3.44 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{12}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{a^5 A}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{9 x^9}-\frac{5 a^3 b (a B+2 A b)}{7 x^7}-\frac{2 a^2 b^2 (a B+A b)}{x^5}-\frac{b^4 (5 a B+A b)}{x}-\frac{5 a b^3 (2 a B+A b)}{3 x^3}+b^5 B x \]

[Out]

-(a^5*A)/(11*x^11) - (a^4*(5*A*b + a*B))/(9*x^9) - (5*a^3*b*(2*A*b + a*B))/(7*x^
7) - (2*a^2*b^2*(A*b + a*B))/x^5 - (5*a*b^3*(A*b + 2*a*B))/(3*x^3) - (b^4*(A*b +
5*a*B))/x + b^5*B*x

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Rubi [A]  time = 0.19647, 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 \[ -\frac{a^5 A}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{9 x^9}-\frac{5 a^3 b (a B+2 A b)}{7 x^7}-\frac{2 a^2 b^2 (a B+A b)}{x^5}-\frac{b^4 (5 a B+A b)}{x}-\frac{5 a b^3 (2 a B+A b)}{3 x^3}+b^5 B x \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^12,x]

[Out]

-(a^5*A)/(11*x^11) - (a^4*(5*A*b + a*B))/(9*x^9) - (5*a^3*b*(2*A*b + a*B))/(7*x^
7) - (2*a^2*b^2*(A*b + a*B))/x^5 - (5*a*b^3*(A*b + 2*a*B))/(3*x^3) - (b^4*(A*b +
5*a*B))/x + b^5*B*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{11 x^{11}} - \frac{a^{4} \left (5 A b + B a\right )}{9 x^{9}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{7 x^{7}} - \frac{2 a^{2} b^{2} \left (A b + B a\right )}{x^{5}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{3 x^{3}} + b^{5} \int B\, dx - \frac{b^{4} \left (A b + 5 B a\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**12,x)

[Out]

-A*a**5/(11*x**11) - a**4*(5*A*b + B*a)/(9*x**9) - 5*a**3*b*(2*A*b + B*a)/(7*x**
7) - 2*a**2*b**2*(A*b + B*a)/x**5 - 5*a*b**3*(A*b + 2*B*a)/(3*x**3) + b**5*Integ
ral(B, x) - b**4*(A*b + 5*B*a)/x

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Mathematica [A]  time = 0.0774845, size = 122, normalized size = 1.13 \[ -\frac{a^5 \left (9 A+11 B x^2\right )}{99 x^{11}}-\frac{5 a^4 b \left (7 A+9 B x^2\right )}{63 x^9}-\frac{2 a^3 b^2 \left (5 A+7 B x^2\right )}{7 x^7}-\frac{2 a^2 b^3 \left (3 A+5 B x^2\right )}{3 x^5}-\frac{5 a b^4 \left (A+3 B x^2\right )}{3 x^3}-\frac{A b^5}{x}+b^5 B x \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^12,x]

[Out]

-((A*b^5)/x) + b^5*B*x - (5*a*b^4*(A + 3*B*x^2))/(3*x^3) - (2*a^2*b^3*(3*A + 5*B
*x^2))/(3*x^5) - (2*a^3*b^2*(5*A + 7*B*x^2))/(7*x^7) - (5*a^4*b*(7*A + 9*B*x^2))
/(63*x^9) - (a^5*(9*A + 11*B*x^2))/(99*x^11)

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Maple [A]  time = 0.01, size = 101, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{11\,{x}^{11}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{9\,{x}^{9}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{7\,{x}^{7}}}-2\,{\frac{{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{{x}^{5}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{3\,{x}^{3}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{x}}+{b}^{5}Bx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^12,x)

[Out]

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

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Maxima [A]  time = 1.34185, size = 161, normalized size = 1.49 \[ B b^{5} x - \frac{693 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 1155 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1386 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 63 \, A a^{5} + 495 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 77 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{693 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^12,x, algorithm="maxima")

[Out]

B*b^5*x - 1/693*(693*(5*B*a*b^4 + A*b^5)*x^10 + 1155*(2*B*a^2*b^3 + A*a*b^4)*x^8
+ 1386*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 63*A*a^5 + 495*(B*a^4*b + 2*A*a^3*b^2)*x^4
+ 77*(B*a^5 + 5*A*a^4*b)*x^2)/x^11

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Fricas [A]  time = 0.230145, size = 163, normalized size = 1.51 \[ \frac{693 \, B b^{5} x^{12} - 693 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} - 1155 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 1386 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 63 \, A a^{5} - 495 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 77 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{693 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^12,x, algorithm="fricas")

[Out]

1/693*(693*B*b^5*x^12 - 693*(5*B*a*b^4 + A*b^5)*x^10 - 1155*(2*B*a^2*b^3 + A*a*b
^4)*x^8 - 1386*(B*a^3*b^2 + A*a^2*b^3)*x^6 - 63*A*a^5 - 495*(B*a^4*b + 2*A*a^3*b
^2)*x^4 - 77*(B*a^5 + 5*A*a^4*b)*x^2)/x^11

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Sympy [A]  time = 25.8644, size = 122, normalized size = 1.13 \[ B b^{5} x - \frac{63 A a^{5} + x^{10} \left (693 A b^{5} + 3465 B a b^{4}\right ) + x^{8} \left (1155 A a b^{4} + 2310 B a^{2} b^{3}\right ) + x^{6} \left (1386 A a^{2} b^{3} + 1386 B a^{3} b^{2}\right ) + x^{4} \left (990 A a^{3} b^{2} + 495 B a^{4} b\right ) + x^{2} \left (385 A a^{4} b + 77 B a^{5}\right )}{693 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5*(B*x**2+A)/x**12,x)

[Out]

B*b**5*x - (63*A*a**5 + x**10*(693*A*b**5 + 3465*B*a*b**4) + x**8*(1155*A*a*b**4
+ 2310*B*a**2*b**3) + x**6*(1386*A*a**2*b**3 + 1386*B*a**3*b**2) + x**4*(990*A*
a**3*b**2 + 495*B*a**4*b) + x**2*(385*A*a**4*b + 77*B*a**5))/(693*x**11)

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GIAC/XCAS [A]  time = 0.226987, size = 169, normalized size = 1.56 \[ B b^{5} x - \frac{3465 \, B a b^{4} x^{10} + 693 \, A b^{5} x^{10} + 2310 \, B a^{2} b^{3} x^{8} + 1155 \, A a b^{4} x^{8} + 1386 \, B a^{3} b^{2} x^{6} + 1386 \, A a^{2} b^{3} x^{6} + 495 \, B a^{4} b x^{4} + 990 \, A a^{3} b^{2} x^{4} + 77 \, B a^{5} x^{2} + 385 \, A a^{4} b x^{2} + 63 \, A a^{5}}{693 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^12,x, algorithm="giac")

[Out]

B*b^5*x - 1/693*(3465*B*a*b^4*x^10 + 693*A*b^5*x^10 + 2310*B*a^2*b^3*x^8 + 1155*
A*a*b^4*x^8 + 1386*B*a^3*b^2*x^6 + 1386*A*a^2*b^3*x^6 + 495*B*a^4*b*x^4 + 990*A*
a^3*b^2*x^4 + 77*B*a^5*x^2 + 385*A*a^4*b*x^2 + 63*A*a^5)/x^11