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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, 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} \begin {gather*} 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} \end {gather*}

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 = 0.00, size = 58, normalized size = 1.00 \begin {gather*} -\frac {a^5}{x}+5 a^4 b \log (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} \end {gather*}

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]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^5/x^2,x]

[Out]

IntegrateAlgebraic[(a + b*x)^5/x^2, x]

________________________________________________________________________________________

fricas [A]  time = 0.97, size = 59, normalized size = 1.02 \begin {gather*} \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 \relax (x) - 12 \, a^{5}}{12 \, x} \end {gather*}

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

________________________________________________________________________________________

giac [A]  time = 1.28, size = 55, normalized size = 0.95 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 55, normalized size = 0.95 \begin {gather*} \frac {b^{5} x^{4}}{4}+\frac {5 a \,b^{4} x^{3}}{3}+5 a^{2} b^{3} x^{2}+5 a^{4} b \ln \relax (x )+10 a^{3} b^{2} x -\frac {a^{5}}{x} \end {gather*}

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)

________________________________________________________________________________________

maxima [A]  time = 1.40, size = 54, normalized size = 0.93 \begin {gather*} \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 \relax (x) - \frac {a^{5}}{x} \end {gather*}

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

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 54, normalized size = 0.93 \begin {gather*} \frac {b^5\,x^4}{4}-\frac {a^5}{x}+10\,a^3\,b^2\,x+\frac {5\,a\,b^4\,x^3}{3}+5\,a^4\,b\,\ln \relax (x)+5\,a^2\,b^3\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/x^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 56, normalized size = 0.97 \begin {gather*} - \frac {a^{5}}{x} + 5 a^{4} b \log {\relax (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} \end {gather*}

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

________________________________________________________________________________________