3.158 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{11}} \, dx\)

Optimal. Leaf size=216 \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{7 x^7}-\frac{5 a^6 b^3 (4 a B+7 A b)}{x^6}-\frac{42 a^5 b^4 (5 a B+6 A b)}{5 x^5}-\frac{21 a^4 b^5 (6 a B+5 A b)}{2 x^4}-\frac{10 a^3 b^6 (7 a B+4 A b)}{x^3}-\frac{15 a^2 b^7 (8 a B+3 A b)}{2 x^2}-\frac{a^9 (a B+10 A b)}{9 x^9}-\frac{5 a^8 b (2 a B+9 A b)}{8 x^8}-\frac{a^{10} A}{10 x^{10}}-\frac{5 a b^8 (9 a B+2 A b)}{x}+b^9 \log (x) (10 a B+A b)+b^{10} B x \]

[Out]

-(a^10*A)/(10*x^10) - (a^9*(10*A*b + a*B))/(9*x^9) - (5*a^8*b*(9*A*b + 2*a*B))/(8*x^8) - (15*a^7*b^2*(8*A*b +
3*a*B))/(7*x^7) - (5*a^6*b^3*(7*A*b + 4*a*B))/x^6 - (42*a^5*b^4*(6*A*b + 5*a*B))/(5*x^5) - (21*a^4*b^5*(5*A*b
+ 6*a*B))/(2*x^4) - (10*a^3*b^6*(4*A*b + 7*a*B))/x^3 - (15*a^2*b^7*(3*A*b + 8*a*B))/(2*x^2) - (5*a*b^8*(2*A*b
+ 9*a*B))/x + b^10*B*x + b^9*(A*b + 10*a*B)*Log[x]

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Rubi [A]  time = 0.144374, antiderivative size = 216, 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} \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{7 x^7}-\frac{5 a^6 b^3 (4 a B+7 A b)}{x^6}-\frac{42 a^5 b^4 (5 a B+6 A b)}{5 x^5}-\frac{21 a^4 b^5 (6 a B+5 A b)}{2 x^4}-\frac{10 a^3 b^6 (7 a B+4 A b)}{x^3}-\frac{15 a^2 b^7 (8 a B+3 A b)}{2 x^2}-\frac{a^9 (a B+10 A b)}{9 x^9}-\frac{5 a^8 b (2 a B+9 A b)}{8 x^8}-\frac{a^{10} A}{10 x^{10}}-\frac{5 a b^8 (9 a B+2 A b)}{x}+b^9 \log (x) (10 a B+A b)+b^{10} B x \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^11,x]

[Out]

-(a^10*A)/(10*x^10) - (a^9*(10*A*b + a*B))/(9*x^9) - (5*a^8*b*(9*A*b + 2*a*B))/(8*x^8) - (15*a^7*b^2*(8*A*b +
3*a*B))/(7*x^7) - (5*a^6*b^3*(7*A*b + 4*a*B))/x^6 - (42*a^5*b^4*(6*A*b + 5*a*B))/(5*x^5) - (21*a^4*b^5*(5*A*b
+ 6*a*B))/(2*x^4) - (10*a^3*b^6*(4*A*b + 7*a*B))/x^3 - (15*a^2*b^7*(3*A*b + 8*a*B))/(2*x^2) - (5*a*b^8*(2*A*b
+ 9*a*B))/x + b^10*B*x + b^9*(A*b + 10*a*B)*Log[x]

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)^{10} (A+B x)}{x^{11}} \, dx &=\int \left (b^{10} B+\frac{a^{10} A}{x^{11}}+\frac{a^9 (10 A b+a B)}{x^{10}}+\frac{5 a^8 b (9 A b+2 a B)}{x^9}+\frac{15 a^7 b^2 (8 A b+3 a B)}{x^8}+\frac{30 a^6 b^3 (7 A b+4 a B)}{x^7}+\frac{42 a^5 b^4 (6 A b+5 a B)}{x^6}+\frac{42 a^4 b^5 (5 A b+6 a B)}{x^5}+\frac{30 a^3 b^6 (4 A b+7 a B)}{x^4}+\frac{15 a^2 b^7 (3 A b+8 a B)}{x^3}+\frac{5 a b^8 (2 A b+9 a B)}{x^2}+\frac{b^9 (A b+10 a B)}{x}\right ) \, dx\\ &=-\frac{a^{10} A}{10 x^{10}}-\frac{a^9 (10 A b+a B)}{9 x^9}-\frac{5 a^8 b (9 A b+2 a B)}{8 x^8}-\frac{15 a^7 b^2 (8 A b+3 a B)}{7 x^7}-\frac{5 a^6 b^3 (7 A b+4 a B)}{x^6}-\frac{42 a^5 b^4 (6 A b+5 a B)}{5 x^5}-\frac{21 a^4 b^5 (5 A b+6 a B)}{2 x^4}-\frac{10 a^3 b^6 (4 A b+7 a B)}{x^3}-\frac{15 a^2 b^7 (3 A b+8 a B)}{2 x^2}-\frac{5 a b^8 (2 A b+9 a B)}{x}+b^{10} B x+b^9 (A b+10 a B) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0926875, size = 209, normalized size = 0.97 \[ -\frac{45 a^8 b^2 (7 A+8 B x)}{56 x^8}-\frac{20 a^7 b^3 (6 A+7 B x)}{7 x^7}-\frac{7 a^6 b^4 (5 A+6 B x)}{x^6}-\frac{63 a^5 b^5 (4 A+5 B x)}{5 x^5}-\frac{35 a^4 b^6 (3 A+4 B x)}{2 x^4}-\frac{20 a^3 b^7 (2 A+3 B x)}{x^3}-\frac{45 a^2 b^8 (A+2 B x)}{2 x^2}-\frac{5 a^9 b (8 A+9 B x)}{36 x^9}-\frac{a^{10} (9 A+10 B x)}{90 x^{10}}+b^9 \log (x) (10 a B+A b)-\frac{10 a A b^9}{x}+b^{10} B x \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^11,x]

[Out]

(-10*a*A*b^9)/x + b^10*B*x - (45*a^2*b^8*(A + 2*B*x))/(2*x^2) - (20*a^3*b^7*(2*A + 3*B*x))/x^3 - (35*a^4*b^6*(
3*A + 4*B*x))/(2*x^4) - (63*a^5*b^5*(4*A + 5*B*x))/(5*x^5) - (7*a^6*b^4*(5*A + 6*B*x))/x^6 - (20*a^7*b^3*(6*A
+ 7*B*x))/(7*x^7) - (45*a^8*b^2*(7*A + 8*B*x))/(56*x^8) - (5*a^9*b*(8*A + 9*B*x))/(36*x^9) - (a^10*(9*A + 10*B
*x))/(90*x^10) + b^9*(A*b + 10*a*B)*Log[x]

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Maple [A]  time = 0.01, size = 240, normalized size = 1.1 \begin{align*}{b}^{10}Bx+A\ln \left ( x \right ){b}^{10}+10\,B\ln \left ( x \right ) a{b}^{9}-40\,{\frac{{a}^{3}{b}^{7}A}{{x}^{3}}}-70\,{\frac{{a}^{4}{b}^{6}B}{{x}^{3}}}-{\frac{252\,{a}^{5}{b}^{5}A}{5\,{x}^{5}}}-42\,{\frac{{a}^{6}{b}^{4}B}{{x}^{5}}}-{\frac{105\,{a}^{4}{b}^{6}A}{2\,{x}^{4}}}-63\,{\frac{{a}^{5}{b}^{5}B}{{x}^{4}}}-{\frac{45\,{a}^{8}{b}^{2}A}{8\,{x}^{8}}}-{\frac{5\,{a}^{9}bB}{4\,{x}^{8}}}-{\frac{45\,{a}^{2}{b}^{8}A}{2\,{x}^{2}}}-60\,{\frac{{a}^{3}{b}^{7}B}{{x}^{2}}}-35\,{\frac{{a}^{6}{b}^{4}A}{{x}^{6}}}-20\,{\frac{{a}^{7}{b}^{3}B}{{x}^{6}}}-{\frac{120\,{a}^{7}{b}^{3}A}{7\,{x}^{7}}}-{\frac{45\,{a}^{8}{b}^{2}B}{7\,{x}^{7}}}-10\,{\frac{a{b}^{9}A}{x}}-45\,{\frac{{a}^{2}{b}^{8}B}{x}}-{\frac{10\,{a}^{9}bA}{9\,{x}^{9}}}-{\frac{{a}^{10}B}{9\,{x}^{9}}}-{\frac{A{a}^{10}}{10\,{x}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^11,x)

[Out]

b^10*B*x+A*ln(x)*b^10+10*B*ln(x)*a*b^9-40*a^3*b^7/x^3*A-70*a^4*b^6/x^3*B-252/5*a^5*b^5/x^5*A-42*a^6*b^4/x^5*B-
105/2*a^4*b^6/x^4*A-63*a^5*b^5/x^4*B-45/8*a^8*b^2/x^8*A-5/4*a^9*b/x^8*B-45/2*a^2*b^8/x^2*A-60*a^3*b^7/x^2*B-35
*a^6*b^4/x^6*A-20*a^7*b^3/x^6*B-120/7*a^7*b^3/x^7*A-45/7*a^8*b^2/x^7*B-10*a*b^9/x*A-45*a^2*b^8/x*B-10/9*a^9/x^
9*A*b-1/9*a^10/x^9*B-1/10*a^10*A/x^10

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Maxima [A]  time = 1.04763, size = 323, normalized size = 1.5 \begin{align*} B b^{10} x +{\left (10 \, B a b^{9} + A b^{10}\right )} \log \left (x\right ) - \frac{252 \, A a^{10} + 12600 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 18900 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 25200 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 26460 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 21168 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 12600 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5400 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1575 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 280 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^11,x, algorithm="maxima")

[Out]

B*b^10*x + (10*B*a*b^9 + A*b^10)*log(x) - 1/2520*(252*A*a^10 + 12600*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 18900*(8*
B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 25200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 26460*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 +
21168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 12600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 5400*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*x^3 + 1575*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 280*(B*a^10 + 10*A*a^9*b)*x)/x^10

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Fricas [A]  time = 1.27044, size = 581, normalized size = 2.69 \begin{align*} \frac{2520 \, B b^{10} x^{11} + 2520 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} \log \left (x\right ) - 252 \, A a^{10} - 12600 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} - 18900 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 25200 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 26460 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 21168 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 12600 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 5400 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 1575 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 280 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^11,x, algorithm="fricas")

[Out]

1/2520*(2520*B*b^10*x^11 + 2520*(10*B*a*b^9 + A*b^10)*x^10*log(x) - 252*A*a^10 - 12600*(9*B*a^2*b^8 + 2*A*a*b^
9)*x^9 - 18900*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 - 25200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 26460*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^6 - 21168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 12600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 5400*(3*B*a
^8*b^2 + 8*A*a^7*b^3)*x^3 - 1575*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 280*(B*a^10 + 10*A*a^9*b)*x)/x^10

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Sympy [A]  time = 22.6507, size = 236, normalized size = 1.09 \begin{align*} B b^{10} x + b^{9} \left (A b + 10 B a\right ) \log{\left (x \right )} - \frac{252 A a^{10} + x^{9} \left (25200 A a b^{9} + 113400 B a^{2} b^{8}\right ) + x^{8} \left (56700 A a^{2} b^{8} + 151200 B a^{3} b^{7}\right ) + x^{7} \left (100800 A a^{3} b^{7} + 176400 B a^{4} b^{6}\right ) + x^{6} \left (132300 A a^{4} b^{6} + 158760 B a^{5} b^{5}\right ) + x^{5} \left (127008 A a^{5} b^{5} + 105840 B a^{6} b^{4}\right ) + x^{4} \left (88200 A a^{6} b^{4} + 50400 B a^{7} b^{3}\right ) + x^{3} \left (43200 A a^{7} b^{3} + 16200 B a^{8} b^{2}\right ) + x^{2} \left (14175 A a^{8} b^{2} + 3150 B a^{9} b\right ) + x \left (2800 A a^{9} b + 280 B a^{10}\right )}{2520 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**11,x)

[Out]

B*b**10*x + b**9*(A*b + 10*B*a)*log(x) - (252*A*a**10 + x**9*(25200*A*a*b**9 + 113400*B*a**2*b**8) + x**8*(567
00*A*a**2*b**8 + 151200*B*a**3*b**7) + x**7*(100800*A*a**3*b**7 + 176400*B*a**4*b**6) + x**6*(132300*A*a**4*b*
*6 + 158760*B*a**5*b**5) + x**5*(127008*A*a**5*b**5 + 105840*B*a**6*b**4) + x**4*(88200*A*a**6*b**4 + 50400*B*
a**7*b**3) + x**3*(43200*A*a**7*b**3 + 16200*B*a**8*b**2) + x**2*(14175*A*a**8*b**2 + 3150*B*a**9*b) + x*(2800
*A*a**9*b + 280*B*a**10))/(2520*x**10)

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Giac [A]  time = 1.29908, size = 324, normalized size = 1.5 \begin{align*} B b^{10} x +{\left (10 \, B a b^{9} + A b^{10}\right )} \log \left ({\left | x \right |}\right ) - \frac{252 \, A a^{10} + 12600 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 18900 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 25200 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 26460 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 21168 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 12600 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5400 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1575 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 280 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^11,x, algorithm="giac")

[Out]

B*b^10*x + (10*B*a*b^9 + A*b^10)*log(abs(x)) - 1/2520*(252*A*a^10 + 12600*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 1890
0*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 25200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 26460*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
x^6 + 21168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 12600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 5400*(3*B*a^8*b^2 + 8*A*
a^7*b^3)*x^3 + 1575*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 280*(B*a^10 + 10*A*a^9*b)*x)/x^10