3.42 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{10}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{a^5 A}{9 x^9}-\frac{a^4 (a B+5 A b)}{7 x^7}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{10 a^2 b^2 (a B+A b)}{3 x^3}+b^4 x (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{x}+\frac{1}{3} b^5 B x^3 \]

[Out]

-(a^5*A)/(9*x^9) - (a^4*(5*A*b + a*B))/(7*x^7) - (a^3*b*(2*A*b + a*B))/x^5 - (10
*a^2*b^2*(A*b + a*B))/(3*x^3) - (5*a*b^3*(A*b + 2*a*B))/x + b^4*(A*b + 5*a*B)*x
+ (b^5*B*x^3)/3

_______________________________________________________________________________________

Rubi [A]  time = 0.201864, antiderivative size = 108, 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 \[ -\frac{a^5 A}{9 x^9}-\frac{a^4 (a B+5 A b)}{7 x^7}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{10 a^2 b^2 (a B+A b)}{3 x^3}+b^4 x (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{x}+\frac{1}{3} b^5 B x^3 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^10,x]

[Out]

-(a^5*A)/(9*x^9) - (a^4*(5*A*b + a*B))/(7*x^7) - (a^3*b*(2*A*b + a*B))/x^5 - (10
*a^2*b^2*(A*b + a*B))/(3*x^3) - (5*a*b^3*(A*b + 2*a*B))/x + b^4*(A*b + 5*a*B)*x
+ (b^5*B*x^3)/3

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{9 x^{9}} + \frac{B b^{5} x^{3}}{3} - \frac{a^{4} \left (5 A b + B a\right )}{7 x^{7}} - \frac{a^{3} b \left (2 A b + B a\right )}{x^{5}} - \frac{10 a^{2} b^{2} \left (A b + B a\right )}{3 x^{3}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{x} + \frac{b^{4} \left (A b + 5 B a\right ) \int A\, dx}{A} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**10,x)

[Out]

-A*a**5/(9*x**9) + B*b**5*x**3/3 - a**4*(5*A*b + B*a)/(7*x**7) - a**3*b*(2*A*b +
B*a)/x**5 - 10*a**2*b**2*(A*b + B*a)/(3*x**3) - 5*a*b**3*(A*b + 2*B*a)/x + b**4
*(A*b + 5*B*a)*Integral(A, x)/A

_______________________________________________________________________________________

Mathematica [A]  time = 0.0634757, size = 115, normalized size = 1.06 \[ -\frac{a^5 \left (7 A+9 B x^2\right )+9 a^4 b x^2 \left (5 A+7 B x^2\right )+42 a^3 b^2 x^4 \left (3 A+5 B x^2\right )+210 a^2 b^3 x^6 \left (A+3 B x^2\right )+315 a b^4 x^8 \left (A-B x^2\right )-21 b^5 x^{10} \left (3 A+B x^2\right )}{63 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^10,x]

[Out]

-(315*a*b^4*x^8*(A - B*x^2) - 21*b^5*x^10*(3*A + B*x^2) + 210*a^2*b^3*x^6*(A + 3
*B*x^2) + 42*a^3*b^2*x^4*(3*A + 5*B*x^2) + 9*a^4*b*x^2*(5*A + 7*B*x^2) + a^5*(7*
A + 9*B*x^2))/(63*x^9)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 102, normalized size = 0.9 \[{\frac{{b}^{5}B{x}^{3}}{3}}+Ax{b}^{5}+5\,Bxa{b}^{4}-{\frac{A{a}^{5}}{9\,{x}^{9}}}-{\frac{10\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{3\,{x}^{3}}}-{\frac{{a}^{3}b \left ( 2\,Ab+Ba \right ) }{{x}^{5}}}-5\,{\frac{a{b}^{3} \left ( Ab+2\,Ba \right ) }{x}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^10,x)

[Out]

1/3*b^5*B*x^3+A*x*b^5+5*B*x*a*b^4-1/9*a^5*A/x^9-10/3*a^2*b^2*(A*b+B*a)/x^3-a^3*b
*(2*A*b+B*a)/x^5-5*a*b^3*(A*b+2*B*a)/x-1/7*a^4*(5*A*b+B*a)/x^7

_______________________________________________________________________________________

Maxima [A]  time = 1.36525, size = 161, normalized size = 1.49 \[ \frac{1}{3} \, B b^{5} x^{3} +{\left (5 \, B a b^{4} + A b^{5}\right )} x - \frac{315 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 210 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 7 \, A a^{5} + 63 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 9 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{63 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^10,x, algorithm="maxima")

[Out]

1/3*B*b^5*x^3 + (5*B*a*b^4 + A*b^5)*x - 1/63*(315*(2*B*a^2*b^3 + A*a*b^4)*x^8 +
210*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 7*A*a^5 + 63*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 9*(
B*a^5 + 5*A*a^4*b)*x^2)/x^9

_______________________________________________________________________________________

Fricas [A]  time = 0.231796, size = 163, normalized size = 1.51 \[ \frac{21 \, B b^{5} x^{12} + 63 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} - 315 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 210 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 7 \, A a^{5} - 63 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 9 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{63 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^10,x, algorithm="fricas")

[Out]

1/63*(21*B*b^5*x^12 + 63*(5*B*a*b^4 + A*b^5)*x^10 - 315*(2*B*a^2*b^3 + A*a*b^4)*
x^8 - 210*(B*a^3*b^2 + A*a^2*b^3)*x^6 - 7*A*a^5 - 63*(B*a^4*b + 2*A*a^3*b^2)*x^4
- 9*(B*a^5 + 5*A*a^4*b)*x^2)/x^9

_______________________________________________________________________________________

Sympy [A]  time = 12.5944, size = 122, normalized size = 1.13 \[ \frac{B b^{5} x^{3}}{3} + x \left (A b^{5} + 5 B a b^{4}\right ) - \frac{7 A a^{5} + x^{8} \left (315 A a b^{4} + 630 B a^{2} b^{3}\right ) + x^{6} \left (210 A a^{2} b^{3} + 210 B a^{3} b^{2}\right ) + x^{4} \left (126 A a^{3} b^{2} + 63 B a^{4} b\right ) + x^{2} \left (45 A a^{4} b + 9 B a^{5}\right )}{63 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5*(B*x**2+A)/x**10,x)

[Out]

B*b**5*x**3/3 + x*(A*b**5 + 5*B*a*b**4) - (7*A*a**5 + x**8*(315*A*a*b**4 + 630*B
*a**2*b**3) + x**6*(210*A*a**2*b**3 + 210*B*a**3*b**2) + x**4*(126*A*a**3*b**2 +
63*B*a**4*b) + x**2*(45*A*a**4*b + 9*B*a**5))/(63*x**9)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.230042, size = 166, normalized size = 1.54 \[ \frac{1}{3} \, B b^{5} x^{3} + 5 \, B a b^{4} x + A b^{5} x - \frac{630 \, B a^{2} b^{3} x^{8} + 315 \, A a b^{4} x^{8} + 210 \, B a^{3} b^{2} x^{6} + 210 \, A a^{2} b^{3} x^{6} + 63 \, B a^{4} b x^{4} + 126 \, A a^{3} b^{2} x^{4} + 9 \, B a^{5} x^{2} + 45 \, A a^{4} b x^{2} + 7 \, A a^{5}}{63 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^10,x, algorithm="giac")

[Out]

1/3*B*b^5*x^3 + 5*B*a*b^4*x + A*b^5*x - 1/63*(630*B*a^2*b^3*x^8 + 315*A*a*b^4*x^
8 + 210*B*a^3*b^2*x^6 + 210*A*a^2*b^3*x^6 + 63*B*a^4*b*x^4 + 126*A*a^3*b^2*x^4 +
9*B*a^5*x^2 + 45*A*a^4*b*x^2 + 7*A*a^5)/x^9