3.120 \(\int \frac{(a+b x)^{10} (A+B x)}{x^4} \, dx\)

Optimal. Leaf size=216 \[ -\frac{a^{10} A}{3 x^3}-\frac{a^9 (a B+10 A b)}{2 x^2}-\frac{5 a^8 b (2 a B+9 A b)}{x}+15 a^7 b^2 \log (x) (3 a B+8 A b)+30 a^6 b^3 x (4 a B+7 A b)+21 a^5 b^4 x^2 (5 a B+6 A b)+14 a^4 b^5 x^3 (6 a B+5 A b)+\frac{15}{2} a^3 b^6 x^4 (7 a B+4 A b)+3 a^2 b^7 x^5 (8 a B+3 A b)+\frac{1}{7} b^9 x^7 (10 a B+A b)+\frac{5}{6} a b^8 x^6 (9 a B+2 A b)+\frac{1}{8} b^{10} B x^8 \]

[Out]

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

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Rubi [A]  time = 0.471398, 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 \[ -\frac{a^{10} A}{3 x^3}-\frac{a^9 (a B+10 A b)}{2 x^2}-\frac{5 a^8 b (2 a B+9 A b)}{x}+15 a^7 b^2 \log (x) (3 a B+8 A b)+30 a^6 b^3 x (4 a B+7 A b)+21 a^5 b^4 x^2 (5 a B+6 A b)+14 a^4 b^5 x^3 (6 a B+5 A b)+\frac{15}{2} a^3 b^6 x^4 (7 a B+4 A b)+3 a^2 b^7 x^5 (8 a B+3 A b)+\frac{1}{7} b^9 x^7 (10 a B+A b)+\frac{5}{6} a b^8 x^6 (9 a B+2 A b)+\frac{1}{8} b^{10} B x^8 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/x^4,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{10}}{3 x^{3}} + \frac{B b^{10} x^{8}}{8} - \frac{a^{9} \left (10 A b + B a\right )}{2 x^{2}} - \frac{5 a^{8} b \left (9 A b + 2 B a\right )}{x} + 15 a^{7} b^{2} \left (8 A b + 3 B a\right ) \log{\left (x \right )} + 120 a^{6} b^{3} x \left (\frac{7 A b}{4} + B a\right ) + 42 a^{5} b^{4} \left (6 A b + 5 B a\right ) \int x\, dx + 14 a^{4} b^{5} x^{3} \left (5 A b + 6 B a\right ) + \frac{15 a^{3} b^{6} x^{4} \left (4 A b + 7 B a\right )}{2} + 3 a^{2} b^{7} x^{5} \left (3 A b + 8 B a\right ) + \frac{5 a b^{8} x^{6} \left (2 A b + 9 B a\right )}{6} + \frac{b^{9} x^{7} \left (A b + 10 B a\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/x**4,x)

[Out]

-A*a**10/(3*x**3) + B*b**10*x**8/8 - a**9*(10*A*b + B*a)/(2*x**2) - 5*a**8*b*(9*
A*b + 2*B*a)/x + 15*a**7*b**2*(8*A*b + 3*B*a)*log(x) + 120*a**6*b**3*x*(7*A*b/4
+ B*a) + 42*a**5*b**4*(6*A*b + 5*B*a)*Integral(x, x) + 14*a**4*b**5*x**3*(5*A*b
+ 6*B*a) + 15*a**3*b**6*x**4*(4*A*b + 7*B*a)/2 + 3*a**2*b**7*x**5*(3*A*b + 8*B*a
) + 5*a*b**8*x**6*(2*A*b + 9*B*a)/6 + b**9*x**7*(A*b + 10*B*a)/7

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Mathematica [A]  time = 0.206323, size = 208, normalized size = 0.96 \[ -\frac{a^{10} (2 A+3 B x)}{6 x^3}-\frac{5 a^9 b (A+2 B x)}{x^2}-\frac{45 a^8 A b^2}{x}+15 a^7 b^2 \log (x) (3 a B+8 A b)+120 a^7 b^3 B x+105 a^6 b^4 x (2 A+B x)+42 a^5 b^5 x^2 (3 A+2 B x)+\frac{35}{2} a^4 b^6 x^3 (4 A+3 B x)+6 a^3 b^7 x^4 (5 A+4 B x)+\frac{3}{2} a^2 b^8 x^5 (6 A+5 B x)+\frac{5}{21} a b^9 x^6 (7 A+6 B x)+\frac{1}{56} b^{10} x^7 (8 A+7 B x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/x^4,x]

[Out]

(-45*a^8*A*b^2)/x + 120*a^7*b^3*B*x + 105*a^6*b^4*x*(2*A + B*x) - (5*a^9*b*(A +
2*B*x))/x^2 + 42*a^5*b^5*x^2*(3*A + 2*B*x) - (a^10*(2*A + 3*B*x))/(6*x^3) + (35*
a^4*b^6*x^3*(4*A + 3*B*x))/2 + 6*a^3*b^7*x^4*(5*A + 4*B*x) + (3*a^2*b^8*x^5*(6*A
+ 5*B*x))/2 + (5*a*b^9*x^6*(7*A + 6*B*x))/21 + (b^10*x^7*(8*A + 7*B*x))/56 + 15
*a^7*b^2*(8*A*b + 3*a*B)*Log[x]

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Maple [A]  time = 0.013, size = 240, normalized size = 1.1 \[{\frac{{b}^{10}B{x}^{8}}{8}}+{\frac{A{x}^{7}{b}^{10}}{7}}+{\frac{10\,B{x}^{7}a{b}^{9}}{7}}+{\frac{5\,A{x}^{6}a{b}^{9}}{3}}+{\frac{15\,B{x}^{6}{a}^{2}{b}^{8}}{2}}+9\,A{x}^{5}{a}^{2}{b}^{8}+24\,B{x}^{5}{a}^{3}{b}^{7}+30\,A{x}^{4}{a}^{3}{b}^{7}+{\frac{105\,B{x}^{4}{a}^{4}{b}^{6}}{2}}+70\,A{x}^{3}{a}^{4}{b}^{6}+84\,B{x}^{3}{a}^{5}{b}^{5}+126\,A{x}^{2}{a}^{5}{b}^{5}+105\,B{x}^{2}{a}^{6}{b}^{4}+210\,Ax{a}^{6}{b}^{4}+120\,Bx{a}^{7}{b}^{3}+120\,A\ln \left ( x \right ){a}^{7}{b}^{3}+45\,B\ln \left ( x \right ){a}^{8}{b}^{2}-5\,{\frac{{a}^{9}bA}{{x}^{2}}}-{\frac{{a}^{10}B}{2\,{x}^{2}}}-45\,{\frac{{a}^{8}{b}^{2}A}{x}}-10\,{\frac{{a}^{9}bB}{x}}-{\frac{A{a}^{10}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/x^4,x)

[Out]

1/8*b^10*B*x^8+1/7*A*x^7*b^10+10/7*B*x^7*a*b^9+5/3*A*x^6*a*b^9+15/2*B*x^6*a^2*b^
8+9*A*x^5*a^2*b^8+24*B*x^5*a^3*b^7+30*A*x^4*a^3*b^7+105/2*B*x^4*a^4*b^6+70*A*x^3
*a^4*b^6+84*B*x^3*a^5*b^5+126*A*x^2*a^5*b^5+105*B*x^2*a^6*b^4+210*A*x*a^6*b^4+12
0*B*x*a^7*b^3+120*A*ln(x)*a^7*b^3+45*B*ln(x)*a^8*b^2-5*a^9/x^2*A*b-1/2*a^10/x^2*
B-45*a^8*b^2/x*A-10*a^9*b/x*B-1/3*a^10*A/x^3

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Maxima [A]  time = 1.36302, size = 325, normalized size = 1.5 \[ \frac{1}{8} \, B b^{10} x^{8} + \frac{1}{7} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{7} + \frac{5}{6} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{6} + 3 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{5} + \frac{15}{2} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{4} + 14 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{3} + 21 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{2} + 30 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x + 15 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} \log \left (x\right ) - \frac{2 \, A a^{10} + 30 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 3 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^4,x, algorithm="maxima")

[Out]

1/8*B*b^10*x^8 + 1/7*(10*B*a*b^9 + A*b^10)*x^7 + 5/6*(9*B*a^2*b^8 + 2*A*a*b^9)*x
^6 + 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^5 + 15/2*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^4 +
14*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^3 + 21*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^2 + 30*(4*
B*a^7*b^3 + 7*A*a^6*b^4)*x + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*log(x) - 1/6*(2*A*a^
10 + 30*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 3*(B*a^10 + 10*A*a^9*b)*x)/x^3

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Fricas [A]  time = 0.207952, size = 331, normalized size = 1.53 \[ \frac{21 \, B b^{10} x^{11} - 56 \, A a^{10} + 24 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 140 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 504 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 1260 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 2352 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 3528 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5040 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2520 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} \log \left (x\right ) - 840 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 84 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^4,x, algorithm="fricas")

[Out]

1/168*(21*B*b^10*x^11 - 56*A*a^10 + 24*(10*B*a*b^9 + A*b^10)*x^10 + 140*(9*B*a^2
*b^8 + 2*A*a*b^9)*x^9 + 504*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 1260*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*x^7 + 2352*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 3528*(5*B*a^6*b^4 +
6*A*a^5*b^5)*x^5 + 5040*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2520*(3*B*a^8*b^2 + 8*
A*a^7*b^3)*x^3*log(x) - 840*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 84*(B*a^10 + 10*A*a^
9*b)*x)/x^3

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Sympy [A]  time = 4.77018, size = 246, normalized size = 1.14 \[ \frac{B b^{10} x^{8}}{8} + 15 a^{7} b^{2} \left (8 A b + 3 B a\right ) \log{\left (x \right )} + x^{7} \left (\frac{A b^{10}}{7} + \frac{10 B a b^{9}}{7}\right ) + x^{6} \left (\frac{5 A a b^{9}}{3} + \frac{15 B a^{2} b^{8}}{2}\right ) + x^{5} \left (9 A a^{2} b^{8} + 24 B a^{3} b^{7}\right ) + x^{4} \left (30 A a^{3} b^{7} + \frac{105 B a^{4} b^{6}}{2}\right ) + x^{3} \left (70 A a^{4} b^{6} + 84 B a^{5} b^{5}\right ) + x^{2} \left (126 A a^{5} b^{5} + 105 B a^{6} b^{4}\right ) + x \left (210 A a^{6} b^{4} + 120 B a^{7} b^{3}\right ) - \frac{2 A a^{10} + x^{2} \left (270 A a^{8} b^{2} + 60 B a^{9} b\right ) + x \left (30 A a^{9} b + 3 B a^{10}\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)/x**4,x)

[Out]

B*b**10*x**8/8 + 15*a**7*b**2*(8*A*b + 3*B*a)*log(x) + x**7*(A*b**10/7 + 10*B*a*
b**9/7) + x**6*(5*A*a*b**9/3 + 15*B*a**2*b**8/2) + x**5*(9*A*a**2*b**8 + 24*B*a*
*3*b**7) + x**4*(30*A*a**3*b**7 + 105*B*a**4*b**6/2) + x**3*(70*A*a**4*b**6 + 84
*B*a**5*b**5) + x**2*(126*A*a**5*b**5 + 105*B*a**6*b**4) + x*(210*A*a**6*b**4 +
120*B*a**7*b**3) - (2*A*a**10 + x**2*(270*A*a**8*b**2 + 60*B*a**9*b) + x*(30*A*a
**9*b + 3*B*a**10))/(6*x**3)

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GIAC/XCAS [A]  time = 0.262339, size = 325, normalized size = 1.5 \[ \frac{1}{8} \, B b^{10} x^{8} + \frac{10}{7} \, B a b^{9} x^{7} + \frac{1}{7} \, A b^{10} x^{7} + \frac{15}{2} \, B a^{2} b^{8} x^{6} + \frac{5}{3} \, A a b^{9} x^{6} + 24 \, B a^{3} b^{7} x^{5} + 9 \, A a^{2} b^{8} x^{5} + \frac{105}{2} \, B a^{4} b^{6} x^{4} + 30 \, A a^{3} b^{7} x^{4} + 84 \, B a^{5} b^{5} x^{3} + 70 \, A a^{4} b^{6} x^{3} + 105 \, B a^{6} b^{4} x^{2} + 126 \, A a^{5} b^{5} x^{2} + 120 \, B a^{7} b^{3} x + 210 \, A a^{6} b^{4} x + 15 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, A a^{10} + 30 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 3 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^4,x, algorithm="giac")

[Out]

1/8*B*b^10*x^8 + 10/7*B*a*b^9*x^7 + 1/7*A*b^10*x^7 + 15/2*B*a^2*b^8*x^6 + 5/3*A*
a*b^9*x^6 + 24*B*a^3*b^7*x^5 + 9*A*a^2*b^8*x^5 + 105/2*B*a^4*b^6*x^4 + 30*A*a^3*
b^7*x^4 + 84*B*a^5*b^5*x^3 + 70*A*a^4*b^6*x^3 + 105*B*a^6*b^4*x^2 + 126*A*a^5*b^
5*x^2 + 120*B*a^7*b^3*x + 210*A*a^6*b^4*x + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*ln(ab
s(x)) - 1/6*(2*A*a^10 + 30*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 3*(B*a^10 + 10*A*a^9*
b)*x)/x^3