3.50 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{18}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{a^5 A}{17 x^{17}}-\frac{a^4 (a B+5 A b)}{15 x^{15}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{5 a b^3 (2 a B+A b)}{9 x^9}-\frac{b^5 B}{5 x^5} \]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(15*x^15) - (5*a^3*b*(2*A*b + a*B))/(13
*x^13) - (10*a^2*b^2*(A*b + a*B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(9*x^9) -
(b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(5*x^5)

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Rubi [A]  time = 0.19153, antiderivative size = 117, 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 \[ -\frac{a^5 A}{17 x^{17}}-\frac{a^4 (a B+5 A b)}{15 x^{15}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{5 a b^3 (2 a B+A b)}{9 x^9}-\frac{b^5 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^18,x]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(15*x^15) - (5*a^3*b*(2*A*b + a*B))/(13
*x^13) - (10*a^2*b^2*(A*b + a*B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(9*x^9) -
(b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(5*x^5)

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Rubi in Sympy [A]  time = 24.4289, size = 116, normalized size = 0.99 \[ - \frac{A a^{5}}{17 x^{17}} - \frac{B b^{5}}{5 x^{5}} - \frac{a^{4} \left (5 A b + B a\right )}{15 x^{15}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{13 x^{13}} - \frac{10 a^{2} b^{2} \left (A b + B a\right )}{11 x^{11}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{9 x^{9}} - \frac{b^{4} \left (A b + 5 B a\right )}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**18,x)

[Out]

-A*a**5/(17*x**17) - B*b**5/(5*x**5) - a**4*(5*A*b + B*a)/(15*x**15) - 5*a**3*b*
(2*A*b + B*a)/(13*x**13) - 10*a**2*b**2*(A*b + B*a)/(11*x**11) - 5*a*b**3*(A*b +
2*B*a)/(9*x**9) - b**4*(A*b + 5*B*a)/(7*x**7)

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Mathematica [A]  time = 0.0839226, size = 117, normalized size = 1. \[ -\frac{a^5 A}{17 x^{17}}-\frac{a^4 (a B+5 A b)}{15 x^{15}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{5 a b^3 (2 a B+A b)}{9 x^9}-\frac{b^5 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^18,x]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(15*x^15) - (5*a^3*b*(2*A*b + a*B))/(13
*x^13) - (10*a^2*b^2*(A*b + a*B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(9*x^9) -
(b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(5*x^5)

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Maple [A]  time = 0.009, size = 104, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{17\,{x}^{17}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{15\,{x}^{15}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{13\,{x}^{13}}}-{\frac{10\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{9\,{x}^{9}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{7\,{x}^{7}}}-{\frac{B{b}^{5}}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^18,x)

[Out]

-1/17*a^5*A/x^17-1/15*a^4*(5*A*b+B*a)/x^15-5/13*a^3*b*(2*A*b+B*a)/x^13-10/11*a^2
*b^2*(A*b+B*a)/x^11-5/9*a*b^3*(A*b+2*B*a)/x^9-1/7*b^4*(A*b+5*B*a)/x^7-1/5*b^5*B/
x^5

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Maxima [A]  time = 1.35182, size = 163, normalized size = 1.39 \[ -\frac{153153 \, B b^{5} x^{12} + 109395 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 425425 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 696150 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 45045 \, A a^{5} + 294525 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 51051 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{765765 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^18,x, algorithm="maxima")

[Out]

-1/765765*(153153*B*b^5*x^12 + 109395*(5*B*a*b^4 + A*b^5)*x^10 + 425425*(2*B*a^2
*b^3 + A*a*b^4)*x^8 + 696150*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 45045*A*a^5 + 294525*
(B*a^4*b + 2*A*a^3*b^2)*x^4 + 51051*(B*a^5 + 5*A*a^4*b)*x^2)/x^17

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Fricas [A]  time = 0.21213, size = 163, normalized size = 1.39 \[ -\frac{153153 \, B b^{5} x^{12} + 109395 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 425425 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 696150 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 45045 \, A a^{5} + 294525 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 51051 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{765765 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^18,x, algorithm="fricas")

[Out]

-1/765765*(153153*B*b^5*x^12 + 109395*(5*B*a*b^4 + A*b^5)*x^10 + 425425*(2*B*a^2
*b^3 + A*a*b^4)*x^8 + 696150*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 45045*A*a^5 + 294525*
(B*a^4*b + 2*A*a^3*b^2)*x^4 + 51051*(B*a^5 + 5*A*a^4*b)*x^2)/x^17

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5*(B*x**2+A)/x**18,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219056, size = 171, normalized size = 1.46 \[ -\frac{153153 \, B b^{5} x^{12} + 546975 \, B a b^{4} x^{10} + 109395 \, A b^{5} x^{10} + 850850 \, B a^{2} b^{3} x^{8} + 425425 \, A a b^{4} x^{8} + 696150 \, B a^{3} b^{2} x^{6} + 696150 \, A a^{2} b^{3} x^{6} + 294525 \, B a^{4} b x^{4} + 589050 \, A a^{3} b^{2} x^{4} + 51051 \, B a^{5} x^{2} + 255255 \, A a^{4} b x^{2} + 45045 \, A a^{5}}{765765 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^18,x, algorithm="giac")

[Out]

-1/765765*(153153*B*b^5*x^12 + 546975*B*a*b^4*x^10 + 109395*A*b^5*x^10 + 850850*
B*a^2*b^3*x^8 + 425425*A*a*b^4*x^8 + 696150*B*a^3*b^2*x^6 + 696150*A*a^2*b^3*x^6
+ 294525*B*a^4*b*x^4 + 589050*A*a^3*b^2*x^4 + 51051*B*a^5*x^2 + 255255*A*a^4*b*
x^2 + 45045*A*a^5)/x^17