3.87 \(\int \frac{(a+b x)^3 (A+B x)}{x^{10}} \, dx\)

Optimal. Leaf size=75 \[ -\frac{a^3 A}{9 x^9}-\frac{a^2 (a B+3 A b)}{8 x^8}-\frac{b^2 (3 a B+A b)}{6 x^6}-\frac{3 a b (a B+A b)}{7 x^7}-\frac{b^3 B}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.0940145, 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 \[ -\frac{a^3 A}{9 x^9}-\frac{a^2 (a B+3 A b)}{8 x^8}-\frac{b^2 (3 a B+A b)}{6 x^6}-\frac{3 a b (a B+A b)}{7 x^7}-\frac{b^3 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^3*(A + B*x))/x^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 22.1189, size = 71, normalized size = 0.95 \[ - \frac{A a^{3}}{9 x^{9}} - \frac{B b^{3}}{5 x^{5}} - \frac{a^{2} \left (3 A b + B a\right )}{8 x^{8}} - \frac{3 a b \left (A b + B a\right )}{7 x^{7}} - \frac{b^{2} \left (A b + 3 B a\right )}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)/x**10,x)

[Out]

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

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Mathematica [A]  time = 0.0341582, size = 69, normalized size = 0.92 \[ -\frac{35 a^3 (8 A+9 B x)+135 a^2 b x (7 A+8 B x)+180 a b^2 x^2 (6 A+7 B x)+84 b^3 x^3 (5 A+6 B x)}{2520 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^3*(A + B*x))/x^10,x]

[Out]

-(84*b^3*x^3*(5*A + 6*B*x) + 180*a*b^2*x^2*(6*A + 7*B*x) + 135*a^2*b*x*(7*A + 8*
B*x) + 35*a^3*(8*A + 9*B*x))/(2520*x^9)

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Maple [A]  time = 0.009, size = 66, normalized size = 0.9 \[ -{\frac{A{a}^{3}}{9\,{x}^{9}}}-{\frac{{a}^{2} \left ( 3\,Ab+Ba \right ) }{8\,{x}^{8}}}-{\frac{3\,ab \left ( Ab+Ba \right ) }{7\,{x}^{7}}}-{\frac{{b}^{2} \left ( Ab+3\,Ba \right ) }{6\,{x}^{6}}}-{\frac{B{b}^{3}}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/x^10,x)

[Out]

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

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Maxima [A]  time = 1.35173, size = 99, normalized size = 1.32 \[ -\frac{504 \, B b^{3} x^{4} + 280 \, A a^{3} + 420 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 1080 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 315 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^10,x, algorithm="maxima")

[Out]

-1/2520*(504*B*b^3*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3)*x^3 + 1080*(B*a^2*b
+ A*a*b^2)*x^2 + 315*(B*a^3 + 3*A*a^2*b)*x)/x^9

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Fricas [A]  time = 0.194826, size = 99, normalized size = 1.32 \[ -\frac{504 \, B b^{3} x^{4} + 280 \, A a^{3} + 420 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 1080 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 315 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^10,x, algorithm="fricas")

[Out]

-1/2520*(504*B*b^3*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3)*x^3 + 1080*(B*a^2*b
+ A*a*b^2)*x^2 + 315*(B*a^3 + 3*A*a^2*b)*x)/x^9

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Sympy [A]  time = 13.206, size = 78, normalized size = 1.04 \[ - \frac{280 A a^{3} + 504 B b^{3} x^{4} + x^{3} \left (420 A b^{3} + 1260 B a b^{2}\right ) + x^{2} \left (1080 A a b^{2} + 1080 B a^{2} b\right ) + x \left (945 A a^{2} b + 315 B a^{3}\right )}{2520 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)/x**10,x)

[Out]

-(280*A*a**3 + 504*B*b**3*x**4 + x**3*(420*A*b**3 + 1260*B*a*b**2) + x**2*(1080*
A*a*b**2 + 1080*B*a**2*b) + x*(945*A*a**2*b + 315*B*a**3))/(2520*x**9)

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GIAC/XCAS [A]  time = 0.328768, size = 101, normalized size = 1.35 \[ -\frac{504 \, B b^{3} x^{4} + 1260 \, B a b^{2} x^{3} + 420 \, A b^{3} x^{3} + 1080 \, B a^{2} b x^{2} + 1080 \, A a b^{2} x^{2} + 315 \, B a^{3} x + 945 \, A a^{2} b x + 280 \, A a^{3}}{2520 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^10,x, algorithm="giac")

[Out]

-1/2520*(504*B*b^3*x^4 + 1260*B*a*b^2*x^3 + 420*A*b^3*x^3 + 1080*B*a^2*b*x^2 + 1
080*A*a*b^2*x^2 + 315*B*a^3*x + 945*A*a^2*b*x + 280*A*a^3)/x^9