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

Optimal. Leaf size=80 \[ a^5 A \log (x)+5 a^4 A b x+5 a^3 A b^2 x^2+\frac{10}{3} a^2 A b^3 x^3+\frac{5}{4} a A b^4 x^4+\frac{B (a+b x)^6}{6 b}+\frac{1}{5} A b^5 x^5 \]

[Out]

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

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Rubi [A]  time = 0.0791376, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ a^5 A \log (x)+5 a^4 A b x+5 a^3 A b^2 x^2+\frac{10}{3} a^2 A b^3 x^3+\frac{5}{4} a A b^4 x^4+\frac{B (a+b x)^6}{6 b}+\frac{1}{5} A b^5 x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**5*log(x) + 5*A*a**4*b*x + 10*A*a**3*b**2*Integral(x, x) + 10*A*a**2*b**3*x*
*3/3 + 5*A*a*b**4*x**4/4 + A*b**5*x**5/5 + B*(a + b*x)**6/(6*b)

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Mathematica [A]  time = 0.0532292, size = 108, normalized size = 1.35 \[ a^5 A \log (x)+a^4 x (a B+5 A b)+\frac{5}{2} a^3 b x^2 (a B+2 A b)+\frac{10}{3} a^2 b^2 x^3 (a B+A b)+\frac{1}{5} b^4 x^5 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{6} b^5 B x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 118, normalized size = 1.5 \[{\frac{B{b}^{5}{x}^{6}}{6}}+{\frac{A{b}^{5}{x}^{5}}{5}}+B{x}^{5}a{b}^{4}+{\frac{5\,aA{b}^{4}{x}^{4}}{4}}+{\frac{5\,B{x}^{4}{a}^{2}{b}^{3}}{2}}+{\frac{10\,{a}^{2}A{b}^{3}{x}^{3}}{3}}+{\frac{10\,B{x}^{3}{a}^{3}{b}^{2}}{3}}+5\,{a}^{3}A{b}^{2}{x}^{2}+{\frac{5\,B{x}^{2}{a}^{4}b}{2}}+5\,{a}^{4}Abx+{a}^{5}Bx+{a}^{5}A\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A)/x,x)

[Out]

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

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Maxima [A]  time = 1.37386, size = 154, normalized size = 1.92 \[ \frac{1}{6} \, B b^{5} x^{6} + A a^{5} \log \left (x\right ) + \frac{1}{5} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + \frac{10}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} +{\left (B a^{5} + 5 \, A a^{4} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.202306, size = 154, normalized size = 1.92 \[ \frac{1}{6} \, B b^{5} x^{6} + A a^{5} \log \left (x\right ) + \frac{1}{5} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + \frac{10}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} +{\left (B a^{5} + 5 \, A a^{4} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.66825, size = 126, normalized size = 1.58 \[ A a^{5} \log{\left (x \right )} + \frac{B b^{5} x^{6}}{6} + x^{5} \left (\frac{A b^{5}}{5} + B a b^{4}\right ) + x^{4} \left (\frac{5 A a b^{4}}{4} + \frac{5 B a^{2} b^{3}}{2}\right ) + x^{3} \left (\frac{10 A a^{2} b^{3}}{3} + \frac{10 B a^{3} b^{2}}{3}\right ) + x^{2} \left (5 A a^{3} b^{2} + \frac{5 B a^{4} b}{2}\right ) + x \left (5 A a^{4} b + B a^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270844, size = 159, normalized size = 1.99 \[ \frac{1}{6} \, B b^{5} x^{6} + B a b^{4} x^{5} + \frac{1}{5} \, A b^{5} x^{5} + \frac{5}{2} \, B a^{2} b^{3} x^{4} + \frac{5}{4} \, A a b^{4} x^{4} + \frac{10}{3} \, B a^{3} b^{2} x^{3} + \frac{10}{3} \, A a^{2} b^{3} x^{3} + \frac{5}{2} \, B a^{4} b x^{2} + 5 \, A a^{3} b^{2} x^{2} + B a^{5} x + 5 \, A a^{4} b x + A a^{5}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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