3.135 \(\int \frac{(a+b x)^5 (A+B x)}{x^{11}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{a^4 (a B+5 A b)}{9 x^9}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{a^5 A}{10 x^{10}}-\frac{5 a b^3 (2 a B+A b)}{6 x^6}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{b^5 B}{4 x^4} \]

[Out]

-(a^5*A)/(10*x^10) - (a^4*(5*A*b + a*B))/(9*x^9) - (5*a^3*b*(2*A*b + a*B))/(8*x^8) - (10*a^2*b^2*(A*b + a*B))/
(7*x^7) - (5*a*b^3*(A*b + 2*a*B))/(6*x^6) - (b^4*(A*b + 5*a*B))/(5*x^5) - (b^5*B)/(4*x^4)

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Rubi [A]  time = 0.0575665, antiderivative size = 117, 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, Rules used = {76} \[ -\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{a^4 (a B+5 A b)}{9 x^9}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{a^5 A}{10 x^{10}}-\frac{5 a b^3 (2 a B+A b)}{6 x^6}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{b^5 B}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^5*(A + B*x))/x^11,x]

[Out]

-(a^5*A)/(10*x^10) - (a^4*(5*A*b + a*B))/(9*x^9) - (5*a^3*b*(2*A*b + a*B))/(8*x^8) - (10*a^2*b^2*(A*b + a*B))/
(7*x^7) - (5*a*b^3*(A*b + 2*a*B))/(6*x^6) - (b^4*(A*b + 5*a*B))/(5*x^5) - (b^5*B)/(4*x^4)

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{(a+b x)^5 (A+B x)}{x^{11}} \, dx &=\int \left (\frac{a^5 A}{x^{11}}+\frac{a^4 (5 A b+a B)}{x^{10}}+\frac{5 a^3 b (2 A b+a B)}{x^9}+\frac{10 a^2 b^2 (A b+a B)}{x^8}+\frac{5 a b^3 (A b+2 a B)}{x^7}+\frac{b^4 (A b+5 a B)}{x^6}+\frac{b^5 B}{x^5}\right ) \, dx\\ &=-\frac{a^5 A}{10 x^{10}}-\frac{a^4 (5 A b+a B)}{9 x^9}-\frac{5 a^3 b (2 A b+a B)}{8 x^8}-\frac{10 a^2 b^2 (A b+a B)}{7 x^7}-\frac{5 a b^3 (A b+2 a B)}{6 x^6}-\frac{b^4 (A b+5 a B)}{5 x^5}-\frac{b^5 B}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0284517, size = 107, normalized size = 0.91 \[ -\frac{450 a^3 b^2 x^2 (7 A+8 B x)+600 a^2 b^3 x^3 (6 A+7 B x)+175 a^4 b x (8 A+9 B x)+28 a^5 (9 A+10 B x)+420 a b^4 x^4 (5 A+6 B x)+126 b^5 x^5 (4 A+5 B x)}{2520 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^11,x]

[Out]

-(126*b^5*x^5*(4*A + 5*B*x) + 420*a*b^4*x^4*(5*A + 6*B*x) + 600*a^2*b^3*x^3*(6*A + 7*B*x) + 450*a^3*b^2*x^2*(7
*A + 8*B*x) + 175*a^4*b*x*(8*A + 9*B*x) + 28*a^5*(9*A + 10*B*x))/(2520*x^10)

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Maple [A]  time = 0.007, size = 104, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{5}}{10\,{x}^{10}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{9\,{x}^{9}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{8\,{x}^{8}}}-{\frac{10\,{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{7\,{x}^{7}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{6\,{x}^{6}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{5}}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(B*x+A)/x^11,x)

[Out]

-1/10*a^5*A/x^10-1/9*a^4*(5*A*b+B*a)/x^9-5/8*a^3*b*(2*A*b+B*a)/x^8-10/7*a^2*b^2*(A*b+B*a)/x^7-5/6*a*b^3*(A*b+2
*B*a)/x^6-1/5*b^4*(A*b+5*B*a)/x^5-1/4*b^5*B/x^4

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Maxima [A]  time = 1.00486, size = 161, normalized size = 1.38 \begin{align*} -\frac{630 \, B b^{5} x^{6} + 252 \, A a^{5} + 504 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 2100 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 3600 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 1575 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 280 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^11,x, algorithm="maxima")

[Out]

-1/2520*(630*B*b^5*x^6 + 252*A*a^5 + 504*(5*B*a*b^4 + A*b^5)*x^5 + 2100*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 3600*(B*
a^3*b^2 + A*a^2*b^3)*x^3 + 1575*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 280*(B*a^5 + 5*A*a^4*b)*x)/x^10

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Fricas [A]  time = 1.63021, size = 277, normalized size = 2.37 \begin{align*} -\frac{630 \, B b^{5} x^{6} + 252 \, A a^{5} + 504 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 2100 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 3600 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 1575 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 280 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^11,x, algorithm="fricas")

[Out]

-1/2520*(630*B*b^5*x^6 + 252*A*a^5 + 504*(5*B*a*b^4 + A*b^5)*x^5 + 2100*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 3600*(B*
a^3*b^2 + A*a^2*b^3)*x^3 + 1575*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 280*(B*a^5 + 5*A*a^4*b)*x)/x^10

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Sympy [A]  time = 11.6305, size = 126, normalized size = 1.08 \begin{align*} - \frac{252 A a^{5} + 630 B b^{5} x^{6} + x^{5} \left (504 A b^{5} + 2520 B a b^{4}\right ) + x^{4} \left (2100 A a b^{4} + 4200 B a^{2} b^{3}\right ) + x^{3} \left (3600 A a^{2} b^{3} + 3600 B a^{3} b^{2}\right ) + x^{2} \left (3150 A a^{3} b^{2} + 1575 B a^{4} b\right ) + x \left (1400 A a^{4} b + 280 B a^{5}\right )}{2520 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(B*x+A)/x**11,x)

[Out]

-(252*A*a**5 + 630*B*b**5*x**6 + x**5*(504*A*b**5 + 2520*B*a*b**4) + x**4*(2100*A*a*b**4 + 4200*B*a**2*b**3) +
x**3*(3600*A*a**2*b**3 + 3600*B*a**3*b**2) + x**2*(3150*A*a**3*b**2 + 1575*B*a**4*b) + x*(1400*A*a**4*b + 280
*B*a**5))/(2520*x**10)

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Giac [A]  time = 1.2056, size = 166, normalized size = 1.42 \begin{align*} -\frac{630 \, B b^{5} x^{6} + 2520 \, B a b^{4} x^{5} + 504 \, A b^{5} x^{5} + 4200 \, B a^{2} b^{3} x^{4} + 2100 \, A a b^{4} x^{4} + 3600 \, B a^{3} b^{2} x^{3} + 3600 \, A a^{2} b^{3} x^{3} + 1575 \, B a^{4} b x^{2} + 3150 \, A a^{3} b^{2} x^{2} + 280 \, B a^{5} x + 1400 \, A a^{4} b x + 252 \, A a^{5}}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^11,x, algorithm="giac")

[Out]

-1/2520*(630*B*b^5*x^6 + 2520*B*a*b^4*x^5 + 504*A*b^5*x^5 + 4200*B*a^2*b^3*x^4 + 2100*A*a*b^4*x^4 + 3600*B*a^3
*b^2*x^3 + 3600*A*a^2*b^3*x^3 + 1575*B*a^4*b*x^2 + 3150*A*a^3*b^2*x^2 + 280*B*a^5*x + 1400*A*a^4*b*x + 252*A*a
^5)/x^10