3.49 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{17}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{b \left (a+b x^2\right )^6 (A b-4 a B)}{336 a^3 x^{12}}+\frac{\left (a+b x^2\right )^6 (A b-4 a B)}{56 a^2 x^{14}}-\frac{A \left (a+b x^2\right )^6}{16 a x^{16}} \]

[Out]

-(A*(a + b*x^2)^6)/(16*a*x^16) + ((A*b - 4*a*B)*(a + b*x^2)^6)/(56*a^2*x^14) - (
b*(A*b - 4*a*B)*(a + b*x^2)^6)/(336*a^3*x^12)

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Rubi [A]  time = 0.171961, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{b \left (a+b x^2\right )^6 (A b-4 a B)}{336 a^3 x^{12}}+\frac{\left (a+b x^2\right )^6 (A b-4 a B)}{56 a^2 x^{14}}-\frac{A \left (a+b x^2\right )^6}{16 a x^{16}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^17,x]

[Out]

-(A*(a + b*x^2)^6)/(16*a*x^16) + ((A*b - 4*a*B)*(a + b*x^2)^6)/(56*a^2*x^14) - (
b*(A*b - 4*a*B)*(a + b*x^2)^6)/(336*a^3*x^12)

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Rubi in Sympy [A]  time = 14.1668, size = 68, normalized size = 0.89 \[ - \frac{A \left (a + b x^{2}\right )^{6}}{16 a x^{16}} + \frac{\left (a + b x^{2}\right )^{6} \left (A b - 4 B a\right )}{56 a^{2} x^{14}} - \frac{b \left (a + b x^{2}\right )^{6} \left (A b - 4 B a\right )}{336 a^{3} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**17,x)

[Out]

-A*(a + b*x**2)**6/(16*a*x**16) + (a + b*x**2)**6*(A*b - 4*B*a)/(56*a**2*x**14)
- b*(a + b*x**2)**6*(A*b - 4*B*a)/(336*a**3*x**12)

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Mathematica [A]  time = 0.0567621, size = 121, normalized size = 1.59 \[ -\frac{3 a^5 \left (7 A+8 B x^2\right )+20 a^4 b x^2 \left (6 A+7 B x^2\right )+56 a^3 b^2 x^4 \left (5 A+6 B x^2\right )+84 a^2 b^3 x^6 \left (4 A+5 B x^2\right )+70 a b^4 x^8 \left (3 A+4 B x^2\right )+28 b^5 x^{10} \left (2 A+3 B x^2\right )}{336 x^{16}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^17,x]

[Out]

-(28*b^5*x^10*(2*A + 3*B*x^2) + 70*a*b^4*x^8*(3*A + 4*B*x^2) + 84*a^2*b^3*x^6*(4
*A + 5*B*x^2) + 56*a^3*b^2*x^4*(5*A + 6*B*x^2) + 20*a^4*b*x^2*(6*A + 7*B*x^2) +
3*a^5*(7*A + 8*B*x^2))/(336*x^16)

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Maple [A]  time = 0.009, size = 104, normalized size = 1.4 \[ -{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{12\,{x}^{12}}}-{\frac{A{a}^{5}}{16\,{x}^{16}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{6\,{x}^{6}}}-{\frac{B{b}^{5}}{4\,{x}^{4}}}-{\frac{{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{{x}^{10}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{8\,{x}^{8}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{14\,{x}^{14}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^17,x)

[Out]

-5/12*a^3*b*(2*A*b+B*a)/x^12-1/16*A*a^5/x^16-1/6*b^4*(A*b+5*B*a)/x^6-1/4*B*b^5/x
^4-a^2*b^2*(A*b+B*a)/x^10-5/8*a*b^3*(A*b+2*B*a)/x^8-1/14*a^4*(5*A*b+B*a)/x^14

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Maxima [A]  time = 1.33512, size = 163, normalized size = 2.14 \[ -\frac{84 \, B b^{5} x^{12} + 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 210 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 336 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 21 \, A a^{5} + 140 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 24 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{336 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^17,x, algorithm="maxima")

[Out]

-1/336*(84*B*b^5*x^12 + 56*(5*B*a*b^4 + A*b^5)*x^10 + 210*(2*B*a^2*b^3 + A*a*b^4
)*x^8 + 336*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 21*A*a^5 + 140*(B*a^4*b + 2*A*a^3*b^2)
*x^4 + 24*(B*a^5 + 5*A*a^4*b)*x^2)/x^16

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Fricas [A]  time = 0.221645, size = 163, normalized size = 2.14 \[ -\frac{84 \, B b^{5} x^{12} + 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 210 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 336 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 21 \, A a^{5} + 140 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 24 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{336 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^17,x, algorithm="fricas")

[Out]

-1/336*(84*B*b^5*x^12 + 56*(5*B*a*b^4 + A*b^5)*x^10 + 210*(2*B*a^2*b^3 + A*a*b^4
)*x^8 + 336*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 21*A*a^5 + 140*(B*a^4*b + 2*A*a^3*b^2)
*x^4 + 24*(B*a^5 + 5*A*a^4*b)*x^2)/x^16

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Sympy [A]  time = 152.951, size = 128, normalized size = 1.68 \[ - \frac{21 A a^{5} + 84 B b^{5} x^{12} + x^{10} \left (56 A b^{5} + 280 B a b^{4}\right ) + x^{8} \left (210 A a b^{4} + 420 B a^{2} b^{3}\right ) + x^{6} \left (336 A a^{2} b^{3} + 336 B a^{3} b^{2}\right ) + x^{4} \left (280 A a^{3} b^{2} + 140 B a^{4} b\right ) + x^{2} \left (120 A a^{4} b + 24 B a^{5}\right )}{336 x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5*(B*x**2+A)/x**17,x)

[Out]

-(21*A*a**5 + 84*B*b**5*x**12 + x**10*(56*A*b**5 + 280*B*a*b**4) + x**8*(210*A*a
*b**4 + 420*B*a**2*b**3) + x**6*(336*A*a**2*b**3 + 336*B*a**3*b**2) + x**4*(280*
A*a**3*b**2 + 140*B*a**4*b) + x**2*(120*A*a**4*b + 24*B*a**5))/(336*x**16)

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GIAC/XCAS [A]  time = 0.230495, size = 171, normalized size = 2.25 \[ -\frac{84 \, B b^{5} x^{12} + 280 \, B a b^{4} x^{10} + 56 \, A b^{5} x^{10} + 420 \, B a^{2} b^{3} x^{8} + 210 \, A a b^{4} x^{8} + 336 \, B a^{3} b^{2} x^{6} + 336 \, A a^{2} b^{3} x^{6} + 140 \, B a^{4} b x^{4} + 280 \, A a^{3} b^{2} x^{4} + 24 \, B a^{5} x^{2} + 120 \, A a^{4} b x^{2} + 21 \, A a^{5}}{336 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^17,x, algorithm="giac")

[Out]

-1/336*(84*B*b^5*x^12 + 280*B*a*b^4*x^10 + 56*A*b^5*x^10 + 420*B*a^2*b^3*x^8 + 2
10*A*a*b^4*x^8 + 336*B*a^3*b^2*x^6 + 336*A*a^2*b^3*x^6 + 140*B*a^4*b*x^4 + 280*A
*a^3*b^2*x^4 + 24*B*a^5*x^2 + 120*A*a^4*b*x^2 + 21*A*a^5)/x^16