3.84 \(\int \frac{(a+b x)^5}{x} \, dx\)

Optimal. Leaf size=59 \[ 5 a^3 b^2 x^2+\frac{10}{3} a^2 b^3 x^3+5 a^4 b x+a^5 \log (x)+\frac{5}{4} a b^4 x^4+\frac{b^5 x^5}{5} \]

[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]

________________________________________________________________________________________

Rubi [A]  time = 0.0175937, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ 5 a^3 b^2 x^2+\frac{10}{3} a^2 b^3 x^3+5 a^4 b x+a^5 \log (x)+\frac{5}{4} a b^4 x^4+\frac{b^5 x^5}{5} \]

Antiderivative was successfully verified.

[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 43

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*} \int \frac{(a+b x)^5}{x} \, dx &=\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]  time = 0.0048367, size = 59, normalized size = 1. \[ 5 a^3 b^2 x^2+\frac{10}{3} a^2 b^3 x^3+5 a^4 b x+a^5 \log (x)+\frac{5}{4} a b^4 x^4+\frac{b^5 x^5}{5} \]

Antiderivative was successfully verified.

[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]  time = 0.002, size = 54, normalized size = 0.9 \begin{align*} 5\,{a}^{4}bx+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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Maxima [A]  time = 1.05842, size = 72, normalized size = 1.22 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 1.58551, size = 120, normalized size = 2.03 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0.360342, size = 60, normalized size = 1.02 \begin{align*} 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [A]  time = 1.19554, size = 73, normalized size = 1.24 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))