3.86 \(\int \frac{(a+b x)^3 (A+B x)}{x^9} \, dx\)

Optimal. Leaf size=75 \[ -\frac{a^3 A}{8 x^8}-\frac{a^2 (a B+3 A b)}{7 x^7}-\frac{b^2 (3 a B+A b)}{5 x^5}-\frac{a b (a B+A b)}{2 x^6}-\frac{b^3 B}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.0971536, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a^3 A}{8 x^8}-\frac{a^2 (a B+3 A b)}{7 x^7}-\frac{b^2 (3 a B+A b)}{5 x^5}-\frac{a b (a B+A b)}{2 x^6}-\frac{b^3 B}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^3*(A + B*x))/x^9,x]

[Out]

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

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Rubi in Sympy [A]  time = 21.6976, size = 70, normalized size = 0.93 \[ - \frac{A a^{3}}{8 x^{8}} - \frac{B b^{3}}{4 x^{4}} - \frac{a^{2} \left (3 A b + B a\right )}{7 x^{7}} - \frac{a b \left (A b + B a\right )}{2 x^{6}} - \frac{b^{2} \left (A b + 3 B a\right )}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)/x**9,x)

[Out]

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

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Mathematica [A]  time = 0.0314434, size = 69, normalized size = 0.92 \[ -\frac{5 a^3 (7 A+8 B x)+20 a^2 b x (6 A+7 B x)+28 a b^2 x^2 (5 A+6 B x)+14 b^3 x^3 (4 A+5 B x)}{280 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^3*(A + B*x))/x^9,x]

[Out]

-(14*b^3*x^3*(4*A + 5*B*x) + 28*a*b^2*x^2*(5*A + 6*B*x) + 20*a^2*b*x*(6*A + 7*B*
x) + 5*a^3*(7*A + 8*B*x))/(280*x^8)

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Maple [A]  time = 0.009, size = 66, normalized size = 0.9 \[ -{\frac{A{a}^{3}}{8\,{x}^{8}}}-{\frac{{a}^{2} \left ( 3\,Ab+Ba \right ) }{7\,{x}^{7}}}-{\frac{ab \left ( Ab+Ba \right ) }{2\,{x}^{6}}}-{\frac{{b}^{2} \left ( Ab+3\,Ba \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{3}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/x^9,x)

[Out]

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

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Maxima [A]  time = 1.35862, size = 99, normalized size = 1.32 \[ -\frac{70 \, B b^{3} x^{4} + 35 \, A a^{3} + 56 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 140 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 40 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^9,x, algorithm="maxima")

[Out]

-1/280*(70*B*b^3*x^4 + 35*A*a^3 + 56*(3*B*a*b^2 + A*b^3)*x^3 + 140*(B*a^2*b + A*
a*b^2)*x^2 + 40*(B*a^3 + 3*A*a^2*b)*x)/x^8

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Fricas [A]  time = 0.197916, size = 99, normalized size = 1.32 \[ -\frac{70 \, B b^{3} x^{4} + 35 \, A a^{3} + 56 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 140 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 40 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^9,x, algorithm="fricas")

[Out]

-1/280*(70*B*b^3*x^4 + 35*A*a^3 + 56*(3*B*a*b^2 + A*b^3)*x^3 + 140*(B*a^2*b + A*
a*b^2)*x^2 + 40*(B*a^3 + 3*A*a^2*b)*x)/x^8

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Sympy [A]  time = 11.2094, size = 78, normalized size = 1.04 \[ - \frac{35 A a^{3} + 70 B b^{3} x^{4} + x^{3} \left (56 A b^{3} + 168 B a b^{2}\right ) + x^{2} \left (140 A a b^{2} + 140 B a^{2} b\right ) + x \left (120 A a^{2} b + 40 B a^{3}\right )}{280 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)/x**9,x)

[Out]

-(35*A*a**3 + 70*B*b**3*x**4 + x**3*(56*A*b**3 + 168*B*a*b**2) + x**2*(140*A*a*b
**2 + 140*B*a**2*b) + x*(120*A*a**2*b + 40*B*a**3))/(280*x**8)

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GIAC/XCAS [A]  time = 0.340221, size = 101, normalized size = 1.35 \[ -\frac{70 \, B b^{3} x^{4} + 168 \, B a b^{2} x^{3} + 56 \, A b^{3} x^{3} + 140 \, B a^{2} b x^{2} + 140 \, A a b^{2} x^{2} + 40 \, B a^{3} x + 120 \, A a^{2} b x + 35 \, A a^{3}}{280 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^9,x, algorithm="giac")

[Out]

-1/280*(70*B*b^3*x^4 + 168*B*a*b^2*x^3 + 56*A*b^3*x^3 + 140*B*a^2*b*x^2 + 140*A*
a*b^2*x^2 + 40*B*a^3*x + 120*A*a^2*b*x + 35*A*a^3)/x^8