3.97 \(\int \frac{(a+b x)^5 (A+B x)}{x^4} \, dx\)

Optimal. Leaf size=108 \[ -\frac{a^5 A}{3 x^3}-\frac{a^4 (a B+5 A b)}{2 x^2}-\frac{5 a^3 b (a B+2 A b)}{x}+10 a^2 b^2 \log (x) (a B+A b)+\frac{1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{3} b^5 B x^3 \]

[Out]

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

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Rubi [A]  time = 0.17002, antiderivative size = 108, 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 \[ -\frac{a^5 A}{3 x^3}-\frac{a^4 (a B+5 A b)}{2 x^2}-\frac{5 a^3 b (a B+2 A b)}{x}+10 a^2 b^2 \log (x) (a B+A b)+\frac{1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{3} b^5 B x^3 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^5*(A + B*x))/x^4,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(B*x+A)/x**4,x)

[Out]

-A*a**5/(3*x**3) + B*b**5*x**3/3 - a**4*(5*A*b + B*a)/(2*x**2) - 5*a**3*b*(2*A*b
+ B*a)/x + 10*a**2*b**2*(A*b + B*a)*log(x) + 5*a*b**3*x*(A*b + 2*B*a) + b**4*(A
*b + 5*B*a)*Integral(x, x)

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Mathematica [A]  time = 0.0441145, size = 109, normalized size = 1.01 \[ \frac{a^5 (-(2 A+3 B x))-15 a^4 b x (A+2 B x)-60 a^3 A b^2 x^2+60 a^2 b^2 x^3 \log (x) (a B+A b)+60 a^2 b^3 B x^4+15 a b^4 x^4 (2 A+B x)+b^5 x^5 (3 A+2 B x)}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^5*(A + B*x))/x^4,x]

[Out]

(-60*a^3*A*b^2*x^2 + 60*a^2*b^3*B*x^4 + 15*a*b^4*x^4*(2*A + B*x) - 15*a^4*b*x*(A
+ 2*B*x) + b^5*x^5*(3*A + 2*B*x) - a^5*(2*A + 3*B*x) + 60*a^2*b^2*(A*b + a*B)*x
^3*Log[x])/(6*x^3)

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Maple [A]  time = 0.01, size = 120, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{3}}{3}}+{\frac{A{x}^{2}{b}^{5}}{2}}+{\frac{5\,B{x}^{2}a{b}^{4}}{2}}+5\,Axa{b}^{4}+10\,Bx{a}^{2}{b}^{3}+10\,A\ln \left ( x \right ){a}^{2}{b}^{3}+10\,B\ln \left ( x \right ){a}^{3}{b}^{2}-{\frac{5\,{a}^{4}bA}{2\,{x}^{2}}}-{\frac{{a}^{5}B}{2\,{x}^{2}}}-10\,{\frac{{a}^{3}{b}^{2}A}{x}}-5\,{\frac{{a}^{4}bB}{x}}-{\frac{A{a}^{5}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A)/x^4,x)

[Out]

1/3*b^5*B*x^3+1/2*A*x^2*b^5+5/2*B*x^2*a*b^4+5*A*x*a*b^4+10*B*x*a^2*b^3+10*A*ln(x
)*a^2*b^3+10*B*ln(x)*a^3*b^2-5/2*a^4/x^2*A*b-1/2*a^5/x^2*B-10*a^3*b^2/x*A-5*a^4*
b/x*B-1/3*a^5*A/x^3

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Maxima [A]  time = 1.34721, size = 158, normalized size = 1.46 \[ \frac{1}{3} \, B b^{5} x^{3} + \frac{1}{2} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{2} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} \log \left (x\right ) - \frac{2 \, A a^{5} + 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^4,x, algorithm="maxima")

[Out]

1/3*B*b^5*x^3 + 1/2*(5*B*a*b^4 + A*b^5)*x^2 + 5*(2*B*a^2*b^3 + A*a*b^4)*x + 10*(
B*a^3*b^2 + A*a^2*b^3)*log(x) - 1/6*(2*A*a^5 + 30*(B*a^4*b + 2*A*a^3*b^2)*x^2 +
3*(B*a^5 + 5*A*a^4*b)*x)/x^3

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Fricas [A]  time = 0.202035, size = 163, normalized size = 1.51 \[ \frac{2 \, B b^{5} x^{6} - 2 \, A a^{5} + 3 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 30 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 60 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} \log \left (x\right ) - 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^4,x, algorithm="fricas")

[Out]

1/6*(2*B*b^5*x^6 - 2*A*a^5 + 3*(5*B*a*b^4 + A*b^5)*x^5 + 30*(2*B*a^2*b^3 + A*a*b
^4)*x^4 + 60*(B*a^3*b^2 + A*a^2*b^3)*x^3*log(x) - 30*(B*a^4*b + 2*A*a^3*b^2)*x^2
- 3*(B*a^5 + 5*A*a^4*b)*x)/x^3

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Sympy [A]  time = 4.08448, size = 119, normalized size = 1.1 \[ \frac{B b^{5} x^{3}}{3} + 10 a^{2} b^{2} \left (A b + B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{5}}{2} + \frac{5 B a b^{4}}{2}\right ) + x \left (5 A a b^{4} + 10 B a^{2} b^{3}\right ) - \frac{2 A a^{5} + x^{2} \left (60 A a^{3} b^{2} + 30 B a^{4} b\right ) + x \left (15 A a^{4} b + 3 B a^{5}\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(B*x+A)/x**4,x)

[Out]

B*b**5*x**3/3 + 10*a**2*b**2*(A*b + B*a)*log(x) + x**2*(A*b**5/2 + 5*B*a*b**4/2)
+ x*(5*A*a*b**4 + 10*B*a**2*b**3) - (2*A*a**5 + x**2*(60*A*a**3*b**2 + 30*B*a**
4*b) + x*(15*A*a**4*b + 3*B*a**5))/(6*x**3)

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GIAC/XCAS [A]  time = 0.24754, size = 159, normalized size = 1.47 \[ \frac{1}{3} \, B b^{5} x^{3} + \frac{5}{2} \, B a b^{4} x^{2} + \frac{1}{2} \, A b^{5} x^{2} + 10 \, B a^{2} b^{3} x + 5 \, A a b^{4} x + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, A a^{5} + 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^4,x, algorithm="giac")

[Out]

1/3*B*b^5*x^3 + 5/2*B*a*b^4*x^2 + 1/2*A*b^5*x^2 + 10*B*a^2*b^3*x + 5*A*a*b^4*x +
10*(B*a^3*b^2 + A*a^2*b^3)*ln(abs(x)) - 1/6*(2*A*a^5 + 30*(B*a^4*b + 2*A*a^3*b^
2)*x^2 + 3*(B*a^5 + 5*A*a^4*b)*x)/x^3