3.85 \(\int \frac{(a+b x)^5}{x^2} \, dx\)

Optimal. Leaf size=58 \[ 5 a^2 b^3 x^2+10 a^3 b^2 x+5 a^4 b \log (x)-\frac{a^5}{x}+\frac{5}{3} a b^4 x^3+\frac{b^5 x^4}{4} \]

[Out]

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

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Rubi [A]  time = 0.0207801, antiderivative size = 58, 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} \[ 5 a^2 b^3 x^2+10 a^3 b^2 x+5 a^4 b \log (x)-\frac{a^5}{x}+\frac{5}{3} a b^4 x^3+\frac{b^5 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0050512, size = 58, normalized size = 1. \[ 5 a^2 b^3 x^2+10 a^3 b^2 x+5 a^4 b \log (x)-\frac{a^5}{x}+\frac{5}{3} a b^4 x^3+\frac{b^5 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^2,x]

[Out]

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

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Maple [A]  time = 0.006, size = 55, normalized size = 1. \begin{align*} -{\frac{{a}^{5}}{x}}+10\,{a}^{3}{b}^{2}x+5\,{a}^{2}{b}^{3}{x}^{2}+{\frac{5\,a{b}^{4}{x}^{3}}{3}}+{\frac{{b}^{5}{x}^{4}}{4}}+5\,{a}^{4}b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^2,x)

[Out]

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

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Maxima [A]  time = 1.03527, size = 73, normalized size = 1.26 \begin{align*} \frac{1}{4} \, b^{5} x^{4} + \frac{5}{3} \, a b^{4} x^{3} + 5 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x + 5 \, a^{4} b \log \left (x\right ) - \frac{a^{5}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.56236, size = 134, normalized size = 2.31 \begin{align*} \frac{3 \, b^{5} x^{5} + 20 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 120 \, a^{3} b^{2} x^{2} + 60 \, a^{4} b x \log \left (x\right ) - 12 \, a^{5}}{12 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^2,x, algorithm="fricas")

[Out]

1/12*(3*b^5*x^5 + 20*a*b^4*x^4 + 60*a^2*b^3*x^3 + 120*a^3*b^2*x^2 + 60*a^4*b*x*log(x) - 12*a^5)/x

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Sympy [A]  time = 0.33742, size = 56, normalized size = 0.97 \begin{align*} - \frac{a^{5}}{x} + 5 a^{4} b \log{\left (x \right )} + 10 a^{3} b^{2} x + 5 a^{2} b^{3} x^{2} + \frac{5 a b^{4} x^{3}}{3} + \frac{b^{5} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**2,x)

[Out]

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

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Giac [A]  time = 1.14791, size = 74, normalized size = 1.28 \begin{align*} \frac{1}{4} \, b^{5} x^{4} + \frac{5}{3} \, a b^{4} x^{3} + 5 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x + 5 \, a^{4} b \log \left ({\left | x \right |}\right ) - \frac{a^{5}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^2,x, algorithm="giac")

[Out]

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