\(\int \frac {(a+b x)^5}{x^5} \, dx\) [88]

Optimal result

Integrand size = 11, antiderivative size = 57 \[ \int \frac {(a+b x)^5}{x^5} \, dx=-\frac {a^5}{4 x^4}-\frac {5 a^4 b}{3 x^3}-\frac {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}+b^5 x+5 a b^4 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^5}{x^5} \, dx=-\frac {a^5}{4 x^4}-\frac {5 a^4 b}{3 x^3}-\frac {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}+5 a b^4 \log (x)+b^5 x \]

[In]

Int[(a + b*x)^5/x^5,x]

[Out]

-1/4*a^5/x^4 - (5*a^4*b)/(3*x^3) - (5*a^3*b^2)/x^2 - (10*a^2*b^3)/x + b^5*x + 5*a*b^4*Log[x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (b^5+\frac {a^5}{x^5}+\frac {5 a^4 b}{x^4}+\frac {10 a^3 b^2}{x^3}+\frac {10 a^2 b^3}{x^2}+\frac {5 a b^4}{x}\right ) \, dx \\ & = -\frac {a^5}{4 x^4}-\frac {5 a^4 b}{3 x^3}-\frac {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}+b^5 x+5 a b^4 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{x^5} \, dx=-\frac {a^5}{4 x^4}-\frac {5 a^4 b}{3 x^3}-\frac {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}+b^5 x+5 a b^4 \log (x) \]

[In]

Integrate[(a + b*x)^5/x^5,x]

[Out]

-1/4*a^5/x^4 - (5*a^4*b)/(3*x^3) - (5*a^3*b^2)/x^2 - (10*a^2*b^3)/x + b^5*x + 5*a*b^4*Log[x]

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {a^{5}}{4 x^{4}}-\frac {5 a^{4} b}{3 x^{3}}-\frac {5 a^{3} b^{2}}{x^{2}}-\frac {10 a^{2} b^{3}}{x}+b^{5} x +5 a \,b^{4} \ln \left (x \right )\)	\(54\)
risch	\(b^{5} x +\frac {-10 a^{2} b^{3} x^{3}-5 a^{3} b^{2} x^{2}-\frac {5}{3} a^{4} b x -\frac {1}{4} a^{5}}{x^{4}}+5 a \,b^{4} \ln \left (x \right )\)	\(54\)
norman	\(\frac {b^{5} x^{5}-\frac {1}{4} a^{5}-10 a^{2} b^{3} x^{3}-5 a^{3} b^{2} x^{2}-\frac {5}{3} a^{4} b x}{x^{4}}+5 a \,b^{4} \ln \left (x \right )\)	\(56\)
parallelrisch	\(\frac {60 a \,b^{4} \ln \left (x \right ) x^{4}+12 b^{5} x^{5}-120 a^{2} b^{3} x^{3}-60 a^{3} b^{2} x^{2}-20 a^{4} b x -3 a^{5}}{12 x^{4}}\)	\(60\)

[In]

int((b*x+a)^5/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^5}{x^5} \, dx=\frac {12 \, b^{5} x^{5} + 60 \, a b^{4} x^{4} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{3} - 60 \, a^{3} b^{2} x^{2} - 20 \, a^{4} b x - 3 \, a^{5}}{12 \, x^{4}} \]

[In]

integrate((b*x+a)^5/x^5,x, algorithm="fricas")

[Out]

1/12*(12*b^5*x^5 + 60*a*b^4*x^4*log(x) - 120*a^2*b^3*x^3 - 60*a^3*b^2*x^2 - 20*a^4*b*x - 3*a^5)/x^4

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5}{x^5} \, dx=5 a b^{4} \log {\left (x \right )} + b^{5} x + \frac {- 3 a^{5} - 20 a^{4} b x - 60 a^{3} b^{2} x^{2} - 120 a^{2} b^{3} x^{3}}{12 x^{4}} \]

[In]

integrate((b*x+a)**5/x**5,x)

[Out]

5*a*b**4*log(x) + b**5*x + (-3*a**5 - 20*a**4*b*x - 60*a**3*b**2*x**2 - 120*a**2*b**3*x**3)/(12*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^5}{x^5} \, dx=b^{5} x + 5 \, a b^{4} \log \left (x\right ) - \frac {120 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} + 20 \, a^{4} b x + 3 \, a^{5}}{12 \, x^{4}} \]

[In]

integrate((b*x+a)^5/x^5,x, algorithm="maxima")

[Out]

b^5*x + 5*a*b^4*log(x) - 1/12*(120*a^2*b^3*x^3 + 60*a^3*b^2*x^2 + 20*a^4*b*x + 3*a^5)/x^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^5}{x^5} \, dx=b^{5} x + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) - \frac {120 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} + 20 \, a^{4} b x + 3 \, a^{5}}{12 \, x^{4}} \]

[In]

integrate((b*x+a)^5/x^5,x, algorithm="giac")

[Out]

b^5*x + 5*a*b^4*log(abs(x)) - 1/12*(120*a^2*b^3*x^3 + 60*a^3*b^2*x^2 + 20*a^4*b*x + 3*a^5)/x^4

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^5}{x^5} \, dx=b^5\,x-\frac {\frac {a^5}{4}+\frac {5\,a^4\,b\,x}{3}+5\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3}{x^4}+5\,a\,b^4\,\ln \left (x\right ) \]

[In]

int((a + b*x)^5/x^5,x)

[Out]

b^5*x - (a^5/4 + 5*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + (5*a^4*b*x)/3)/x^4 + 5*a*b^4*log(x)