3.536 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^9} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.134425, 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 \[ -\frac{a^4 A}{8 x^8}-\frac{a^3 (a B+4 A b)}{7 x^7}-\frac{a^2 b (2 a B+3 A b)}{3 x^6}-\frac{b^3 (4 a B+A b)}{4 x^4}-\frac{2 a b^2 (3 a B+2 A b)}{5 x^5}-\frac{b^4 B}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^9,x]

[Out]

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

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Rubi in Sympy [A]  time = 35.7571, size = 97, normalized size = 0.98 \[ - \frac{A a^{4}}{8 x^{8}} - \frac{B b^{4}}{3 x^{3}} - \frac{a^{3} \left (4 A b + B a\right )}{7 x^{7}} - \frac{a^{2} b \left (3 A b + 2 B a\right )}{3 x^{6}} - \frac{2 a b^{2} \left (2 A b + 3 B a\right )}{5 x^{5}} - \frac{b^{3} \left (A b + 4 B a\right )}{4 x^{4}} \]

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**9,x)

[Out]

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

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Mathematica [A]  time = 0.0465873, size = 88, normalized size = 0.89 \[ -\frac{15 a^4 (7 A+8 B x)+80 a^3 b x (6 A+7 B x)+168 a^2 b^2 x^2 (5 A+6 B x)+168 a b^3 x^3 (4 A+5 B x)+70 b^4 x^4 (3 A+4 B x)}{840 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^9,x]

[Out]

-(70*b^4*x^4*(3*A + 4*B*x) + 168*a*b^3*x^3*(4*A + 5*B*x) + 168*a^2*b^2*x^2*(5*A
+ 6*B*x) + 80*a^3*b*x*(6*A + 7*B*x) + 15*a^4*(7*A + 8*B*x))/(840*x^8)

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

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^9,x)

[Out]

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

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Maxima [A]  time = 0.680152, size = 134, normalized size = 1.35 \[ -\frac{280 \, B b^{4} x^{5} + 105 \, A a^{4} + 210 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 336 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 280 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 120 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{840 \, x^{8}} \]

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^9,x, algorithm="maxima")

[Out]

-1/840*(280*B*b^4*x^5 + 105*A*a^4 + 210*(4*B*a*b^3 + A*b^4)*x^4 + 336*(3*B*a^2*b
^2 + 2*A*a*b^3)*x^3 + 280*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 120*(B*a^4 + 4*A*a^3*b
)*x)/x^8

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Fricas [A]  time = 0.263777, size = 134, normalized size = 1.35 \[ -\frac{280 \, B b^{4} x^{5} + 105 \, A a^{4} + 210 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 336 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 280 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 120 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{840 \, x^{8}} \]

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^9,x, algorithm="fricas")

[Out]

-1/840*(280*B*b^4*x^5 + 105*A*a^4 + 210*(4*B*a*b^3 + A*b^4)*x^4 + 336*(3*B*a^2*b
^2 + 2*A*a*b^3)*x^3 + 280*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 120*(B*a^4 + 4*A*a^3*b
)*x)/x^8

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Sympy [A]  time = 14.5983, size = 102, normalized size = 1.03 \[ - \frac{105 A a^{4} + 280 B b^{4} x^{5} + x^{4} \left (210 A b^{4} + 840 B a b^{3}\right ) + x^{3} \left (672 A a b^{3} + 1008 B a^{2} b^{2}\right ) + x^{2} \left (840 A a^{2} b^{2} + 560 B a^{3} b\right ) + x \left (480 A a^{3} b + 120 B a^{4}\right )}{840 x^{8}} \]

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**9,x)

[Out]

-(105*A*a**4 + 280*B*b**4*x**5 + x**4*(210*A*b**4 + 840*B*a*b**3) + x**3*(672*A*
a*b**3 + 1008*B*a**2*b**2) + x**2*(840*A*a**2*b**2 + 560*B*a**3*b) + x*(480*A*a*
*3*b + 120*B*a**4))/(840*x**8)

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GIAC/XCAS [A]  time = 0.270099, size = 134, normalized size = 1.35 \[ -\frac{280 \, B b^{4} x^{5} + 840 \, B a b^{3} x^{4} + 210 \, A b^{4} x^{4} + 1008 \, B a^{2} b^{2} x^{3} + 672 \, A a b^{3} x^{3} + 560 \, B a^{3} b x^{2} + 840 \, A a^{2} b^{2} x^{2} + 120 \, B a^{4} x + 480 \, A a^{3} b x + 105 \, A a^{4}}{840 \, x^{8}} \]

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^9,x, algorithm="giac")

[Out]

-1/840*(280*B*b^4*x^5 + 840*B*a*b^3*x^4 + 210*A*b^4*x^4 + 1008*B*a^2*b^2*x^3 + 6
72*A*a*b^3*x^3 + 560*B*a^3*b*x^2 + 840*A*a^2*b^2*x^2 + 120*B*a^4*x + 480*A*a^3*b
*x + 105*A*a^4)/x^8