3.119 \(\int \frac{(a+b x)^{10} (A+B x)}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.464411, 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 \[ -\frac{a^{10} A}{2 x^2}-\frac{a^9 (a B+10 A b)}{x}+5 a^8 b \log (x) (2 a B+9 A b)+15 a^7 b^2 x (3 a B+8 A b)+15 a^6 b^3 x^2 (4 a B+7 A b)+14 a^5 b^4 x^3 (5 a B+6 A b)+\frac{21}{2} a^4 b^5 x^4 (6 a B+5 A b)+6 a^3 b^6 x^5 (7 a B+4 A b)+\frac{5}{2} a^2 b^7 x^6 (8 a B+3 A b)+\frac{1}{8} b^9 x^8 (10 a B+A b)+\frac{5}{7} a b^8 x^7 (9 a B+2 A b)+\frac{1}{9} b^{10} B x^9 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/x^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{10}}{2 x^{2}} + \frac{B b^{10} x^{9}}{9} - \frac{a^{9} \left (10 A b + B a\right )}{x} + 5 a^{8} b \left (9 A b + 2 B a\right ) \log{\left (x \right )} + 45 a^{7} b^{2} x \left (\frac{8 A b}{3} + B a\right ) + 30 a^{6} b^{3} \left (7 A b + 4 B a\right ) \int x\, dx + 14 a^{5} b^{4} x^{3} \left (6 A b + 5 B a\right ) + \frac{21 a^{4} b^{5} x^{4} \left (5 A b + 6 B a\right )}{2} + 6 a^{3} b^{6} x^{5} \left (4 A b + 7 B a\right ) + \frac{5 a^{2} b^{7} x^{6} \left (3 A b + 8 B a\right )}{2} + \frac{5 a b^{8} x^{7} \left (2 A b + 9 B a\right )}{7} + \frac{b^{9} x^{8} \left (A b + 10 B a\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/x**3,x)

[Out]

-A*a**10/(2*x**2) + B*b**10*x**9/9 - a**9*(10*A*b + B*a)/x + 5*a**8*b*(9*A*b + 2
*B*a)*log(x) + 45*a**7*b**2*x*(8*A*b/3 + B*a) + 30*a**6*b**3*(7*A*b + 4*B*a)*Int
egral(x, x) + 14*a**5*b**4*x**3*(6*A*b + 5*B*a) + 21*a**4*b**5*x**4*(5*A*b + 6*B
*a)/2 + 6*a**3*b**6*x**5*(4*A*b + 7*B*a) + 5*a**2*b**7*x**6*(3*A*b + 8*B*a)/2 +
5*a*b**8*x**7*(2*A*b + 9*B*a)/7 + b**9*x**8*(A*b + 10*B*a)/8

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Mathematica [A]  time = 0.201315, size = 206, normalized size = 0.95 \[ -\frac{a^{10} (A+2 B x)}{2 x^2}-\frac{10 a^9 A b}{x}+5 a^8 b \log (x) (2 a B+9 A b)+45 a^8 b^2 B x+60 a^7 b^3 x (2 A+B x)+35 a^6 b^4 x^2 (3 A+2 B x)+21 a^5 b^5 x^3 (4 A+3 B x)+\frac{21}{2} a^4 b^6 x^4 (5 A+4 B x)+4 a^3 b^7 x^5 (6 A+5 B x)+\frac{15}{14} a^2 b^8 x^6 (7 A+6 B x)+\frac{5}{28} a b^9 x^7 (8 A+7 B x)+\frac{1}{72} b^{10} x^8 (9 A+8 B x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/x^3,x]

[Out]

(-10*a^9*A*b)/x + 45*a^8*b^2*B*x + 60*a^7*b^3*x*(2*A + B*x) - (a^10*(A + 2*B*x))
/(2*x^2) + 35*a^6*b^4*x^2*(3*A + 2*B*x) + 21*a^5*b^5*x^3*(4*A + 3*B*x) + (21*a^4
*b^6*x^4*(5*A + 4*B*x))/2 + 4*a^3*b^7*x^5*(6*A + 5*B*x) + (15*a^2*b^8*x^6*(7*A +
6*B*x))/14 + (5*a*b^9*x^7*(8*A + 7*B*x))/28 + (b^10*x^8*(9*A + 8*B*x))/72 + 5*a
^8*b*(9*A*b + 2*a*B)*Log[x]

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Maple [A]  time = 0.012, size = 240, normalized size = 1.1 \[{\frac{{b}^{10}B{x}^{9}}{9}}+{\frac{A{x}^{8}{b}^{10}}{8}}+{\frac{5\,B{x}^{8}a{b}^{9}}{4}}+{\frac{10\,A{x}^{7}a{b}^{9}}{7}}+{\frac{45\,B{x}^{7}{a}^{2}{b}^{8}}{7}}+{\frac{15\,A{x}^{6}{a}^{2}{b}^{8}}{2}}+20\,B{x}^{6}{a}^{3}{b}^{7}+24\,A{x}^{5}{a}^{3}{b}^{7}+42\,B{x}^{5}{a}^{4}{b}^{6}+{\frac{105\,A{x}^{4}{a}^{4}{b}^{6}}{2}}+63\,B{x}^{4}{a}^{5}{b}^{5}+84\,A{x}^{3}{a}^{5}{b}^{5}+70\,B{x}^{3}{a}^{6}{b}^{4}+105\,A{x}^{2}{a}^{6}{b}^{4}+60\,B{x}^{2}{a}^{7}{b}^{3}+120\,Ax{a}^{7}{b}^{3}+45\,Bx{a}^{8}{b}^{2}+45\,A\ln \left ( x \right ){a}^{8}{b}^{2}+10\,B\ln \left ( x \right ){a}^{9}b-{\frac{A{a}^{10}}{2\,{x}^{2}}}-10\,{\frac{{a}^{9}bA}{x}}-{\frac{{a}^{10}B}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/x^3,x)

[Out]

1/9*b^10*B*x^9+1/8*A*x^8*b^10+5/4*B*x^8*a*b^9+10/7*A*x^7*a*b^9+45/7*B*x^7*a^2*b^
8+15/2*A*x^6*a^2*b^8+20*B*x^6*a^3*b^7+24*A*x^5*a^3*b^7+42*B*x^5*a^4*b^6+105/2*A*
x^4*a^4*b^6+63*B*x^4*a^5*b^5+84*A*x^3*a^5*b^5+70*B*x^3*a^6*b^4+105*A*x^2*a^6*b^4
+60*B*x^2*a^7*b^3+120*A*x*a^7*b^3+45*B*x*a^8*b^2+45*A*ln(x)*a^8*b^2+10*B*ln(x)*a
^9*b-1/2*a^10*A/x^2-10*a^9/x*A*b-a^10/x*B

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Maxima [A]  time = 1.34725, size = 324, normalized size = 1.5 \[ \frac{1}{9} \, B b^{10} x^{9} + \frac{1}{8} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{8} + \frac{5}{7} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{7} + \frac{5}{2} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{6} + 6 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{5} + \frac{21}{2} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{4} + 14 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{3} + 15 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{2} + 15 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x + 5 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} \log \left (x\right ) - \frac{A a^{10} + 2 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^3,x, algorithm="maxima")

[Out]

1/9*B*b^10*x^9 + 1/8*(10*B*a*b^9 + A*b^10)*x^8 + 5/7*(9*B*a^2*b^8 + 2*A*a*b^9)*x
^7 + 5/2*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^6 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^5 + 2
1/2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^4 + 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^3 + 15*(4
*B*a^7*b^3 + 7*A*a^6*b^4)*x^2 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x + 5*(2*B*a^9*b
+ 9*A*a^8*b^2)*log(x) - 1/2*(A*a^10 + 2*(B*a^10 + 10*A*a^9*b)*x)/x^2

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Fricas [A]  time = 0.206324, size = 331, normalized size = 1.53 \[ \frac{56 \, B b^{10} x^{11} - 252 \, A a^{10} + 63 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 360 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 1260 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 3024 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 5292 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7056 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 7560 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 7560 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 2520 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} \log \left (x\right ) - 504 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^3,x, algorithm="fricas")

[Out]

1/504*(56*B*b^10*x^11 - 252*A*a^10 + 63*(10*B*a*b^9 + A*b^10)*x^10 + 360*(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 + 3024*(7*B*a^4*b^
6 + 4*A*a^3*b^7)*x^7 + 5292*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 7056*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*x^5 + 7560*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 7560*(3*B*a^8*b^2 +
8*A*a^7*b^3)*x^3 + 2520*(2*B*a^9*b + 9*A*a^8*b^2)*x^2*log(x) - 504*(B*a^10 + 10*
A*a^9*b)*x)/x^2

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Sympy [A]  time = 3.49357, size = 246, normalized size = 1.14 \[ \frac{B b^{10} x^{9}}{9} + 5 a^{8} b \left (9 A b + 2 B a\right ) \log{\left (x \right )} + x^{8} \left (\frac{A b^{10}}{8} + \frac{5 B a b^{9}}{4}\right ) + x^{7} \left (\frac{10 A a b^{9}}{7} + \frac{45 B a^{2} b^{8}}{7}\right ) + x^{6} \left (\frac{15 A a^{2} b^{8}}{2} + 20 B a^{3} b^{7}\right ) + x^{5} \left (24 A a^{3} b^{7} + 42 B a^{4} b^{6}\right ) + x^{4} \left (\frac{105 A a^{4} b^{6}}{2} + 63 B a^{5} b^{5}\right ) + x^{3} \left (84 A a^{5} b^{5} + 70 B a^{6} b^{4}\right ) + x^{2} \left (105 A a^{6} b^{4} + 60 B a^{7} b^{3}\right ) + x \left (120 A a^{7} b^{3} + 45 B a^{8} b^{2}\right ) - \frac{A a^{10} + x \left (20 A a^{9} b + 2 B a^{10}\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)/x**3,x)

[Out]

B*b**10*x**9/9 + 5*a**8*b*(9*A*b + 2*B*a)*log(x) + x**8*(A*b**10/8 + 5*B*a*b**9/
4) + x**7*(10*A*a*b**9/7 + 45*B*a**2*b**8/7) + x**6*(15*A*a**2*b**8/2 + 20*B*a**
3*b**7) + x**5*(24*A*a**3*b**7 + 42*B*a**4*b**6) + x**4*(105*A*a**4*b**6/2 + 63*
B*a**5*b**5) + x**3*(84*A*a**5*b**5 + 70*B*a**6*b**4) + x**2*(105*A*a**6*b**4 +
60*B*a**7*b**3) + x*(120*A*a**7*b**3 + 45*B*a**8*b**2) - (A*a**10 + x*(20*A*a**9
*b + 2*B*a**10))/(2*x**2)

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GIAC/XCAS [A]  time = 0.2957, size = 324, normalized size = 1.5 \[ \frac{1}{9} \, B b^{10} x^{9} + \frac{5}{4} \, B a b^{9} x^{8} + \frac{1}{8} \, A b^{10} x^{8} + \frac{45}{7} \, B a^{2} b^{8} x^{7} + \frac{10}{7} \, A a b^{9} x^{7} + 20 \, B a^{3} b^{7} x^{6} + \frac{15}{2} \, A a^{2} b^{8} x^{6} + 42 \, B a^{4} b^{6} x^{5} + 24 \, A a^{3} b^{7} x^{5} + 63 \, B a^{5} b^{5} x^{4} + \frac{105}{2} \, A a^{4} b^{6} x^{4} + 70 \, B a^{6} b^{4} x^{3} + 84 \, A a^{5} b^{5} x^{3} + 60 \, B a^{7} b^{3} x^{2} + 105 \, A a^{6} b^{4} x^{2} + 45 \, B a^{8} b^{2} x + 120 \, A a^{7} b^{3} x + 5 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a^{10} + 2 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^3,x, algorithm="giac")

[Out]

1/9*B*b^10*x^9 + 5/4*B*a*b^9*x^8 + 1/8*A*b^10*x^8 + 45/7*B*a^2*b^8*x^7 + 10/7*A*
a*b^9*x^7 + 20*B*a^3*b^7*x^6 + 15/2*A*a^2*b^8*x^6 + 42*B*a^4*b^6*x^5 + 24*A*a^3*
b^7*x^5 + 63*B*a^5*b^5*x^4 + 105/2*A*a^4*b^6*x^4 + 70*B*a^6*b^4*x^3 + 84*A*a^5*b
^5*x^3 + 60*B*a^7*b^3*x^2 + 105*A*a^6*b^4*x^2 + 45*B*a^8*b^2*x + 120*A*a^7*b^3*x
+ 5*(2*B*a^9*b + 9*A*a^8*b^2)*ln(abs(x)) - 1/2*(A*a^10 + 2*(B*a^10 + 10*A*a^9*b
)*x)/x^2