\(\int \frac {(a+b x^n)^5}{x} \, dx\) [2555]

Optimal result

Integrand size = 13, antiderivative size = 84 \[ \int \frac {\left (a+b x^n\right )^5}{x} \, dx=\frac {5 a^4 b x^n}{n}+\frac {5 a^3 b^2 x^{2 n}}{n}+\frac {10 a^2 b^3 x^{3 n}}{3 n}+\frac {5 a b^4 x^{4 n}}{4 n}+\frac {b^5 x^{5 n}}{5 n}+a^5 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {\left (a+b x^n\right )^5}{x} \, dx=a^5 \log (x)+\frac {5 a^4 b x^n}{n}+\frac {5 a^3 b^2 x^{2 n}}{n}+\frac {10 a^2 b^3 x^{3 n}}{3 n}+\frac {5 a b^4 x^{4 n}}{4 n}+\frac {b^5 x^{5 n}}{5 n} \]

[In]

Int[(a + b*x^n)^5/x,x]

[Out]

(5*a^4*b*x^n)/n + (5*a^3*b^2*x^(2*n))/n + (10*a^2*b^3*x^(3*n))/(3*n) + (5*a*b^4*x^(4*n))/(4*n) + (b^5*x^(5*n))
/(5*n) + a^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])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^n\right )^5}{x} \, dx=\frac {b x^n \left (300 a^4+300 a^3 b x^n+200 a^2 b^2 x^{2 n}+75 a b^3 x^{3 n}+12 b^4 x^{4 n}\right )}{60 n}+\frac {a^5 \log \left (x^n\right )}{n} \]

[In]

Integrate[(a + b*x^n)^5/x,x]

[Out]

(b*x^n*(300*a^4 + 300*a^3*b*x^n + 200*a^2*b^2*x^(2*n) + 75*a*b^3*x^(3*n) + 12*b^4*x^(4*n)))/(60*n) + (a^5*Log[
x^n])/n

Maple [A] (verified)

Time = 3.96 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\frac {b^{5} x^{5 n}}{5}+\frac {5 a \,b^{4} x^{4 n}}{4}+\frac {10 a^{2} b^{3} x^{3 n}}{3}+5 a^{3} b^{2} x^{2 n}+5 a^{4} b \,x^{n}+a^{5} \ln \left (x^{n}\right )}{n}\)	\(70\)
default	\(\frac {\frac {b^{5} x^{5 n}}{5}+\frac {5 a \,b^{4} x^{4 n}}{4}+\frac {10 a^{2} b^{3} x^{3 n}}{3}+5 a^{3} b^{2} x^{2 n}+5 a^{4} b \,x^{n}+a^{5} \ln \left (x^{n}\right )}{n}\)	\(70\)
parallelrisch	\(\frac {12 b^{5} x^{5 n}+75 a \,b^{4} x^{4 n}+200 a^{2} b^{3} x^{3 n}+60 a^{5} \ln \left (x \right ) n +300 a^{3} b^{2} x^{2 n}+300 a^{4} b \,x^{n}}{60 n}\)	\(71\)
risch	\(\frac {5 a^{4} b \,x^{n}}{n}+\frac {5 a^{3} b^{2} x^{2 n}}{n}+\frac {10 a^{2} b^{3} x^{3 n}}{3 n}+\frac {5 a \,b^{4} x^{4 n}}{4 n}+\frac {b^{5} x^{5 n}}{5 n}+a^{5} \ln \left (x \right )\)	\(79\)
norman	\(a^{5} \ln \left (x \right )+\frac {b^{5} {\mathrm e}^{5 n \ln \left (x \right )}}{5 n}+\frac {5 a \,b^{4} {\mathrm e}^{4 n \ln \left (x \right )}}{4 n}+\frac {10 a^{2} b^{3} {\mathrm e}^{3 n \ln \left (x \right )}}{3 n}+\frac {5 a^{3} b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{n}+\frac {5 a^{4} b \,{\mathrm e}^{n \ln \left (x \right )}}{n}\)	\(89\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^n\right )^5}{x} \, dx=\frac {60 \, a^{5} n \log \left (x\right ) + 12 \, b^{5} x^{5 \, n} + 75 \, a b^{4} x^{4 \, n} + 200 \, a^{2} b^{3} x^{3 \, n} + 300 \, a^{3} b^{2} x^{2 \, n} + 300 \, a^{4} b x^{n}}{60 \, n} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^n\right )^5}{x} \, dx=\begin {cases} a^{5} \log {\left (x \right )} + \frac {5 a^{4} b x^{n}}{n} + \frac {5 a^{3} b^{2} x^{2 n}}{n} + \frac {10 a^{2} b^{3} x^{3 n}}{3 n} + \frac {5 a b^{4} x^{4 n}}{4 n} + \frac {b^{5} x^{5 n}}{5 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{5} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**5*log(x) + 5*a**4*b*x**n/n + 5*a**3*b**2*x**(2*n)/n + 10*a**2*b**3*x**(3*n)/(3*n) + 5*a*b**4*x**
(4*n)/(4*n) + b**5*x**(5*n)/(5*n), Ne(n, 0)), ((a + b)**5*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^n\right )^5}{x} \, dx=\frac {a^{5} \log \left (x^{n}\right )}{n} + \frac {12 \, b^{5} x^{5 \, n} + 75 \, a b^{4} x^{4 \, n} + 200 \, a^{2} b^{3} x^{3 \, n} + 300 \, a^{3} b^{2} x^{2 \, n} + 300 \, a^{4} b x^{n}}{60 \, n} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b x^n\right )^5}{x} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{5}}{x} \,d x } \]

[In]

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

[Out]

integrate((b*x^n + a)^5/x, x)

Mupad [B] (verification not implemented)

Time = 5.68 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^n\right )^5}{x} \, dx=a^5\,\ln \left (x\right )+\frac {b^5\,x^{5\,n}}{5\,n}+\frac {5\,a^3\,b^2\,x^{2\,n}}{n}+\frac {10\,a^2\,b^3\,x^{3\,n}}{3\,n}+\frac {5\,a^4\,b\,x^n}{n}+\frac {5\,a\,b^4\,x^{4\,n}}{4\,n} \]

[In]

int((a + b*x^n)^5/x,x)

[Out]

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