3.103 \(\int \frac{(a+b x)^5 (A+B x)}{x^{10}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{a^5 A}{9 x^9}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{5 a^3 b (a B+2 A b)}{7 x^7}-\frac{5 a^2 b^2 (a B+A b)}{3 x^6}-\frac{b^4 (5 a B+A b)}{4 x^4}-\frac{a b^3 (2 a B+A b)}{x^5}-\frac{b^5 B}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.18986, antiderivative size = 115, 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 \[ -\frac{a^5 A}{9 x^9}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{5 a^3 b (a B+2 A b)}{7 x^7}-\frac{5 a^2 b^2 (a B+A b)}{3 x^6}-\frac{b^4 (5 a B+A b)}{4 x^4}-\frac{a b^3 (2 a B+A b)}{x^5}-\frac{b^5 B}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^5*(A + B*x))/x^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 33.9683, size = 112, normalized size = 0.97 \[ - \frac{A a^{5}}{9 x^{9}} - \frac{B b^{5}}{3 x^{3}} - \frac{a^{4} \left (5 A b + B a\right )}{8 x^{8}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{7 x^{7}} - \frac{5 a^{2} b^{2} \left (A b + B a\right )}{3 x^{6}} - \frac{a b^{3} \left (A b + 2 B a\right )}{x^{5}} - \frac{b^{4} \left (A b + 5 B a\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(B*x+A)/x**10,x)

[Out]

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

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Mathematica [A]  time = 0.0489532, size = 107, normalized size = 0.93 \[ -\frac{7 a^5 (8 A+9 B x)+45 a^4 b x (7 A+8 B x)+120 a^3 b^2 x^2 (6 A+7 B x)+168 a^2 b^3 x^3 (5 A+6 B x)+126 a b^4 x^4 (4 A+5 B x)+42 b^5 x^5 (3 A+4 B x)}{504 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^5*(A + B*x))/x^10,x]

[Out]

-(42*b^5*x^5*(3*A + 4*B*x) + 126*a*b^4*x^4*(4*A + 5*B*x) + 168*a^2*b^3*x^3*(5*A
+ 6*B*x) + 120*a^3*b^2*x^2*(6*A + 7*B*x) + 45*a^4*b*x*(7*A + 8*B*x) + 7*a^5*(8*A
+ 9*B*x))/(504*x^9)

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Maple [A]  time = 0.008, size = 104, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{9\,{x}^{9}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{8\,{x}^{8}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{7\,{x}^{7}}}-{\frac{5\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{3\,{x}^{6}}}-{\frac{a{b}^{3} \left ( Ab+2\,Ba \right ) }{{x}^{5}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{4\,{x}^{4}}}-{\frac{B{b}^{5}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A)/x^10,x)

[Out]

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

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Maxima [A]  time = 1.36017, size = 161, normalized size = 1.4 \[ -\frac{168 \, B b^{5} x^{6} + 56 \, A a^{5} + 126 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 504 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 840 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 360 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 63 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^10,x, algorithm="maxima")

[Out]

-1/504*(168*B*b^5*x^6 + 56*A*a^5 + 126*(5*B*a*b^4 + A*b^5)*x^5 + 504*(2*B*a^2*b^
3 + A*a*b^4)*x^4 + 840*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 360*(B*a^4*b + 2*A*a^3*b^2)
*x^2 + 63*(B*a^5 + 5*A*a^4*b)*x)/x^9

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Fricas [A]  time = 0.194952, size = 161, normalized size = 1.4 \[ -\frac{168 \, B b^{5} x^{6} + 56 \, A a^{5} + 126 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 504 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 840 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 360 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 63 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^10,x, algorithm="fricas")

[Out]

-1/504*(168*B*b^5*x^6 + 56*A*a^5 + 126*(5*B*a*b^4 + A*b^5)*x^5 + 504*(2*B*a^2*b^
3 + A*a*b^4)*x^4 + 840*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 360*(B*a^4*b + 2*A*a^3*b^2)
*x^2 + 63*(B*a^5 + 5*A*a^4*b)*x)/x^9

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Sympy [A]  time = 26.0589, size = 126, normalized size = 1.1 \[ - \frac{56 A a^{5} + 168 B b^{5} x^{6} + x^{5} \left (126 A b^{5} + 630 B a b^{4}\right ) + x^{4} \left (504 A a b^{4} + 1008 B a^{2} b^{3}\right ) + x^{3} \left (840 A a^{2} b^{3} + 840 B a^{3} b^{2}\right ) + x^{2} \left (720 A a^{3} b^{2} + 360 B a^{4} b\right ) + x \left (315 A a^{4} b + 63 B a^{5}\right )}{504 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(B*x+A)/x**10,x)

[Out]

-(56*A*a**5 + 168*B*b**5*x**6 + x**5*(126*A*b**5 + 630*B*a*b**4) + x**4*(504*A*a
*b**4 + 1008*B*a**2*b**3) + x**3*(840*A*a**2*b**3 + 840*B*a**3*b**2) + x**2*(720
*A*a**3*b**2 + 360*B*a**4*b) + x*(315*A*a**4*b + 63*B*a**5))/(504*x**9)

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GIAC/XCAS [A]  time = 0.284362, size = 166, normalized size = 1.44 \[ -\frac{168 \, B b^{5} x^{6} + 630 \, B a b^{4} x^{5} + 126 \, A b^{5} x^{5} + 1008 \, B a^{2} b^{3} x^{4} + 504 \, A a b^{4} x^{4} + 840 \, B a^{3} b^{2} x^{3} + 840 \, A a^{2} b^{3} x^{3} + 360 \, B a^{4} b x^{2} + 720 \, A a^{3} b^{2} x^{2} + 63 \, B a^{5} x + 315 \, A a^{4} b x + 56 \, A a^{5}}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^10,x, algorithm="giac")

[Out]

-1/504*(168*B*b^5*x^6 + 630*B*a*b^4*x^5 + 126*A*b^5*x^5 + 1008*B*a^2*b^3*x^4 + 5
04*A*a*b^4*x^4 + 840*B*a^3*b^2*x^3 + 840*A*a^2*b^3*x^3 + 360*B*a^4*b*x^2 + 720*A
*a^3*b^2*x^2 + 63*B*a^5*x + 315*A*a^4*b*x + 56*A*a^5)/x^9