\(\int \frac {(a+b x)^5}{x^6} \, dx\) [89]

Optimal result

Integrand size = 11, antiderivative size = 61 \[ \int \frac {(a+b x)^5}{x^6} \, dx=-\frac {a^5}{5 x^5}-\frac {5 a^4 b}{4 x^4}-\frac {10 a^3 b^2}{3 x^3}-\frac {5 a^2 b^3}{x^2}-\frac {5 a b^4}{x}+b^5 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, 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} \[ \int \frac {(a+b x)^5}{x^6} \, dx=-\frac {a^5}{5 x^5}-\frac {5 a^4 b}{4 x^4}-\frac {10 a^3 b^2}{3 x^3}-\frac {5 a^2 b^3}{x^2}-\frac {5 a b^4}{x}+b^5 \log (x) \]

[In]

Int[(a + b*x)^5/x^6,x]

[Out]

-1/5*a^5/x^5 - (5*a^4*b)/(4*x^4) - (10*a^3*b^2)/(3*x^3) - (5*a^2*b^3)/x^2 - (5*a*b^4)/x + b^5*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 (\frac {a^5}{x^6}+\frac {5 a^4 b}{x^5}+\frac {10 a^3 b^2}{x^4}+\frac {10 a^2 b^3}{x^3}+\frac {5 a b^4}{x^2}+\frac {b^5}{x}\right ) \, dx \\ & = -\frac {a^5}{5 x^5}-\frac {5 a^4 b}{4 x^4}-\frac {10 a^3 b^2}{3 x^3}-\frac {5 a^2 b^3}{x^2}-\frac {5 a b^4}{x}+b^5 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{x^6} \, dx=-\frac {a^5}{5 x^5}-\frac {5 a^4 b}{4 x^4}-\frac {10 a^3 b^2}{3 x^3}-\frac {5 a^2 b^3}{x^2}-\frac {5 a b^4}{x}+b^5 \log (x) \]

[In]

Integrate[(a + b*x)^5/x^6,x]

[Out]

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {a^{5}}{5 x^{5}}-\frac {5 a^{4} b}{4 x^{4}}-\frac {10 a^{3} b^{2}}{3 x^{3}}-\frac {5 a^{2} b^{3}}{x^{2}}-\frac {5 a \,b^{4}}{x}+b^{5} \ln \left (x \right )\)	\(56\)
norman	\(\frac {-\frac {1}{5} a^{5}-5 a \,b^{4} x^{4}-5 a^{2} b^{3} x^{3}-\frac {10}{3} a^{3} b^{2} x^{2}-\frac {5}{4} a^{4} b x}{x^{5}}+b^{5} \ln \left (x \right )\)	\(56\)
risch	\(\frac {-\frac {1}{5} a^{5}-5 a \,b^{4} x^{4}-5 a^{2} b^{3} x^{3}-\frac {10}{3} a^{3} b^{2} x^{2}-\frac {5}{4} a^{4} b x}{x^{5}}+b^{5} \ln \left (x \right )\)	\(56\)
parallelrisch	\(\frac {60 b^{5} \ln \left (x \right ) x^{5}-300 a \,b^{4} x^{4}-300 a^{2} b^{3} x^{3}-200 a^{3} b^{2} x^{2}-75 a^{4} b x -12 a^{5}}{60 x^{5}}\)	\(60\)

[In]

int((b*x+a)^5/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^5}{x^6} \, dx=\frac {60 \, b^{5} x^{5} \log \left (x\right ) - 300 \, a b^{4} x^{4} - 300 \, a^{2} b^{3} x^{3} - 200 \, a^{3} b^{2} x^{2} - 75 \, a^{4} b x - 12 \, a^{5}}{60 \, x^{5}} \]

[In]

integrate((b*x+a)^5/x^6,x, algorithm="fricas")

[Out]

1/60*(60*b^5*x^5*log(x) - 300*a*b^4*x^4 - 300*a^2*b^3*x^3 - 200*a^3*b^2*x^2 - 75*a^4*b*x - 12*a^5)/x^5

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^5}{x^6} \, dx=b^{5} \log {\left (x \right )} + \frac {- 12 a^{5} - 75 a^{4} b x - 200 a^{3} b^{2} x^{2} - 300 a^{2} b^{3} x^{3} - 300 a b^{4} x^{4}}{60 x^{5}} \]

[In]

integrate((b*x+a)**5/x**6,x)

[Out]

b**5*log(x) + (-12*a**5 - 75*a**4*b*x - 200*a**3*b**2*x**2 - 300*a**2*b**3*x**3 - 300*a*b**4*x**4)/(60*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^5}{x^6} \, dx=b^{5} \log \left (x\right ) - \frac {300 \, a b^{4} x^{4} + 300 \, a^{2} b^{3} x^{3} + 200 \, a^{3} b^{2} x^{2} + 75 \, a^{4} b x + 12 \, a^{5}}{60 \, x^{5}} \]

[In]

integrate((b*x+a)^5/x^6,x, algorithm="maxima")

[Out]

b^5*log(x) - 1/60*(300*a*b^4*x^4 + 300*a^2*b^3*x^3 + 200*a^3*b^2*x^2 + 75*a^4*b*x + 12*a^5)/x^5

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^5}{x^6} \, dx=b^{5} \log \left ({\left | x \right |}\right ) - \frac {300 \, a b^{4} x^{4} + 300 \, a^{2} b^{3} x^{3} + 200 \, a^{3} b^{2} x^{2} + 75 \, a^{4} b x + 12 \, a^{5}}{60 \, x^{5}} \]

[In]

integrate((b*x+a)^5/x^6,x, algorithm="giac")

[Out]

b^5*log(abs(x)) - 1/60*(300*a*b^4*x^4 + 300*a^2*b^3*x^3 + 200*a^3*b^2*x^2 + 75*a^4*b*x + 12*a^5)/x^5

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^5}{x^6} \, dx=b^5\,\ln \left (x\right )-\frac {\frac {a^5}{5}+\frac {5\,a^4\,b\,x}{4}+\frac {10\,a^3\,b^2\,x^2}{3}+5\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4}{x^5} \]

[In]

int((a + b*x)^5/x^6,x)

[Out]

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