3.154 \(\int \frac{(a+b x)^{10} (A+B x)}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.143427, 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} \[ -\frac{5 a^7 b^2 (3 a B+8 A b)}{x^3}-\frac{15 a^6 b^3 (4 a B+7 A b)}{x^2}+\frac{15}{2} a^2 b^7 x^2 (8 a B+3 A b)-\frac{42 a^5 b^4 (5 a B+6 A b)}{x}+30 a^3 b^6 x (7 a B+4 A b)+42 a^4 b^5 \log (x) (6 a B+5 A b)-\frac{a^9 (a B+10 A b)}{5 x^5}-\frac{5 a^8 b (2 a B+9 A b)}{4 x^4}-\frac{a^{10} A}{6 x^6}+\frac{5}{3} a b^8 x^3 (9 a B+2 A b)+\frac{1}{4} b^9 x^4 (10 a B+A b)+\frac{1}{5} b^{10} B x^5 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^7,x]

[Out]

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

Mathematica [A]  time = 0.0792477, size = 210, normalized size = 0.96 \[ -\frac{15 a^8 b^2 (3 A+4 B x)}{4 x^4}-\frac{20 a^7 b^3 (2 A+3 B x)}{x^3}-\frac{105 a^6 b^4 (A+2 B x)}{x^2}+\frac{15}{2} a^2 b^8 x^2 (3 A+2 B x)+60 a^3 b^7 x (2 A+B x)+42 a^4 b^5 \log (x) (6 a B+5 A b)-\frac{252 a^5 A b^5}{x}-\frac{a^9 b (4 A+5 B x)}{2 x^5}-\frac{a^{10} (5 A+6 B x)}{30 x^6}+210 a^4 b^6 B x+\frac{5}{6} a b^9 x^3 (4 A+3 B x)+\frac{1}{20} b^{10} x^4 (5 A+4 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^7,x]

[Out]

(-252*a^5*A*b^5)/x + 210*a^4*b^6*B*x + 60*a^3*b^7*x*(2*A + B*x) - (105*a^6*b^4*(A + 2*B*x))/x^2 + (15*a^2*b^8*
x^2*(3*A + 2*B*x))/2 - (20*a^7*b^3*(2*A + 3*B*x))/x^3 + (5*a*b^9*x^3*(4*A + 3*B*x))/6 - (15*a^8*b^2*(3*A + 4*B
*x))/(4*x^4) + (b^10*x^4*(5*A + 4*B*x))/20 - (a^9*b*(4*A + 5*B*x))/(2*x^5) - (a^10*(5*A + 6*B*x))/(30*x^6) + 4
2*a^4*b^5*(5*A*b + 6*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}^{5}}{5}}+{\frac{A{x}^{4}{b}^{10}}{4}}+{\frac{5\,B{x}^{4}a{b}^{9}}{2}}+{\frac{10\,A{x}^{3}a{b}^{9}}{3}}+15\,B{x}^{3}{a}^{2}{b}^{8}+{\frac{45\,A{x}^{2}{a}^{2}{b}^{8}}{2}}+60\,B{x}^{2}{a}^{3}{b}^{7}+120\,{a}^{3}{b}^{7}Ax+210\,{a}^{4}{b}^{6}Bx+210\,A\ln \left ( x \right ){a}^{4}{b}^{6}+252\,B\ln \left ( x \right ){a}^{5}{b}^{5}-40\,{\frac{{a}^{7}{b}^{3}A}{{x}^{3}}}-15\,{\frac{{a}^{8}{b}^{2}B}{{x}^{3}}}-2\,{\frac{{a}^{9}bA}{{x}^{5}}}-{\frac{{a}^{10}B}{5\,{x}^{5}}}-{\frac{45\,{a}^{8}{b}^{2}A}{4\,{x}^{4}}}-{\frac{5\,{a}^{9}bB}{2\,{x}^{4}}}-105\,{\frac{{a}^{6}{b}^{4}A}{{x}^{2}}}-60\,{\frac{{a}^{7}{b}^{3}B}{{x}^{2}}}-{\frac{A{a}^{10}}{6\,{x}^{6}}}-252\,{\frac{{a}^{5}{b}^{5}A}{x}}-210\,{\frac{{a}^{6}{b}^{4}B}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^7,x)

[Out]

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

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Maxima [A]  time = 1.0547, size = 325, normalized size = 1.49 \begin{align*} \frac{1}{5} \, B b^{10} x^{5} + \frac{1}{4} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{4} + \frac{5}{3} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{3} + \frac{15}{2} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{2} + 30 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x + 42 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} \log \left (x\right ) - \frac{10 \, A a^{10} + 2520 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 900 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 300 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 75 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 12 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^7,x, algorithm="maxima")

[Out]

1/5*B*b^10*x^5 + 1/4*(10*B*a*b^9 + A*b^10)*x^4 + 5/3*(9*B*a^2*b^8 + 2*A*a*b^9)*x^3 + 15/2*(8*B*a^3*b^7 + 3*A*a
^2*b^8)*x^2 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*log(x) - 1/60*(10*A*a^10 + 252
0*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 900*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 300*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3
+ 75*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 12*(B*a^10 + 10*A*a^9*b)*x)/x^6

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Fricas [A]  time = 1.40746, size = 552, normalized size = 2.53 \begin{align*} \frac{12 \, B b^{10} x^{11} - 10 \, A a^{10} + 15 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 100 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 450 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 1800 \,{\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} \log \left (x\right ) - 2520 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 900 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 300 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 75 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 12 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.43906, size = 245, normalized size = 1.12 \begin{align*} \frac{B b^{10} x^{5}}{5} + 42 a^{4} b^{5} \left (5 A b + 6 B a\right ) \log{\left (x \right )} + x^{4} \left (\frac{A b^{10}}{4} + \frac{5 B a b^{9}}{2}\right ) + x^{3} \left (\frac{10 A a b^{9}}{3} + 15 B a^{2} b^{8}\right ) + x^{2} \left (\frac{45 A a^{2} b^{8}}{2} + 60 B a^{3} b^{7}\right ) + x \left (120 A a^{3} b^{7} + 210 B a^{4} b^{6}\right ) - \frac{10 A a^{10} + x^{5} \left (15120 A a^{5} b^{5} + 12600 B a^{6} b^{4}\right ) + x^{4} \left (6300 A a^{6} b^{4} + 3600 B a^{7} b^{3}\right ) + x^{3} \left (2400 A a^{7} b^{3} + 900 B a^{8} b^{2}\right ) + x^{2} \left (675 A a^{8} b^{2} + 150 B a^{9} b\right ) + x \left (120 A a^{9} b + 12 B a^{10}\right )}{60 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**7,x)

[Out]

B*b**10*x**5/5 + 42*a**4*b**5*(5*A*b + 6*B*a)*log(x) + x**4*(A*b**10/4 + 5*B*a*b**9/2) + x**3*(10*A*a*b**9/3 +
15*B*a**2*b**8) + x**2*(45*A*a**2*b**8/2 + 60*B*a**3*b**7) + x*(120*A*a**3*b**7 + 210*B*a**4*b**6) - (10*A*a*
*10 + x**5*(15120*A*a**5*b**5 + 12600*B*a**6*b**4) + x**4*(6300*A*a**6*b**4 + 3600*B*a**7*b**3) + x**3*(2400*A
*a**7*b**3 + 900*B*a**8*b**2) + x**2*(675*A*a**8*b**2 + 150*B*a**9*b) + x*(120*A*a**9*b + 12*B*a**10))/(60*x**
6)

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Giac [A]  time = 1.19125, size = 325, normalized size = 1.49 \begin{align*} \frac{1}{5} \, B b^{10} x^{5} + \frac{5}{2} \, B a b^{9} x^{4} + \frac{1}{4} \, A b^{10} x^{4} + 15 \, B a^{2} b^{8} x^{3} + \frac{10}{3} \, A a b^{9} x^{3} + 60 \, B a^{3} b^{7} x^{2} + \frac{45}{2} \, A a^{2} b^{8} x^{2} + 210 \, B a^{4} b^{6} x + 120 \, A a^{3} b^{7} x + 42 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} \log \left ({\left | x \right |}\right ) - \frac{10 \, A a^{10} + 2520 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 900 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 300 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 75 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 12 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^7,x, algorithm="giac")

[Out]

1/5*B*b^10*x^5 + 5/2*B*a*b^9*x^4 + 1/4*A*b^10*x^4 + 15*B*a^2*b^8*x^3 + 10/3*A*a*b^9*x^3 + 60*B*a^3*b^7*x^2 + 4
5/2*A*a^2*b^8*x^2 + 210*B*a^4*b^6*x + 120*A*a^3*b^7*x + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*log(abs(x)) - 1/60*(10*
A*a^10 + 2520*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 900*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 300*(3*B*a^8*b^2 + 8*A*a
^7*b^3)*x^3 + 75*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 12*(B*a^10 + 10*A*a^9*b)*x)/x^6