\(\int \frac {(a+b x)^5}{x} \, dx\) [84]

Optimal result

Integrand size = 11, antiderivative size = 59 \[ \int \frac {(a+b x)^5}{x} \, dx=5 a^4 b x+5 a^3 b^2 x^2+\frac {10}{3} a^2 b^3 x^3+\frac {5}{4} a b^4 x^4+\frac {b^5 x^5}{5}+a^5 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 59, 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} \[ \int \frac {(a+b x)^5}{x} \, dx=a^5 \log (x)+5 a^4 b x+5 a^3 b^2 x^2+\frac {10}{3} a^2 b^3 x^3+\frac {5}{4} a b^4 x^4+\frac {b^5 x^5}{5} \]

[In]

Int[(a + b*x)^5/x,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{x} \, dx=5 a^4 b x+5 a^3 b^2 x^2+\frac {10}{3} a^2 b^3 x^3+\frac {5}{4} a b^4 x^4+\frac {b^5 x^5}{5}+a^5 \log (x) \]

[In]

Integrate[(a + b*x)^5/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^5}{x} \, dx=\frac {1}{5} \, b^{5} x^{5} + \frac {5}{4} \, a b^{4} x^{4} + \frac {10}{3} \, a^{2} b^{3} x^{3} + 5 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5} \log \left (x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5}{x} \, dx=a^{5} \log {\left (x \right )} + 5 a^{4} b x + 5 a^{3} b^{2} x^{2} + \frac {10 a^{2} b^{3} x^{3}}{3} + \frac {5 a b^{4} x^{4}}{4} + \frac {b^{5} x^{5}}{5} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^5}{x} \, dx=\frac {1}{5} \, b^{5} x^{5} + \frac {5}{4} \, a b^{4} x^{4} + \frac {10}{3} \, a^{2} b^{3} x^{3} + 5 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5} \log \left (x\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^5}{x} \, dx=\frac {1}{5} \, b^{5} x^{5} + \frac {5}{4} \, a b^{4} x^{4} + \frac {10}{3} \, a^{2} b^{3} x^{3} + 5 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5} \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^5}{x} \, dx=a^5\,\ln \left (x\right )+\frac {b^5\,x^5}{5}+\frac {5\,a\,b^4\,x^4}{4}+5\,a^3\,b^2\,x^2+\frac {10\,a^2\,b^3\,x^3}{3}+5\,a^4\,b\,x \]

[In]

int((a + b*x)^5/x,x)

[Out]

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