3.153 \(\int \frac{(a+b x)^{10} (A+B x)}{x^6} \, dx\)

Optimal. Leaf size=218 \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{2 x^2}+15 a^3 b^6 x^2 (7 a B+4 A b)+5 a^2 b^7 x^3 (8 a B+3 A b)-\frac{30 a^6 b^3 (4 a B+7 A b)}{x}+42 a^4 b^5 x (6 a B+5 A b)+42 a^5 b^4 \log (x) (5 a B+6 A b)-\frac{a^9 (a B+10 A b)}{4 x^4}-\frac{5 a^8 b (2 a B+9 A b)}{3 x^3}-\frac{a^{10} A}{5 x^5}+\frac{5}{4} a b^8 x^4 (9 a B+2 A b)+\frac{1}{5} b^9 x^5 (10 a B+A b)+\frac{1}{6} b^{10} B x^6 \]

[Out]

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

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Rubi [A]  time = 0.14394, antiderivative size = 218, 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} \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{2 x^2}+15 a^3 b^6 x^2 (7 a B+4 A b)+5 a^2 b^7 x^3 (8 a B+3 A b)-\frac{30 a^6 b^3 (4 a B+7 A b)}{x}+42 a^4 b^5 x (6 a B+5 A b)+42 a^5 b^4 \log (x) (5 a B+6 A b)-\frac{a^9 (a B+10 A b)}{4 x^4}-\frac{5 a^8 b (2 a B+9 A b)}{3 x^3}-\frac{a^{10} A}{5 x^5}+\frac{5}{4} a b^8 x^4 (9 a B+2 A b)+\frac{1}{5} b^9 x^5 (10 a B+A b)+\frac{1}{6} b^{10} B x^6 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^6,x]

[Out]

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

Mathematica [A]  time = 0.0795937, size = 210, normalized size = 0.96 \[ -\frac{15 a^8 b^2 (2 A+3 B x)}{2 x^3}-\frac{60 a^7 b^3 (A+2 B x)}{x^2}+20 a^3 b^7 x^2 (3 A+2 B x)+\frac{15}{4} a^2 b^8 x^3 (4 A+3 B x)+105 a^4 b^6 x (2 A+B x)+42 a^5 b^4 \log (x) (5 a B+6 A b)-\frac{210 a^6 A b^4}{x}-\frac{5 a^9 b (3 A+4 B x)}{6 x^4}-\frac{a^{10} (4 A+5 B x)}{20 x^5}+252 a^5 b^5 B x+\frac{1}{2} a b^9 x^4 (5 A+4 B x)+\frac{1}{30} b^{10} x^5 (6 A+5 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^6,x]

[Out]

(-210*a^6*A*b^4)/x + 252*a^5*b^5*B*x + 105*a^4*b^6*x*(2*A + B*x) - (60*a^7*b^3*(A + 2*B*x))/x^2 + 20*a^3*b^7*x
^2*(3*A + 2*B*x) - (15*a^8*b^2*(2*A + 3*B*x))/(2*x^3) + (15*a^2*b^8*x^3*(4*A + 3*B*x))/4 - (5*a^9*b*(3*A + 4*B
*x))/(6*x^4) + (a*b^9*x^4*(5*A + 4*B*x))/2 - (a^10*(4*A + 5*B*x))/(20*x^5) + (b^10*x^5*(6*A + 5*B*x))/30 + 42*
a^5*b^4*(6*A*b + 5*a*B)*Log[x]

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Maple [A]  time = 0.007, size = 240, normalized size = 1.1 \begin{align*}{\frac{{b}^{10}B{x}^{6}}{6}}+{\frac{A{x}^{5}{b}^{10}}{5}}+2\,B{x}^{5}a{b}^{9}+{\frac{5\,A{x}^{4}a{b}^{9}}{2}}+{\frac{45\,B{x}^{4}{a}^{2}{b}^{8}}{4}}+15\,A{x}^{3}{a}^{2}{b}^{8}+40\,B{x}^{3}{a}^{3}{b}^{7}+60\,A{x}^{2}{a}^{3}{b}^{7}+105\,B{x}^{2}{a}^{4}{b}^{6}+210\,{a}^{4}{b}^{6}Ax+252\,{a}^{5}{b}^{5}Bx+252\,A\ln \left ( x \right ){a}^{5}{b}^{5}+210\,B\ln \left ( x \right ){a}^{6}{b}^{4}-15\,{\frac{{a}^{8}{b}^{2}A}{{x}^{3}}}-{\frac{10\,{a}^{9}bB}{3\,{x}^{3}}}-{\frac{A{a}^{10}}{5\,{x}^{5}}}-{\frac{5\,{a}^{9}bA}{2\,{x}^{4}}}-{\frac{{a}^{10}B}{4\,{x}^{4}}}-60\,{\frac{{a}^{7}{b}^{3}A}{{x}^{2}}}-{\frac{45\,{a}^{8}{b}^{2}B}{2\,{x}^{2}}}-210\,{\frac{{a}^{6}{b}^{4}A}{x}}-120\,{\frac{{a}^{7}{b}^{3}B}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^6,x)

[Out]

1/6*b^10*B*x^6+1/5*A*x^5*b^10+2*B*x^5*a*b^9+5/2*A*x^4*a*b^9+45/4*B*x^4*a^2*b^8+15*A*x^3*a^2*b^8+40*B*x^3*a^3*b
^7+60*A*x^2*a^3*b^7+105*B*x^2*a^4*b^6+210*a^4*b^6*A*x+252*a^5*b^5*B*x+252*A*ln(x)*a^5*b^5+210*B*ln(x)*a^6*b^4-
15*a^8*b^2/x^3*A-10/3*a^9*b/x^3*B-1/5*a^10*A/x^5-5/2*a^9/x^4*A*b-1/4*a^10/x^4*B-60*a^7*b^3/x^2*A-45/2*a^8*b^2/
x^2*B-210*a^6*b^4/x*A-120*a^7*b^3/x*B

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Maxima [A]  time = 1.01622, size = 325, normalized size = 1.49 \begin{align*} \frac{1}{6} \, B b^{10} x^{6} + \frac{1}{5} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{5} + \frac{5}{4} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{4} + 5 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{3} + 15 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{2} + 42 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x + 42 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} \log \left (x\right ) - \frac{12 \, A a^{10} + 1800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 450 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 100 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 15 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^6,x, algorithm="maxima")

[Out]

1/6*B*b^10*x^6 + 1/5*(10*B*a*b^9 + A*b^10)*x^5 + 5/4*(9*B*a^2*b^8 + 2*A*a*b^9)*x^4 + 5*(8*B*a^3*b^7 + 3*A*a^2*
b^8)*x^3 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^2 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x + 42*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*log(x) - 1/60*(12*A*a^10 + 1800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 450*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 10
0*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 15*(B*a^10 + 10*A*a^9*b)*x)/x^5

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Fricas [A]  time = 1.44275, size = 552, normalized size = 2.53 \begin{align*} \frac{10 \, B b^{10} x^{11} - 12 \, A a^{10} + 12 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 75 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 300 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 900 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 2520 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2520 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} \log \left (x\right ) - 1800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 450 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 100 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 15 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^6,x, algorithm="fricas")

[Out]

1/60*(10*B*b^10*x^11 - 12*A*a^10 + 12*(10*B*a*b^9 + A*b^10)*x^10 + 75*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 300*(8*B
*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 900*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 2520*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 252
0*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5*log(x) - 1800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 450*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*x^3 - 100*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 15*(B*a^10 + 10*A*a^9*b)*x)/x^5

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Sympy [A]  time = 3.02947, size = 243, normalized size = 1.11 \begin{align*} \frac{B b^{10} x^{6}}{6} + 42 a^{5} b^{4} \left (6 A b + 5 B a\right ) \log{\left (x \right )} + x^{5} \left (\frac{A b^{10}}{5} + 2 B a b^{9}\right ) + x^{4} \left (\frac{5 A a b^{9}}{2} + \frac{45 B a^{2} b^{8}}{4}\right ) + x^{3} \left (15 A a^{2} b^{8} + 40 B a^{3} b^{7}\right ) + x^{2} \left (60 A a^{3} b^{7} + 105 B a^{4} b^{6}\right ) + x \left (210 A a^{4} b^{6} + 252 B a^{5} b^{5}\right ) - \frac{12 A a^{10} + x^{4} \left (12600 A a^{6} b^{4} + 7200 B a^{7} b^{3}\right ) + x^{3} \left (3600 A a^{7} b^{3} + 1350 B a^{8} b^{2}\right ) + x^{2} \left (900 A a^{8} b^{2} + 200 B a^{9} b\right ) + x \left (150 A a^{9} b + 15 B a^{10}\right )}{60 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**6,x)

[Out]

B*b**10*x**6/6 + 42*a**5*b**4*(6*A*b + 5*B*a)*log(x) + x**5*(A*b**10/5 + 2*B*a*b**9) + x**4*(5*A*a*b**9/2 + 45
*B*a**2*b**8/4) + x**3*(15*A*a**2*b**8 + 40*B*a**3*b**7) + x**2*(60*A*a**3*b**7 + 105*B*a**4*b**6) + x*(210*A*
a**4*b**6 + 252*B*a**5*b**5) - (12*A*a**10 + x**4*(12600*A*a**6*b**4 + 7200*B*a**7*b**3) + x**3*(3600*A*a**7*b
**3 + 1350*B*a**8*b**2) + x**2*(900*A*a**8*b**2 + 200*B*a**9*b) + x*(150*A*a**9*b + 15*B*a**10))/(60*x**5)

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Giac [A]  time = 1.1794, size = 325, normalized size = 1.49 \begin{align*} \frac{1}{6} \, B b^{10} x^{6} + 2 \, B a b^{9} x^{5} + \frac{1}{5} \, A b^{10} x^{5} + \frac{45}{4} \, B a^{2} b^{8} x^{4} + \frac{5}{2} \, A a b^{9} x^{4} + 40 \, B a^{3} b^{7} x^{3} + 15 \, A a^{2} b^{8} x^{3} + 105 \, B a^{4} b^{6} x^{2} + 60 \, A a^{3} b^{7} x^{2} + 252 \, B a^{5} b^{5} x + 210 \, A a^{4} b^{6} x + 42 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} \log \left ({\left | x \right |}\right ) - \frac{12 \, A a^{10} + 1800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 450 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 100 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 15 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^6,x, algorithm="giac")

[Out]

1/6*B*b^10*x^6 + 2*B*a*b^9*x^5 + 1/5*A*b^10*x^5 + 45/4*B*a^2*b^8*x^4 + 5/2*A*a*b^9*x^4 + 40*B*a^3*b^7*x^3 + 15
*A*a^2*b^8*x^3 + 105*B*a^4*b^6*x^2 + 60*A*a^3*b^7*x^2 + 252*B*a^5*b^5*x + 210*A*a^4*b^6*x + 42*(5*B*a^6*b^4 +
6*A*a^5*b^5)*log(abs(x)) - 1/60*(12*A*a^10 + 1800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 450*(3*B*a^8*b^2 + 8*A*a^7
*b^3)*x^3 + 100*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 15*(B*a^10 + 10*A*a^9*b)*x)/x^5