3.87 \(\int \frac{(a+b x)^5}{x^4} \, dx\)

Optimal. Leaf size=60 \[ -\frac{10 a^3 b^2}{x}+10 a^2 b^3 \log (x)-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{3 x^3}+5 a b^4 x+\frac{b^5 x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.0216694, antiderivative size = 60, 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} \[ -\frac{10 a^3 b^2}{x}+10 a^2 b^3 \log (x)-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{3 x^3}+5 a b^4 x+\frac{b^5 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^4,x]

[Out]

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

Mathematica [A]  time = 0.0058558, size = 60, normalized size = 1. \[ -\frac{10 a^3 b^2}{x}+10 a^2 b^3 \log (x)-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{3 x^3}+5 a b^4 x+\frac{b^5 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^4,x]

[Out]

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

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Maple [A]  time = 0.005, size = 55, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{3\,{x}^{3}}}-{\frac{5\,{a}^{4}b}{2\,{x}^{2}}}-10\,{\frac{{a}^{3}{b}^{2}}{x}}+5\,a{b}^{4}x+{\frac{{b}^{5}{x}^{2}}{2}}+10\,{a}^{2}{b}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^4,x)

[Out]

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

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Maxima [A]  time = 1.02227, size = 74, normalized size = 1.23 \begin{align*} \frac{1}{2} \, b^{5} x^{2} + 5 \, a b^{4} x + 10 \, a^{2} b^{3} \log \left (x\right ) - \frac{60 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x + 2 \, a^{5}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^4,x, algorithm="maxima")

[Out]

1/2*b^5*x^2 + 5*a*b^4*x + 10*a^2*b^3*log(x) - 1/6*(60*a^3*b^2*x^2 + 15*a^4*b*x + 2*a^5)/x^3

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Fricas [A]  time = 1.48722, size = 132, normalized size = 2.2 \begin{align*} \frac{3 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} \log \left (x\right ) - 60 \, a^{3} b^{2} x^{2} - 15 \, a^{4} b x - 2 \, a^{5}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^4,x, algorithm="fricas")

[Out]

1/6*(3*b^5*x^5 + 30*a*b^4*x^4 + 60*a^2*b^3*x^3*log(x) - 60*a^3*b^2*x^2 - 15*a^4*b*x - 2*a^5)/x^3

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Sympy [A]  time = 0.481205, size = 58, normalized size = 0.97 \begin{align*} 10 a^{2} b^{3} \log{\left (x \right )} + 5 a b^{4} x + \frac{b^{5} x^{2}}{2} - \frac{2 a^{5} + 15 a^{4} b x + 60 a^{3} b^{2} x^{2}}{6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**4,x)

[Out]

10*a**2*b**3*log(x) + 5*a*b**4*x + b**5*x**2/2 - (2*a**5 + 15*a**4*b*x + 60*a**3*b**2*x**2)/(6*x**3)

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Giac [A]  time = 1.2318, size = 76, normalized size = 1.27 \begin{align*} \frac{1}{2} \, b^{5} x^{2} + 5 \, a b^{4} x + 10 \, a^{2} b^{3} \log \left ({\left | x \right |}\right ) - \frac{60 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x + 2 \, a^{5}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^4,x, algorithm="giac")

[Out]

1/2*b^5*x^2 + 5*a*b^4*x + 10*a^2*b^3*log(abs(x)) - 1/6*(60*a^3*b^2*x^2 + 15*a^4*b*x + 2*a^5)/x^3