3.156 \(\int \frac{(a+b x)^{10} (A+B x)}{x^9} \, dx\)

Optimal. Leaf size=216 \[ -\frac{3 a^7 b^2 (3 a B+8 A b)}{x^5}-\frac{15 a^6 b^3 (4 a B+7 A b)}{2 x^4}-\frac{14 a^5 b^4 (5 a B+6 A b)}{x^3}-\frac{21 a^4 b^5 (6 a B+5 A b)}{x^2}-\frac{30 a^3 b^6 (7 a B+4 A b)}{x}+15 a^2 b^7 \log (x) (8 a B+3 A b)-\frac{a^9 (a B+10 A b)}{7 x^7}-\frac{5 a^8 b (2 a B+9 A b)}{6 x^6}-\frac{a^{10} A}{8 x^8}+\frac{1}{2} b^9 x^2 (10 a B+A b)+5 a b^8 x (9 a B+2 A b)+\frac{1}{3} b^{10} B x^3 \]

[Out]

-(a^10*A)/(8*x^8) - (a^9*(10*A*b + a*B))/(7*x^7) - (5*a^8*b*(9*A*b + 2*a*B))/(6*x^6) - (3*a^7*b^2*(8*A*b + 3*a
*B))/x^5 - (15*a^6*b^3*(7*A*b + 4*a*B))/(2*x^4) - (14*a^5*b^4*(6*A*b + 5*a*B))/x^3 - (21*a^4*b^5*(5*A*b + 6*a*
B))/x^2 - (30*a^3*b^6*(4*A*b + 7*a*B))/x + 5*a*b^8*(2*A*b + 9*a*B)*x + (b^9*(A*b + 10*a*B)*x^2)/2 + (b^10*B*x^
3)/3 + 15*a^2*b^7*(3*A*b + 8*a*B)*Log[x]

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Rubi [A]  time = 0.147677, 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} \[ -\frac{3 a^7 b^2 (3 a B+8 A b)}{x^5}-\frac{15 a^6 b^3 (4 a B+7 A b)}{2 x^4}-\frac{14 a^5 b^4 (5 a B+6 A b)}{x^3}-\frac{21 a^4 b^5 (6 a B+5 A b)}{x^2}-\frac{30 a^3 b^6 (7 a B+4 A b)}{x}+15 a^2 b^7 \log (x) (8 a B+3 A b)-\frac{a^9 (a B+10 A b)}{7 x^7}-\frac{5 a^8 b (2 a B+9 A b)}{6 x^6}-\frac{a^{10} A}{8 x^8}+\frac{1}{2} b^9 x^2 (10 a B+A b)+5 a b^8 x (9 a B+2 A b)+\frac{1}{3} b^{10} B x^3 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^9,x]

[Out]

-(a^10*A)/(8*x^8) - (a^9*(10*A*b + a*B))/(7*x^7) - (5*a^8*b*(9*A*b + 2*a*B))/(6*x^6) - (3*a^7*b^2*(8*A*b + 3*a
*B))/x^5 - (15*a^6*b^3*(7*A*b + 4*a*B))/(2*x^4) - (14*a^5*b^4*(6*A*b + 5*a*B))/x^3 - (21*a^4*b^5*(5*A*b + 6*a*
B))/x^2 - (30*a^3*b^6*(4*A*b + 7*a*B))/x + 5*a*b^8*(2*A*b + 9*a*B)*x + (b^9*(A*b + 10*a*B)*x^2)/2 + (b^10*B*x^
3)/3 + 15*a^2*b^7*(3*A*b + 8*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^9} \, dx &=\int \left (5 a b^8 (2 A b+9 a B)+\frac{a^{10} A}{x^9}+\frac{a^9 (10 A b+a B)}{x^8}+\frac{5 a^8 b (9 A b+2 a B)}{x^7}+\frac{15 a^7 b^2 (8 A b+3 a B)}{x^6}+\frac{30 a^6 b^3 (7 A b+4 a B)}{x^5}+\frac{42 a^5 b^4 (6 A b+5 a B)}{x^4}+\frac{42 a^4 b^5 (5 A b+6 a B)}{x^3}+\frac{30 a^3 b^6 (4 A b+7 a B)}{x^2}+\frac{15 a^2 b^7 (3 A b+8 a B)}{x}+b^9 (A b+10 a B) x+b^{10} B x^2\right ) \, dx\\ &=-\frac{a^{10} A}{8 x^8}-\frac{a^9 (10 A b+a B)}{7 x^7}-\frac{5 a^8 b (9 A b+2 a B)}{6 x^6}-\frac{3 a^7 b^2 (8 A b+3 a B)}{x^5}-\frac{15 a^6 b^3 (7 A b+4 a B)}{2 x^4}-\frac{14 a^5 b^4 (6 A b+5 a B)}{x^3}-\frac{21 a^4 b^5 (5 A b+6 a B)}{x^2}-\frac{30 a^3 b^6 (4 A b+7 a B)}{x}+5 a b^8 (2 A b+9 a B) x+\frac{1}{2} b^9 (A b+10 a B) x^2+\frac{1}{3} b^{10} B x^3+15 a^2 b^7 (3 A b+8 a B) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.083696, size = 208, normalized size = 0.96 \[ -\frac{3 a^8 b^2 (5 A+6 B x)}{2 x^6}-\frac{6 a^7 b^3 (4 A+5 B x)}{x^5}-\frac{35 a^6 b^4 (3 A+4 B x)}{2 x^4}-\frac{42 a^5 b^5 (2 A+3 B x)}{x^3}-\frac{105 a^4 b^6 (A+2 B x)}{x^2}+15 a^2 b^7 \log (x) (8 a B+3 A b)-\frac{120 a^3 A b^7}{x}-\frac{5 a^9 b (6 A+7 B x)}{21 x^7}-\frac{a^{10} (7 A+8 B x)}{56 x^8}+45 a^2 b^8 B x+5 a b^9 x (2 A+B x)+\frac{1}{6} b^{10} x^2 (3 A+2 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^9,x]

[Out]

(-120*a^3*A*b^7)/x + 45*a^2*b^8*B*x + 5*a*b^9*x*(2*A + B*x) - (105*a^4*b^6*(A + 2*B*x))/x^2 + (b^10*x^2*(3*A +
2*B*x))/6 - (42*a^5*b^5*(2*A + 3*B*x))/x^3 - (35*a^6*b^4*(3*A + 4*B*x))/(2*x^4) - (6*a^7*b^3*(4*A + 5*B*x))/x
^5 - (3*a^8*b^2*(5*A + 6*B*x))/(2*x^6) - (5*a^9*b*(6*A + 7*B*x))/(21*x^7) - (a^10*(7*A + 8*B*x))/(56*x^8) + 15
*a^2*b^7*(3*A*b + 8*a*B)*Log[x]

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Maple [A]  time = 0.009, size = 240, normalized size = 1.1 \begin{align*}{\frac{{b}^{10}B{x}^{3}}{3}}+{\frac{A{x}^{2}{b}^{10}}{2}}+5\,B{x}^{2}a{b}^{9}+10\,a{b}^{9}Ax+45\,{a}^{2}{b}^{8}Bx+45\,A\ln \left ( x \right ){a}^{2}{b}^{8}+120\,B\ln \left ( x \right ){a}^{3}{b}^{7}-84\,{\frac{{a}^{5}{b}^{5}A}{{x}^{3}}}-70\,{\frac{{a}^{6}{b}^{4}B}{{x}^{3}}}-24\,{\frac{{a}^{7}{b}^{3}A}{{x}^{5}}}-9\,{\frac{{a}^{8}{b}^{2}B}{{x}^{5}}}-{\frac{105\,{a}^{6}{b}^{4}A}{2\,{x}^{4}}}-30\,{\frac{{a}^{7}{b}^{3}B}{{x}^{4}}}-{\frac{A{a}^{10}}{8\,{x}^{8}}}-105\,{\frac{{a}^{4}{b}^{6}A}{{x}^{2}}}-126\,{\frac{{a}^{5}{b}^{5}B}{{x}^{2}}}-{\frac{15\,{a}^{8}{b}^{2}A}{2\,{x}^{6}}}-{\frac{5\,{a}^{9}bB}{3\,{x}^{6}}}-{\frac{10\,{a}^{9}bA}{7\,{x}^{7}}}-{\frac{{a}^{10}B}{7\,{x}^{7}}}-120\,{\frac{{a}^{3}{b}^{7}A}{x}}-210\,{\frac{{a}^{4}{b}^{6}B}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^9,x)

[Out]

1/3*b^10*B*x^3+1/2*A*x^2*b^10+5*B*x^2*a*b^9+10*a*b^9*A*x+45*a^2*b^8*B*x+45*A*ln(x)*a^2*b^8+120*B*ln(x)*a^3*b^7
-84*a^5*b^5/x^3*A-70*a^6*b^4/x^3*B-24*a^7*b^3/x^5*A-9*a^8*b^2/x^5*B-105/2*a^6*b^4/x^4*A-30*a^7*b^3/x^4*B-1/8*a
^10*A/x^8-105*a^4*b^6/x^2*A-126*a^5*b^5/x^2*B-15/2*a^8*b^2/x^6*A-5/3*a^9*b/x^6*B-10/7*a^9/x^7*A*b-1/7*a^10/x^7
*B-120*a^3*b^7/x*A-210*a^4*b^6/x*B

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Maxima [A]  time = 1.03368, size = 325, normalized size = 1.5 \begin{align*} \frac{1}{3} \, B b^{10} x^{3} + \frac{1}{2} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{2} + 5 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x + 15 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} \log \left (x\right ) - \frac{21 \, A a^{10} + 5040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 3528 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2352 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 1260 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 504 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 140 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 24 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^9,x, algorithm="maxima")

[Out]

1/3*B*b^10*x^3 + 1/2*(10*B*a*b^9 + A*b^10)*x^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*x + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8
)*log(x) - 1/168*(21*A*a^10 + 5040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 23
52*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 1260*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 504*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^
3 + 140*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 24*(B*a^10 + 10*A*a^9*b)*x)/x^8

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Fricas [A]  time = 1.39061, size = 558, normalized size = 2.58 \begin{align*} \frac{56 \, B b^{10} x^{11} - 21 \, A a^{10} + 84 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 840 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 2520 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} \log \left (x\right ) - 5040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 3528 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 2352 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 1260 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 504 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 140 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 24 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^9,x, algorithm="fricas")

[Out]

1/168*(56*B*b^10*x^11 - 21*A*a^10 + 84*(10*B*a*b^9 + A*b^10)*x^10 + 840*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 2520*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8*log(x) - 5040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6
)*x^6 - 2352*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 1260*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 504*(3*B*a^8*b^2 + 8*A*a
^7*b^3)*x^3 - 140*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 24*(B*a^10 + 10*A*a^9*b)*x)/x^8

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Sympy [A]  time = 9.76607, size = 240, normalized size = 1.11 \begin{align*} \frac{B b^{10} x^{3}}{3} + 15 a^{2} b^{7} \left (3 A b + 8 B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{10}}{2} + 5 B a b^{9}\right ) + x \left (10 A a b^{9} + 45 B a^{2} b^{8}\right ) - \frac{21 A a^{10} + x^{7} \left (20160 A a^{3} b^{7} + 35280 B a^{4} b^{6}\right ) + x^{6} \left (17640 A a^{4} b^{6} + 21168 B a^{5} b^{5}\right ) + x^{5} \left (14112 A a^{5} b^{5} + 11760 B a^{6} b^{4}\right ) + x^{4} \left (8820 A a^{6} b^{4} + 5040 B a^{7} b^{3}\right ) + x^{3} \left (4032 A a^{7} b^{3} + 1512 B a^{8} b^{2}\right ) + x^{2} \left (1260 A a^{8} b^{2} + 280 B a^{9} b\right ) + x \left (240 A a^{9} b + 24 B a^{10}\right )}{168 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**9,x)

[Out]

B*b**10*x**3/3 + 15*a**2*b**7*(3*A*b + 8*B*a)*log(x) + x**2*(A*b**10/2 + 5*B*a*b**9) + x*(10*A*a*b**9 + 45*B*a
**2*b**8) - (21*A*a**10 + x**7*(20160*A*a**3*b**7 + 35280*B*a**4*b**6) + x**6*(17640*A*a**4*b**6 + 21168*B*a**
5*b**5) + x**5*(14112*A*a**5*b**5 + 11760*B*a**6*b**4) + x**4*(8820*A*a**6*b**4 + 5040*B*a**7*b**3) + x**3*(40
32*A*a**7*b**3 + 1512*B*a**8*b**2) + x**2*(1260*A*a**8*b**2 + 280*B*a**9*b) + x*(240*A*a**9*b + 24*B*a**10))/(
168*x**8)

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Giac [A]  time = 1.16758, size = 325, normalized size = 1.5 \begin{align*} \frac{1}{3} \, B b^{10} x^{3} + 5 \, B a b^{9} x^{2} + \frac{1}{2} \, A b^{10} x^{2} + 45 \, B a^{2} b^{8} x + 10 \, A a b^{9} x + 15 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} \log \left ({\left | x \right |}\right ) - \frac{21 \, A a^{10} + 5040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 3528 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2352 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 1260 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 504 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 140 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 24 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^9,x, algorithm="giac")

[Out]

1/3*B*b^10*x^3 + 5*B*a*b^9*x^2 + 1/2*A*b^10*x^2 + 45*B*a^2*b^8*x + 10*A*a*b^9*x + 15*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*log(abs(x)) - 1/168*(21*A*a^10 + 5040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^
6 + 2352*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 1260*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 504*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*x^3 + 140*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 24*(B*a^10 + 10*A*a^9*b)*x)/x^8