3.86 \(\int \frac{(a+b x)^5}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0216892, 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} \[ 10 a^2 b^3 x+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{x}-\frac{a^5}{2 x^2}+\frac{5}{2} a b^4 x^2+\frac{b^5 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0059594, size = 60, normalized size = 1. \[ 10 a^2 b^3 x+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{x}-\frac{a^5}{2 x^2}+\frac{5}{2} a b^4 x^2+\frac{b^5 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^3,x]

[Out]

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

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Maple [A]  time = 0.006, size = 55, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{2\,{x}^{2}}}-5\,{\frac{{a}^{4}b}{x}}+10\,{a}^{2}{b}^{3}x+{\frac{5\,a{b}^{4}{x}^{2}}{2}}+{\frac{{b}^{5}{x}^{3}}{3}}+10\,{a}^{3}{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^3,x)

[Out]

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

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Maxima [A]  time = 1.03899, size = 72, normalized size = 1.2 \begin{align*} \frac{1}{3} \, b^{5} x^{3} + \frac{5}{2} \, a b^{4} x^{2} + 10 \, a^{2} b^{3} x + 10 \, a^{3} b^{2} \log \left (x\right ) - \frac{10 \, a^{4} b x + a^{5}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47676, size = 132, normalized size = 2.2 \begin{align*} \frac{2 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} \log \left (x\right ) - 30 \, a^{4} b x - 3 \, a^{5}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.382662, size = 58, normalized size = 0.97 \begin{align*} 10 a^{3} b^{2} \log{\left (x \right )} + 10 a^{2} b^{3} x + \frac{5 a b^{4} x^{2}}{2} + \frac{b^{5} x^{3}}{3} - \frac{a^{5} + 10 a^{4} b x}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**3,x)

[Out]

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

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Giac [A]  time = 1.18271, size = 73, normalized size = 1.22 \begin{align*} \frac{1}{3} \, b^{5} x^{3} + \frac{5}{2} \, a b^{4} x^{2} + 10 \, a^{2} b^{3} x + 10 \, a^{3} b^{2} \log \left ({\left | x \right |}\right ) - \frac{10 \, a^{4} b x + a^{5}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^3,x, algorithm="giac")

[Out]

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