3.102 \(\int \frac{(a+b x)^5 (A+B x)}{x^9} \, dx\)

Optimal. Leaf size=70 \[ -\frac{b (a+b x)^6 (A b-4 a B)}{168 a^3 x^6}+\frac{(a+b x)^6 (A b-4 a B)}{28 a^2 x^7}-\frac{A (a+b x)^6}{8 a x^8} \]

[Out]

-(A*(a + b*x)^6)/(8*a*x^8) + ((A*b - 4*a*B)*(a + b*x)^6)/(28*a^2*x^7) - (b*(A*b
- 4*a*B)*(a + b*x)^6)/(168*a^3*x^6)

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Rubi [A]  time = 0.098301, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{b (a+b x)^6 (A b-4 a B)}{168 a^3 x^6}+\frac{(a+b x)^6 (A b-4 a B)}{28 a^2 x^7}-\frac{A (a+b x)^6}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^5*(A + B*x))/x^9,x]

[Out]

-(A*(a + b*x)^6)/(8*a*x^8) + ((A*b - 4*a*B)*(a + b*x)^6)/(28*a^2*x^7) - (b*(A*b
- 4*a*B)*(a + b*x)^6)/(168*a^3*x^6)

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Rubi in Sympy [A]  time = 13.371, size = 63, normalized size = 0.9 \[ - \frac{A \left (a + b x\right )^{6}}{8 a x^{8}} + \frac{\left (a + b x\right )^{6} \left (A b - 4 B a\right )}{28 a^{2} x^{7}} - \frac{b \left (a + b x\right )^{6} \left (A b - 4 B a\right )}{168 a^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(B*x+A)/x**9,x)

[Out]

-A*(a + b*x)**6/(8*a*x**8) + (a + b*x)**6*(A*b - 4*B*a)/(28*a**2*x**7) - b*(a +
b*x)**6*(A*b - 4*B*a)/(168*a**3*x**6)

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Mathematica [A]  time = 0.0476819, size = 107, normalized size = 1.53 \[ -\frac{3 a^5 (7 A+8 B x)+20 a^4 b x (6 A+7 B x)+56 a^3 b^2 x^2 (5 A+6 B x)+84 a^2 b^3 x^3 (4 A+5 B x)+70 a b^4 x^4 (3 A+4 B x)+28 b^5 x^5 (2 A+3 B x)}{168 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^5*(A + B*x))/x^9,x]

[Out]

-(28*b^5*x^5*(2*A + 3*B*x) + 70*a*b^4*x^4*(3*A + 4*B*x) + 84*a^2*b^3*x^3*(4*A +
5*B*x) + 56*a^3*b^2*x^2*(5*A + 6*B*x) + 20*a^4*b*x*(6*A + 7*B*x) + 3*a^5*(7*A +
8*B*x))/(168*x^8)

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Maple [A]  time = 0.008, size = 104, normalized size = 1.5 \[ -{\frac{A{a}^{5}}{8\,{x}^{8}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{7\,{x}^{7}}}-{\frac{B{b}^{5}}{2\,{x}^{2}}}-2\,{\frac{{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{{x}^{5}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{3\,{x}^{3}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{4\,{x}^{4}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{6\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A)/x^9,x)

[Out]

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

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Maxima [A]  time = 1.39534, size = 161, normalized size = 2.3 \[ -\frac{84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^9,x, algorithm="maxima")

[Out]

-1/168*(84*B*b^5*x^6 + 21*A*a^5 + 56*(5*B*a*b^4 + A*b^5)*x^5 + 210*(2*B*a^2*b^3
+ A*a*b^4)*x^4 + 336*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 140*(B*a^4*b + 2*A*a^3*b^2)*x
^2 + 24*(B*a^5 + 5*A*a^4*b)*x)/x^8

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Fricas [A]  time = 0.194873, size = 161, normalized size = 2.3 \[ -\frac{84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^9,x, algorithm="fricas")

[Out]

-1/168*(84*B*b^5*x^6 + 21*A*a^5 + 56*(5*B*a*b^4 + A*b^5)*x^5 + 210*(2*B*a^2*b^3
+ A*a*b^4)*x^4 + 336*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 140*(B*a^4*b + 2*A*a^3*b^2)*x
^2 + 24*(B*a^5 + 5*A*a^4*b)*x)/x^8

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Sympy [A]  time = 19.2484, size = 126, normalized size = 1.8 \[ - \frac{21 A a^{5} + 84 B b^{5} x^{6} + x^{5} \left (56 A b^{5} + 280 B a b^{4}\right ) + x^{4} \left (210 A a b^{4} + 420 B a^{2} b^{3}\right ) + x^{3} \left (336 A a^{2} b^{3} + 336 B a^{3} b^{2}\right ) + x^{2} \left (280 A a^{3} b^{2} + 140 B a^{4} b\right ) + x \left (120 A a^{4} b + 24 B a^{5}\right )}{168 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(B*x+A)/x**9,x)

[Out]

-(21*A*a**5 + 84*B*b**5*x**6 + x**5*(56*A*b**5 + 280*B*a*b**4) + x**4*(210*A*a*b
**4 + 420*B*a**2*b**3) + x**3*(336*A*a**2*b**3 + 336*B*a**3*b**2) + x**2*(280*A*
a**3*b**2 + 140*B*a**4*b) + x*(120*A*a**4*b + 24*B*a**5))/(168*x**8)

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GIAC/XCAS [A]  time = 0.273669, size = 166, normalized size = 2.37 \[ -\frac{84 \, B b^{5} x^{6} + 280 \, B a b^{4} x^{5} + 56 \, A b^{5} x^{5} + 420 \, B a^{2} b^{3} x^{4} + 210 \, A a b^{4} x^{4} + 336 \, B a^{3} b^{2} x^{3} + 336 \, A a^{2} b^{3} x^{3} + 140 \, B a^{4} b x^{2} + 280 \, A a^{3} b^{2} x^{2} + 24 \, B a^{5} x + 120 \, A a^{4} b x + 21 \, A a^{5}}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^9,x, algorithm="giac")

[Out]

-1/168*(84*B*b^5*x^6 + 280*B*a*b^4*x^5 + 56*A*b^5*x^5 + 420*B*a^2*b^3*x^4 + 210*
A*a*b^4*x^4 + 336*B*a^3*b^2*x^3 + 336*A*a^2*b^3*x^3 + 140*B*a^4*b*x^2 + 280*A*a^
3*b^2*x^2 + 24*B*a^5*x + 120*A*a^4*b*x + 21*A*a^5)/x^8