3.159 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{12}} \, dx\)

Optimal. Leaf size=153 \[ -\frac{45 a^2 b^8 B}{2 x^2}-\frac{40 a^3 b^7 B}{x^3}-\frac{105 a^4 b^6 B}{2 x^4}-\frac{252 a^5 b^5 B}{5 x^5}-\frac{35 a^6 b^4 B}{x^6}-\frac{120 a^7 b^3 B}{7 x^7}-\frac{45 a^8 b^2 B}{8 x^8}-\frac{10 a^9 b B}{9 x^9}-\frac{a^{10} B}{10 x^{10}}-\frac{A (a+b x)^{11}}{11 a x^{11}}-\frac{10 a b^9 B}{x}+b^{10} B \log (x) \]

[Out]

-(a^10*B)/(10*x^10) - (10*a^9*b*B)/(9*x^9) - (45*a^8*b^2*B)/(8*x^8) - (120*a^7*b^3*B)/(7*x^7) - (35*a^6*b^4*B)
/x^6 - (252*a^5*b^5*B)/(5*x^5) - (105*a^4*b^6*B)/(2*x^4) - (40*a^3*b^7*B)/x^3 - (45*a^2*b^8*B)/(2*x^2) - (10*a
*b^9*B)/x - (A*(a + b*x)^11)/(11*a*x^11) + b^10*B*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0818423, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 43} \[ -\frac{45 a^2 b^8 B}{2 x^2}-\frac{40 a^3 b^7 B}{x^3}-\frac{105 a^4 b^6 B}{2 x^4}-\frac{252 a^5 b^5 B}{5 x^5}-\frac{35 a^6 b^4 B}{x^6}-\frac{120 a^7 b^3 B}{7 x^7}-\frac{45 a^8 b^2 B}{8 x^8}-\frac{10 a^9 b B}{9 x^9}-\frac{a^{10} B}{10 x^{10}}-\frac{A (a+b x)^{11}}{11 a x^{11}}-\frac{10 a b^9 B}{x}+b^{10} B \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^12,x]

[Out]

-(a^10*B)/(10*x^10) - (10*a^9*b*B)/(9*x^9) - (45*a^8*b^2*B)/(8*x^8) - (120*a^7*b^3*B)/(7*x^7) - (35*a^6*b^4*B)
/x^6 - (252*a^5*b^5*B)/(5*x^5) - (105*a^4*b^6*B)/(2*x^4) - (40*a^3*b^7*B)/x^3 - (45*a^2*b^8*B)/(2*x^2) - (10*a
*b^9*B)/x - (A*(a + b*x)^11)/(11*a*x^11) + b^10*B*Log[x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^{10} (A+B x)}{x^{12}} \, dx &=-\frac{A (a+b x)^{11}}{11 a x^{11}}+B \int \frac{(a+b x)^{10}}{x^{11}} \, dx\\ &=-\frac{A (a+b x)^{11}}{11 a x^{11}}+B \int \left (\frac{a^{10}}{x^{11}}+\frac{10 a^9 b}{x^{10}}+\frac{45 a^8 b^2}{x^9}+\frac{120 a^7 b^3}{x^8}+\frac{210 a^6 b^4}{x^7}+\frac{252 a^5 b^5}{x^6}+\frac{210 a^4 b^6}{x^5}+\frac{120 a^3 b^7}{x^4}+\frac{45 a^2 b^8}{x^3}+\frac{10 a b^9}{x^2}+\frac{b^{10}}{x}\right ) \, dx\\ &=-\frac{a^{10} B}{10 x^{10}}-\frac{10 a^9 b B}{9 x^9}-\frac{45 a^8 b^2 B}{8 x^8}-\frac{120 a^7 b^3 B}{7 x^7}-\frac{35 a^6 b^4 B}{x^6}-\frac{252 a^5 b^5 B}{5 x^5}-\frac{105 a^4 b^6 B}{2 x^4}-\frac{40 a^3 b^7 B}{x^3}-\frac{45 a^2 b^8 B}{2 x^2}-\frac{10 a b^9 B}{x}-\frac{A (a+b x)^{11}}{11 a x^{11}}+b^{10} B \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0933029, size = 212, normalized size = 1.39 \[ -\frac{5 a^8 b^2 (8 A+9 B x)}{8 x^9}-\frac{15 a^7 b^3 (7 A+8 B x)}{7 x^8}-\frac{5 a^6 b^4 (6 A+7 B x)}{x^7}-\frac{42 a^5 b^5 (5 A+6 B x)}{5 x^6}-\frac{21 a^4 b^6 (4 A+5 B x)}{2 x^5}-\frac{10 a^3 b^7 (3 A+4 B x)}{x^4}-\frac{15 a^2 b^8 (2 A+3 B x)}{2 x^3}-\frac{a^9 b (9 A+10 B x)}{9 x^{10}}-\frac{a^{10} (10 A+11 B x)}{110 x^{11}}-\frac{5 a b^9 (A+2 B x)}{x^2}-\frac{A b^{10}}{x}+b^{10} B \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^12,x]

[Out]

-((A*b^10)/x) - (5*a*b^9*(A + 2*B*x))/x^2 - (15*a^2*b^8*(2*A + 3*B*x))/(2*x^3) - (10*a^3*b^7*(3*A + 4*B*x))/x^
4 - (21*a^4*b^6*(4*A + 5*B*x))/(2*x^5) - (42*a^5*b^5*(5*A + 6*B*x))/(5*x^6) - (5*a^6*b^4*(6*A + 7*B*x))/x^7 -
(15*a^7*b^3*(7*A + 8*B*x))/(7*x^8) - (5*a^8*b^2*(8*A + 9*B*x))/(8*x^9) - (a^9*b*(9*A + 10*B*x))/(9*x^10) - (a^
10*(10*A + 11*B*x))/(110*x^11) + b^10*B*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 244, normalized size = 1.6 \begin{align*}{b}^{10}B\ln \left ( x \right ) -15\,{\frac{{a}^{2}{b}^{8}A}{{x}^{3}}}-40\,{\frac{{a}^{3}{b}^{7}B}{{x}^{3}}}-42\,{\frac{{a}^{4}{b}^{6}A}{{x}^{5}}}-{\frac{252\,{a}^{5}{b}^{5}B}{5\,{x}^{5}}}-{\frac{A{a}^{10}}{11\,{x}^{11}}}-30\,{\frac{{a}^{3}{b}^{7}A}{{x}^{4}}}-{\frac{105\,{a}^{4}{b}^{6}B}{2\,{x}^{4}}}-15\,{\frac{{a}^{7}{b}^{3}A}{{x}^{8}}}-{\frac{45\,{a}^{8}{b}^{2}B}{8\,{x}^{8}}}-5\,{\frac{a{b}^{9}A}{{x}^{2}}}-{\frac{45\,{a}^{2}{b}^{8}B}{2\,{x}^{2}}}-42\,{\frac{{a}^{5}{b}^{5}A}{{x}^{6}}}-35\,{\frac{{a}^{6}{b}^{4}B}{{x}^{6}}}-30\,{\frac{{a}^{6}{b}^{4}A}{{x}^{7}}}-{\frac{120\,{a}^{7}{b}^{3}B}{7\,{x}^{7}}}-{\frac{{b}^{10}A}{x}}-10\,{\frac{a{b}^{9}B}{x}}-5\,{\frac{{a}^{8}{b}^{2}A}{{x}^{9}}}-{\frac{10\,{a}^{9}bB}{9\,{x}^{9}}}-{\frac{{a}^{9}bA}{{x}^{10}}}-{\frac{{a}^{10}B}{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^12,x)

[Out]

b^10*B*ln(x)-15*a^2*b^8/x^3*A-40*a^3*b^7*B/x^3-42*a^4*b^6/x^5*A-252/5*a^5*b^5*B/x^5-1/11*A*a^10/x^11-30*a^3*b^
7/x^4*A-105/2*a^4*b^6*B/x^4-15*a^7*b^3/x^8*A-45/8*a^8*b^2*B/x^8-5*a*b^9/x^2*A-45/2*a^2*b^8*B/x^2-42*a^5*b^5/x^
6*A-35*a^6*b^4*B/x^6-30*a^6*b^4/x^7*A-120/7*a^7*b^3*B/x^7-b^10/x*A-10*a*b^9*B/x-5*a^8*b^2/x^9*A-10/9*a^9*b*B/x
^9-a^9/x^10*A*b-1/10*a^10*B/x^10

________________________________________________________________________________________

Maxima [A]  time = 1.03987, size = 327, normalized size = 2.14 \begin{align*} B b^{10} \log \left (x\right ) - \frac{2520 \, A a^{10} + 27720 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 69300 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 138600 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 207900 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 232848 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 194040 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 118800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 51975 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 15400 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 2772 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^12,x, algorithm="maxima")

[Out]

B*b^10*log(x) - 1/27720*(2520*A*a^10 + 27720*(10*B*a*b^9 + A*b^10)*x^10 + 69300*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9
+ 138600*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 207900*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 232848*(6*B*a^5*b^5 + 5*A*
a^4*b^6)*x^6 + 194040*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 118800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 51975*(3*B*a^
8*b^2 + 8*A*a^7*b^3)*x^3 + 15400*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 2772*(B*a^10 + 10*A*a^9*b)*x)/x^11

________________________________________________________________________________________

Fricas [A]  time = 1.55056, size = 597, normalized size = 3.9 \begin{align*} \frac{27720 \, B b^{10} x^{11} \log \left (x\right ) - 2520 \, A a^{10} - 27720 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} - 69300 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} - 138600 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 207900 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 232848 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 194040 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 118800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 51975 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 15400 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 2772 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^12,x, algorithm="fricas")

[Out]

1/27720*(27720*B*b^10*x^11*log(x) - 2520*A*a^10 - 27720*(10*B*a*b^9 + A*b^10)*x^10 - 69300*(9*B*a^2*b^8 + 2*A*
a*b^9)*x^9 - 138600*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 - 207900*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 232848*(6*B*a^5
*b^5 + 5*A*a^4*b^6)*x^6 - 194040*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 118800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 51
975*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 - 15400*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 2772*(B*a^10 + 10*A*a^9*b)*x)/x^11

________________________________________________________________________________________

Sympy [A]  time = 34.0386, size = 241, normalized size = 1.58 \begin{align*} B b^{10} \log{\left (x \right )} - \frac{2520 A a^{10} + x^{10} \left (27720 A b^{10} + 277200 B a b^{9}\right ) + x^{9} \left (138600 A a b^{9} + 623700 B a^{2} b^{8}\right ) + x^{8} \left (415800 A a^{2} b^{8} + 1108800 B a^{3} b^{7}\right ) + x^{7} \left (831600 A a^{3} b^{7} + 1455300 B a^{4} b^{6}\right ) + x^{6} \left (1164240 A a^{4} b^{6} + 1397088 B a^{5} b^{5}\right ) + x^{5} \left (1164240 A a^{5} b^{5} + 970200 B a^{6} b^{4}\right ) + x^{4} \left (831600 A a^{6} b^{4} + 475200 B a^{7} b^{3}\right ) + x^{3} \left (415800 A a^{7} b^{3} + 155925 B a^{8} b^{2}\right ) + x^{2} \left (138600 A a^{8} b^{2} + 30800 B a^{9} b\right ) + x \left (27720 A a^{9} b + 2772 B a^{10}\right )}{27720 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**12,x)

[Out]

B*b**10*log(x) - (2520*A*a**10 + x**10*(27720*A*b**10 + 277200*B*a*b**9) + x**9*(138600*A*a*b**9 + 623700*B*a*
*2*b**8) + x**8*(415800*A*a**2*b**8 + 1108800*B*a**3*b**7) + x**7*(831600*A*a**3*b**7 + 1455300*B*a**4*b**6) +
x**6*(1164240*A*a**4*b**6 + 1397088*B*a**5*b**5) + x**5*(1164240*A*a**5*b**5 + 970200*B*a**6*b**4) + x**4*(83
1600*A*a**6*b**4 + 475200*B*a**7*b**3) + x**3*(415800*A*a**7*b**3 + 155925*B*a**8*b**2) + x**2*(138600*A*a**8*
b**2 + 30800*B*a**9*b) + x*(27720*A*a**9*b + 2772*B*a**10))/(27720*x**11)

________________________________________________________________________________________

Giac [A]  time = 1.19494, size = 328, normalized size = 2.14 \begin{align*} B b^{10} \log \left ({\left | x \right |}\right ) - \frac{2520 \, A a^{10} + 27720 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 69300 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 138600 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 207900 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 232848 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 194040 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 118800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 51975 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 15400 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 2772 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^12,x, algorithm="giac")

[Out]

B*b^10*log(abs(x)) - 1/27720*(2520*A*a^10 + 27720*(10*B*a*b^9 + A*b^10)*x^10 + 69300*(9*B*a^2*b^8 + 2*A*a*b^9)
*x^9 + 138600*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 207900*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 232848*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^6 + 194040*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 118800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 51975*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 15400*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 2772*(B*a^10 + 10*A*a^9*b)*x)/x^11