, B b^{5} x^{18} + 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 700 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 2800 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 350 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 28 \, A a^{5} - 40 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{280 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^11,x, algorithm="fricas")

[Out]

1/280*(35*B*b^5*x^18 + 56*(5*B*a*b^4 + A*b^5)*x^15 + 700*(2*B*a^2*b^3 + A*a*b^4)
*x^12 - 2800*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 350*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 28*
A*a^5 - 40*(B*a^5 + 5*A*a^4*b)*x^3)/x^10

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Sympy [A]  time = 11.7063, size = 126, normalized size = 1.1 \[ \frac{B b^{5} x^{8}}{8} + x^{5} \left (\frac{A b^{5}}{5} + B a b^{4}\right ) + x^{2} \left (\frac{5 A a b^{4}}{2} + 5 B a^{2} b^{3}\right ) - \frac{14 A a^{5} + x^{9} \left (1400 A a^{2} b^{3} + 1400 B a^{3} b^{2}\right ) + x^{6} \left (350 A a^{3} b^{2} + 175 B a^{4} b\right ) + x^{3} \left (100 A a^{4} b + 20 B a^{5}\right )}{140 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5*(B*x**3+A)/x**11,x)

[Out]

B*b**5*x**8/8 + x**5*(A*b**5/5 + B*a*b**4) + x**2*(5*A*a*b**4/2 + 5*B*a**2*b**3)
- (14*A*a**5 + x**9*(1400*A*a**2*b**3 + 1400*B*a**3*b**2) + x**6*(350*A*a**3*b*
*2 + 175*B*a**4*b) + x**3*(100*A*a**4*b + 20*B*a**5))/(140*x**10)

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GIAC/XCAS [A]  time = 0.218673, size = 171, normalized size = 1.49 \[ \frac{1}{8} \, B b^{5} x^{8} + B a b^{4} x^{5} + \frac{1}{5} \, A b^{5} x^{5} + 5 \, B a^{2} b^{3} x^{2} + \frac{5}{2} \, A a b^{4} x^{2} - \frac{1400 \, B a^{3} b^{2} x^{9} + 1400 \, A a^{2} b^{3} x^{9} + 175 \, B a^{4} b x^{6} + 350 \, A a^{3} b^{2} x^{6} + 20 \, B a^{5} x^{3} + 100 \, A a^{4} b x^{3} + 14 \, A a^{5}}{140 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^11,x, algorithm="giac")

[Out]

1/8*B*b^5*x^8 + B*a*b^4*x^5 + 1/5*A*b^5*x^5 + 5*B*a^2*b^3*x^2 + 5/2*A*a*b^4*x^2
- 1/140*(1400*B*a^3*b^2*x^9 + 1400*A*a^2*b^3*x^9 + 175*B*a^4*b*x^6 + 350*A*a^3*b
^2*x^6 + 20*B*a^5*x^3 + 100*A*a^4*b*x^3 + 14*A*a^5)/x^10