3.538 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{11}} \, dx\)

Optimal. Leaf size=99 \[ -\frac{a^4 A}{10 x^{10}}-\frac{a^3 (a B+4 A b)}{9 x^9}-\frac{a^2 b (2 a B+3 A b)}{4 x^8}-\frac{b^3 (4 a B+A b)}{6 x^6}-\frac{2 a b^2 (3 a B+2 A b)}{7 x^7}-\frac{b^4 B}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.13159, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{a^4 A}{10 x^{10}}-\frac{a^3 (a B+4 A b)}{9 x^9}-\frac{a^2 b (2 a B+3 A b)}{4 x^8}-\frac{b^3 (4 a B+A b)}{6 x^6}-\frac{2 a b^2 (3 a B+2 A b)}{7 x^7}-\frac{b^4 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^11,x]

[Out]

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

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Rubi in Sympy [A]  time = 35.9394, size = 97, normalized size = 0.98 \[ - \frac{A a^{4}}{10 x^{10}} - \frac{B b^{4}}{5 x^{5}} - \frac{a^{3} \left (4 A b + B a\right )}{9 x^{9}} - \frac{a^{2} b \left (3 A b + 2 B a\right )}{4 x^{8}} - \frac{2 a b^{2} \left (2 A b + 3 B a\right )}{7 x^{7}} - \frac{b^{3} \left (A b + 4 B a\right )}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/x**11,x)

[Out]

-A*a**4/(10*x**10) - B*b**4/(5*x**5) - a**3*(4*A*b + B*a)/(9*x**9) - a**2*b*(3*A
*b + 2*B*a)/(4*x**8) - 2*a*b**2*(2*A*b + 3*B*a)/(7*x**7) - b**3*(A*b + 4*B*a)/(6
*x**6)

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Mathematica [A]  time = 0.0461348, size = 88, normalized size = 0.89 \[ -\frac{14 a^4 (9 A+10 B x)+70 a^3 b x (8 A+9 B x)+135 a^2 b^2 x^2 (7 A+8 B x)+120 a b^3 x^3 (6 A+7 B x)+42 b^4 x^4 (5 A+6 B x)}{1260 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^11,x]

[Out]

-(42*b^4*x^4*(5*A + 6*B*x) + 120*a*b^3*x^3*(6*A + 7*B*x) + 135*a^2*b^2*x^2*(7*A
+ 8*B*x) + 70*a^3*b*x*(8*A + 9*B*x) + 14*a^4*(9*A + 10*B*x))/(1260*x^10)

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Maple [A]  time = 0.009, size = 88, normalized size = 0.9 \[ -{\frac{A{a}^{4}}{10\,{x}^{10}}}-{\frac{{a}^{3} \left ( 4\,Ab+Ba \right ) }{9\,{x}^{9}}}-{\frac{{a}^{2}b \left ( 3\,Ab+2\,Ba \right ) }{4\,{x}^{8}}}-{\frac{2\,a{b}^{2} \left ( 2\,Ab+3\,Ba \right ) }{7\,{x}^{7}}}-{\frac{{b}^{3} \left ( Ab+4\,Ba \right ) }{6\,{x}^{6}}}-{\frac{{b}^{4}B}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^11,x)

[Out]

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

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Maxima [A]  time = 0.677381, size = 134, normalized size = 1.35 \[ -\frac{252 \, B b^{4} x^{5} + 126 \, A a^{4} + 210 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 360 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 315 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 140 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{1260 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x^11,x, algorithm="maxima")

[Out]

-1/1260*(252*B*b^4*x^5 + 126*A*a^4 + 210*(4*B*a*b^3 + A*b^4)*x^4 + 360*(3*B*a^2*
b^2 + 2*A*a*b^3)*x^3 + 315*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 140*(B*a^4 + 4*A*a^3*
b)*x)/x^10

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Fricas [A]  time = 0.303153, size = 134, normalized size = 1.35 \[ -\frac{252 \, B b^{4} x^{5} + 126 \, A a^{4} + 210 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 360 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 315 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 140 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{1260 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x^11,x, algorithm="fricas")

[Out]

-1/1260*(252*B*b^4*x^5 + 126*A*a^4 + 210*(4*B*a*b^3 + A*b^4)*x^4 + 360*(3*B*a^2*
b^2 + 2*A*a*b^3)*x^3 + 315*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 140*(B*a^4 + 4*A*a^3*
b)*x)/x^10

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Sympy [A]  time = 23.6786, size = 102, normalized size = 1.03 \[ - \frac{126 A a^{4} + 252 B b^{4} x^{5} + x^{4} \left (210 A b^{4} + 840 B a b^{3}\right ) + x^{3} \left (720 A a b^{3} + 1080 B a^{2} b^{2}\right ) + x^{2} \left (945 A a^{2} b^{2} + 630 B a^{3} b\right ) + x \left (560 A a^{3} b + 140 B a^{4}\right )}{1260 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/x**11,x)

[Out]

-(126*A*a**4 + 252*B*b**4*x**5 + x**4*(210*A*b**4 + 840*B*a*b**3) + x**3*(720*A*
a*b**3 + 1080*B*a**2*b**2) + x**2*(945*A*a**2*b**2 + 630*B*a**3*b) + x*(560*A*a*
*3*b + 140*B*a**4))/(1260*x**10)

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GIAC/XCAS [A]  time = 0.271597, size = 134, normalized size = 1.35 \[ -\frac{252 \, B b^{4} x^{5} + 840 \, B a b^{3} x^{4} + 210 \, A b^{4} x^{4} + 1080 \, B a^{2} b^{2} x^{3} + 720 \, A a b^{3} x^{3} + 630 \, B a^{3} b x^{2} + 945 \, A a^{2} b^{2} x^{2} + 140 \, B a^{4} x + 560 \, A a^{3} b x + 126 \, A a^{4}}{1260 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x^11,x, algorithm="giac")

[Out]

-1/1260*(252*B*b^4*x^5 + 840*B*a*b^3*x^4 + 210*A*b^4*x^4 + 1080*B*a^2*b^2*x^3 +
720*A*a*b^3*x^3 + 630*B*a^3*b*x^2 + 945*A*a^2*b^2*x^2 + 140*B*a^4*x + 560*A*a^3*
b*x + 126*A*a^4)/x^10