3.40 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^8} \, dx\)

Optimal. Leaf size=111 \[ -\frac{a^5 A}{7 x^7}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}-\frac{10 a^2 b^2 (a B+A b)}{x}+\frac{1}{3} b^4 x^3 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{5} b^5 B x^5 \]

[Out]

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

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Rubi [A]  time = 0.192304, antiderivative size = 111, 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 \[ -\frac{a^5 A}{7 x^7}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}-\frac{10 a^2 b^2 (a B+A b)}{x}+\frac{1}{3} b^4 x^3 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{5} b^5 B x^5 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 27.008, size = 107, normalized size = 0.96 \[ - \frac{A a^{5}}{7 x^{7}} + \frac{B b^{5} x^{5}}{5} - \frac{a^{4} \left (5 A b + B a\right )}{5 x^{5}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{3 x^{3}} - \frac{10 a^{2} b^{2} \left (A b + B a\right )}{x} + 5 a b^{3} x \left (A b + 2 B a\right ) + \frac{b^{4} x^{3} \left (A b + 5 B a\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.0728265, size = 111, normalized size = 1. \[ -\frac{a^5 A}{7 x^7}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}-\frac{10 a^2 b^2 (a B+A b)}{x}+\frac{1}{3} b^4 x^3 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{5} b^5 B x^5 \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^8,x]

[Out]

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

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Maple [A]  time = 0.008, size = 108, normalized size = 1. \[{\frac{{b}^{5}B{x}^{5}}{5}}+{\frac{A{x}^{3}{b}^{5}}{3}}+{\frac{5\,B{x}^{3}a{b}^{4}}{3}}+5\,Axa{b}^{4}+10\,Bx{a}^{2}{b}^{3}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{3\,{x}^{3}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{5\,{x}^{5}}}-10\,{\frac{{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{x}}-{\frac{A{a}^{5}}{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^8,x)

[Out]

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

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Maxima [A]  time = 1.35688, size = 162, normalized size = 1.46 \[ \frac{1}{5} \, B b^{5} x^{5} + \frac{1}{3} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{3} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x - \frac{1050 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 15 \, A a^{5} + 175 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 21 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^8,x, algorithm="maxima")

[Out]

1/5*B*b^5*x^5 + 1/3*(5*B*a*b^4 + A*b^5)*x^3 + 5*(2*B*a^2*b^3 + A*a*b^4)*x - 1/10
5*(1050*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 15*A*a^5 + 175*(B*a^4*b + 2*A*a^3*b^2)*x^4
+ 21*(B*a^5 + 5*A*a^4*b)*x^2)/x^7

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Fricas [A]  time = 0.215868, size = 163, normalized size = 1.47 \[ \frac{21 \, B b^{5} x^{12} + 35 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 525 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 1050 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 15 \, A a^{5} - 175 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 21 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^8,x, algorithm="fricas")

[Out]

1/105*(21*B*b^5*x^12 + 35*(5*B*a*b^4 + A*b^5)*x^10 + 525*(2*B*a^2*b^3 + A*a*b^4)
*x^8 - 1050*(B*a^3*b^2 + A*a^2*b^3)*x^6 - 15*A*a^5 - 175*(B*a^4*b + 2*A*a^3*b^2)
*x^4 - 21*(B*a^5 + 5*A*a^4*b)*x^2)/x^7

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Sympy [A]  time = 6.67653, size = 126, normalized size = 1.14 \[ \frac{B b^{5} x^{5}}{5} + x^{3} \left (\frac{A b^{5}}{3} + \frac{5 B a b^{4}}{3}\right ) + x \left (5 A a b^{4} + 10 B a^{2} b^{3}\right ) - \frac{15 A a^{5} + x^{6} \left (1050 A a^{2} b^{3} + 1050 B a^{3} b^{2}\right ) + x^{4} \left (350 A a^{3} b^{2} + 175 B a^{4} b\right ) + x^{2} \left (105 A a^{4} b + 21 B a^{5}\right )}{105 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5*(B*x**2+A)/x**8,x)

[Out]

B*b**5*x**5/5 + x**3*(A*b**5/3 + 5*B*a*b**4/3) + x*(5*A*a*b**4 + 10*B*a**2*b**3)
- (15*A*a**5 + x**6*(1050*A*a**2*b**3 + 1050*B*a**3*b**2) + x**4*(350*A*a**3*b*
*2 + 175*B*a**4*b) + x**2*(105*A*a**4*b + 21*B*a**5))/(105*x**7)

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GIAC/XCAS [A]  time = 0.235117, size = 167, normalized size = 1.5 \[ \frac{1}{5} \, B b^{5} x^{5} + \frac{5}{3} \, B a b^{4} x^{3} + \frac{1}{3} \, A b^{5} x^{3} + 10 \, B a^{2} b^{3} x + 5 \, A a b^{4} x - \frac{1050 \, B a^{3} b^{2} x^{6} + 1050 \, A a^{2} b^{3} x^{6} + 175 \, B a^{4} b x^{4} + 350 \, A a^{3} b^{2} x^{4} + 21 \, B a^{5} x^{2} + 105 \, A a^{4} b x^{2} + 15 \, A a^{5}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^8,x, algorithm="giac")

[Out]

1/5*B*b^5*x^5 + 5/3*B*a*b^4*x^3 + 1/3*A*b^5*x^3 + 10*B*a^2*b^3*x + 5*A*a*b^4*x -
1/105*(1050*B*a^3*b^2*x^6 + 1050*A*a^2*b^3*x^6 + 175*B*a^4*b*x^4 + 350*A*a^3*b^
2*x^4 + 21*B*a^5*x^2 + 105*A*a^4*b*x^2 + 15*A*a^5)/x^7