3.101 \(\int \frac{(a+b x)^5 (A+B x)}{x^8} \, dx\)

Optimal. Leaf size=44 \[ \frac{(a+b x)^6 (A b-7 a B)}{42 a^2 x^6}-\frac{A (a+b x)^6}{7 a x^7} \]

[Out]

-(A*(a + b*x)^6)/(7*a*x^7) + ((A*b - 7*a*B)*(a + b*x)^6)/(42*a^2*x^6)

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Rubi [A]  time = 0.0568072, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(a+b x)^6 (A b-7 a B)}{42 a^2 x^6}-\frac{A (a+b x)^6}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^5*(A + B*x))/x^8,x]

[Out]

-(A*(a + b*x)^6)/(7*a*x^7) + ((A*b - 7*a*B)*(a + b*x)^6)/(42*a^2*x^6)

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Rubi in Sympy [A]  time = 9.11375, size = 37, normalized size = 0.84 \[ - \frac{A \left (a + b x\right )^{6}}{7 a x^{7}} + \frac{\left (a + b x\right )^{6} \left (A b - 7 B a\right )}{42 a^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(B*x+A)/x**8,x)

[Out]

-A*(a + b*x)**6/(7*a*x**7) + (a + b*x)**6*(A*b - 7*B*a)/(42*a**2*x**6)

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Mathematica [B]  time = 0.0480026, size = 104, normalized size = 2.36 \[ -\frac{a^5 (6 A+7 B x)+7 a^4 b x (5 A+6 B x)+21 a^3 b^2 x^2 (4 A+5 B x)+35 a^2 b^3 x^3 (3 A+4 B x)+35 a b^4 x^4 (2 A+3 B x)+21 b^5 x^5 (A+2 B x)}{42 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^5*(A + B*x))/x^8,x]

[Out]

-(21*b^5*x^5*(A + 2*B*x) + 35*a*b^4*x^4*(2*A + 3*B*x) + 35*a^2*b^3*x^3*(3*A + 4*
B*x) + 21*a^3*b^2*x^2*(4*A + 5*B*x) + 7*a^4*b*x*(5*A + 6*B*x) + a^5*(6*A + 7*B*x
))/(42*x^7)

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Maple [B]  time = 0.009, size = 104, normalized size = 2.4 \[ -{\frac{A{a}^{5}}{7\,{x}^{7}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{2\,{x}^{2}}}-{\frac{{a}^{3}b \left ( 2\,Ab+Ba \right ) }{{x}^{5}}}-{\frac{B{b}^{5}}{x}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{3\,{x}^{3}}}-{\frac{5\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{2\,{x}^{4}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{6\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A)/x^8,x)

[Out]

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

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Maxima [A]  time = 1.36453, size = 161, normalized size = 3.66 \[ -\frac{42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^8,x, algorithm="maxima")

[Out]

-1/42*(42*B*b^5*x^6 + 6*A*a^5 + 21*(5*B*a*b^4 + A*b^5)*x^5 + 70*(2*B*a^2*b^3 + A
*a*b^4)*x^4 + 105*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^2 +
7*(B*a^5 + 5*A*a^4*b)*x)/x^7

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Fricas [A]  time = 0.195514, size = 161, normalized size = 3.66 \[ -\frac{42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^8,x, algorithm="fricas")

[Out]

-1/42*(42*B*b^5*x^6 + 6*A*a^5 + 21*(5*B*a*b^4 + A*b^5)*x^5 + 70*(2*B*a^2*b^3 + A
*a*b^4)*x^4 + 105*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^2 +
7*(B*a^5 + 5*A*a^4*b)*x)/x^7

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Sympy [A]  time = 16.2177, size = 126, normalized size = 2.86 \[ - \frac{6 A a^{5} + 42 B b^{5} x^{6} + x^{5} \left (21 A b^{5} + 105 B a b^{4}\right ) + x^{4} \left (70 A a b^{4} + 140 B a^{2} b^{3}\right ) + x^{3} \left (105 A a^{2} b^{3} + 105 B a^{3} b^{2}\right ) + x^{2} \left (84 A a^{3} b^{2} + 42 B a^{4} b\right ) + x \left (35 A a^{4} b + 7 B a^{5}\right )}{42 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(B*x+A)/x**8,x)

[Out]

-(6*A*a**5 + 42*B*b**5*x**6 + x**5*(21*A*b**5 + 105*B*a*b**4) + x**4*(70*A*a*b**
4 + 140*B*a**2*b**3) + x**3*(105*A*a**2*b**3 + 105*B*a**3*b**2) + x**2*(84*A*a**
3*b**2 + 42*B*a**4*b) + x*(35*A*a**4*b + 7*B*a**5))/(42*x**7)

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GIAC/XCAS [A]  time = 0.263366, size = 166, normalized size = 3.77 \[ -\frac{42 \, B b^{5} x^{6} + 105 \, B a b^{4} x^{5} + 21 \, A b^{5} x^{5} + 140 \, B a^{2} b^{3} x^{4} + 70 \, A a b^{4} x^{4} + 105 \, B a^{3} b^{2} x^{3} + 105 \, A a^{2} b^{3} x^{3} + 42 \, B a^{4} b x^{2} + 84 \, A a^{3} b^{2} x^{2} + 7 \, B a^{5} x + 35 \, A a^{4} b x + 6 \, A a^{5}}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^8,x, algorithm="giac")

[Out]

-1/42*(42*B*b^5*x^6 + 105*B*a*b^4*x^5 + 21*A*b^5*x^5 + 140*B*a^2*b^3*x^4 + 70*A*
a*b^4*x^4 + 105*B*a^3*b^2*x^3 + 105*A*a^2*b^3*x^3 + 42*B*a^4*b*x^2 + 84*A*a^3*b^
2*x^2 + 7*B*a^5*x + 35*A*a^4*b*x + 6*A*a^5)/x^7