3.155 \(\int \frac{(a+b x)^{10} (A+B x)}{x^8} \, dx\)

Optimal. Leaf size=216 \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{4 x^4}-\frac{10 a^6 b^3 (4 a B+7 A b)}{x^3}-\frac{21 a^5 b^4 (5 a B+6 A b)}{x^2}-\frac{42 a^4 b^5 (6 a B+5 A b)}{x}+15 a^2 b^7 x (8 a B+3 A b)+30 a^3 b^6 \log (x) (7 a B+4 A b)-\frac{a^9 (a B+10 A b)}{6 x^6}-\frac{a^8 b (2 a B+9 A b)}{x^5}-\frac{a^{10} A}{7 x^7}+\frac{5}{2} a b^8 x^2 (9 a B+2 A b)+\frac{1}{3} b^9 x^3 (10 a B+A b)+\frac{1}{4} b^{10} B x^4 \]

[Out]

-(a^10*A)/(7*x^7) - (a^9*(10*A*b + a*B))/(6*x^6) - (a^8*b*(9*A*b + 2*a*B))/x^5 - (15*a^7*b^2*(8*A*b + 3*a*B))/
(4*x^4) - (10*a^6*b^3*(7*A*b + 4*a*B))/x^3 - (21*a^5*b^4*(6*A*b + 5*a*B))/x^2 - (42*a^4*b^5*(5*A*b + 6*a*B))/x
+ 15*a^2*b^7*(3*A*b + 8*a*B)*x + (5*a*b^8*(2*A*b + 9*a*B)*x^2)/2 + (b^9*(A*b + 10*a*B)*x^3)/3 + (b^10*B*x^4)/
4 + 30*a^3*b^6*(4*A*b + 7*a*B)*Log[x]

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Rubi [A]  time = 0.147658, 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} \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{4 x^4}-\frac{10 a^6 b^3 (4 a B+7 A b)}{x^3}-\frac{21 a^5 b^4 (5 a B+6 A b)}{x^2}-\frac{42 a^4 b^5 (6 a B+5 A b)}{x}+15 a^2 b^7 x (8 a B+3 A b)+30 a^3 b^6 \log (x) (7 a B+4 A b)-\frac{a^9 (a B+10 A b)}{6 x^6}-\frac{a^8 b (2 a B+9 A b)}{x^5}-\frac{a^{10} A}{7 x^7}+\frac{5}{2} a b^8 x^2 (9 a B+2 A b)+\frac{1}{3} b^9 x^3 (10 a B+A b)+\frac{1}{4} b^{10} B x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^8,x]

[Out]

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

Mathematica [A]  time = 0.0787461, size = 210, normalized size = 0.97 \[ -\frac{9 a^8 b^2 (4 A+5 B x)}{4 x^5}-\frac{10 a^7 b^3 (3 A+4 B x)}{x^4}-\frac{35 a^6 b^4 (2 A+3 B x)}{x^3}-\frac{126 a^5 b^5 (A+2 B x)}{x^2}+\frac{45}{2} a^2 b^8 x (2 A+B x)+30 a^3 b^6 \log (x) (7 a B+4 A b)-\frac{210 a^4 A b^6}{x}-\frac{a^9 b (5 A+6 B x)}{3 x^6}-\frac{a^{10} (6 A+7 B x)}{42 x^7}+120 a^3 b^7 B x+\frac{5}{3} a b^9 x^2 (3 A+2 B x)+\frac{1}{12} b^{10} x^3 (4 A+3 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^8,x]

[Out]

(-210*a^4*A*b^6)/x + 120*a^3*b^7*B*x + (45*a^2*b^8*x*(2*A + B*x))/2 - (126*a^5*b^5*(A + 2*B*x))/x^2 + (5*a*b^9
*x^2*(3*A + 2*B*x))/3 - (35*a^6*b^4*(2*A + 3*B*x))/x^3 + (b^10*x^3*(4*A + 3*B*x))/12 - (10*a^7*b^3*(3*A + 4*B*
x))/x^4 - (9*a^8*b^2*(4*A + 5*B*x))/(4*x^5) - (a^9*b*(5*A + 6*B*x))/(3*x^6) - (a^10*(6*A + 7*B*x))/(42*x^7) +
30*a^3*b^6*(4*A*b + 7*a*B)*Log[x]

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Maple [A]  time = 0.008, size = 240, normalized size = 1.1 \begin{align*}{\frac{{b}^{10}B{x}^{4}}{4}}+{\frac{A{x}^{3}{b}^{10}}{3}}+{\frac{10\,B{x}^{3}a{b}^{9}}{3}}+5\,A{x}^{2}a{b}^{9}+{\frac{45\,B{x}^{2}{a}^{2}{b}^{8}}{2}}+45\,{a}^{2}{b}^{8}Ax+120\,{a}^{3}{b}^{7}Bx+120\,A\ln \left ( x \right ){a}^{3}{b}^{7}+210\,B\ln \left ( x \right ){a}^{4}{b}^{6}-70\,{\frac{{a}^{6}{b}^{4}A}{{x}^{3}}}-40\,{\frac{{a}^{7}{b}^{3}B}{{x}^{3}}}-9\,{\frac{{a}^{8}{b}^{2}A}{{x}^{5}}}-2\,{\frac{{a}^{9}bB}{{x}^{5}}}-30\,{\frac{{a}^{7}{b}^{3}A}{{x}^{4}}}-{\frac{45\,{a}^{8}{b}^{2}B}{4\,{x}^{4}}}-126\,{\frac{{a}^{5}{b}^{5}A}{{x}^{2}}}-105\,{\frac{{a}^{6}{b}^{4}B}{{x}^{2}}}-{\frac{5\,{a}^{9}bA}{3\,{x}^{6}}}-{\frac{{a}^{10}B}{6\,{x}^{6}}}-{\frac{A{a}^{10}}{7\,{x}^{7}}}-210\,{\frac{{a}^{4}{b}^{6}A}{x}}-252\,{\frac{{a}^{5}{b}^{5}B}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^8,x)

[Out]

1/4*b^10*B*x^4+1/3*A*x^3*b^10+10/3*B*x^3*a*b^9+5*A*x^2*a*b^9+45/2*B*x^2*a^2*b^8+45*a^2*b^8*A*x+120*a^3*b^7*B*x
+120*A*ln(x)*a^3*b^7+210*B*ln(x)*a^4*b^6-70*a^6*b^4/x^3*A-40*a^7*b^3/x^3*B-9*a^8*b^2/x^5*A-2*a^9*b/x^5*B-30*a^
7*b^3/x^4*A-45/4*a^8*b^2/x^4*B-126*a^5*b^5/x^2*A-105*a^6*b^4/x^2*B-5/3*a^9/x^6*A*b-1/6*a^10/x^6*B-1/7*a^10*A/x
^7-210*a^4*b^6/x*A-252*a^5*b^5/x*B

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Maxima [A]  time = 1.0702, size = 325, normalized size = 1.5 \begin{align*} \frac{1}{4} \, B b^{10} x^{4} + \frac{1}{3} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{3} + \frac{5}{2} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{2} + 15 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x + 30 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} \log \left (x\right ) - \frac{12 \, A a^{10} + 3528 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 1764 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 840 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 315 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 84 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 14 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{84 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

1/4*B*b^10*x^4 + 1/3*(10*B*a*b^9 + A*b^10)*x^3 + 5/2*(9*B*a^2*b^8 + 2*A*a*b^9)*x^2 + 15*(8*B*a^3*b^7 + 3*A*a^2
*b^8)*x + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*log(x) - 1/84*(12*A*a^10 + 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 176
4*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 840*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3
+ 84*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 14*(B*a^10 + 10*A*a^9*b)*x)/x^7

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Fricas [A]  time = 1.45091, size = 554, normalized size = 2.56 \begin{align*} \frac{21 \, B b^{10} x^{11} - 12 \, A a^{10} + 28 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 210 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 1260 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 2520 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} \log \left (x\right ) - 3528 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 1764 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 840 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 315 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 84 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 14 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{84 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

1/84*(21*B*b^10*x^11 - 12*A*a^10 + 28*(10*B*a*b^9 + A*b^10)*x^10 + 210*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 1260*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 2520*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7*log(x) - 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*x^6 - 1764*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 840*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 315*(3*B*a^8*b^2 + 8*A*a^7
*b^3)*x^3 - 84*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 14*(B*a^10 + 10*A*a^9*b)*x)/x^7

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Sympy [A]  time = 6.69506, size = 243, normalized size = 1.12 \begin{align*} \frac{B b^{10} x^{4}}{4} + 30 a^{3} b^{6} \left (4 A b + 7 B a\right ) \log{\left (x \right )} + x^{3} \left (\frac{A b^{10}}{3} + \frac{10 B a b^{9}}{3}\right ) + x^{2} \left (5 A a b^{9} + \frac{45 B a^{2} b^{8}}{2}\right ) + x \left (45 A a^{2} b^{8} + 120 B a^{3} b^{7}\right ) - \frac{12 A a^{10} + x^{6} \left (17640 A a^{4} b^{6} + 21168 B a^{5} b^{5}\right ) + x^{5} \left (10584 A a^{5} b^{5} + 8820 B a^{6} b^{4}\right ) + x^{4} \left (5880 A a^{6} b^{4} + 3360 B a^{7} b^{3}\right ) + x^{3} \left (2520 A a^{7} b^{3} + 945 B a^{8} b^{2}\right ) + x^{2} \left (756 A a^{8} b^{2} + 168 B a^{9} b\right ) + x \left (140 A a^{9} b + 14 B a^{10}\right )}{84 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**8,x)

[Out]

B*b**10*x**4/4 + 30*a**3*b**6*(4*A*b + 7*B*a)*log(x) + x**3*(A*b**10/3 + 10*B*a*b**9/3) + x**2*(5*A*a*b**9 + 4
5*B*a**2*b**8/2) + x*(45*A*a**2*b**8 + 120*B*a**3*b**7) - (12*A*a**10 + x**6*(17640*A*a**4*b**6 + 21168*B*a**5
*b**5) + x**5*(10584*A*a**5*b**5 + 8820*B*a**6*b**4) + x**4*(5880*A*a**6*b**4 + 3360*B*a**7*b**3) + x**3*(2520
*A*a**7*b**3 + 945*B*a**8*b**2) + x**2*(756*A*a**8*b**2 + 168*B*a**9*b) + x*(140*A*a**9*b + 14*B*a**10))/(84*x
**7)

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Giac [A]  time = 1.26935, size = 325, normalized size = 1.5 \begin{align*} \frac{1}{4} \, B b^{10} x^{4} + \frac{10}{3} \, B a b^{9} x^{3} + \frac{1}{3} \, A b^{10} x^{3} + \frac{45}{2} \, B a^{2} b^{8} x^{2} + 5 \, A a b^{9} x^{2} + 120 \, B a^{3} b^{7} x + 45 \, A a^{2} b^{8} x + 30 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} \log \left ({\left | x \right |}\right ) - \frac{12 \, A a^{10} + 3528 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 1764 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 840 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 315 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 84 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 14 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{84 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^8,x, algorithm="giac")

[Out]

1/4*B*b^10*x^4 + 10/3*B*a*b^9*x^3 + 1/3*A*b^10*x^3 + 45/2*B*a^2*b^8*x^2 + 5*A*a*b^9*x^2 + 120*B*a^3*b^7*x + 45
*A*a^2*b^8*x + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*log(abs(x)) - 1/84*(12*A*a^10 + 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*x^6 + 1764*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 840*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 315*(3*B*a^8*b^2 + 8*A*a^7
*b^3)*x^3 + 84*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 14*(B*a^10 + 10*A*a^9*b)*x)/x^7