\(\int \frac {(a+b \sqrt {x})^5}{x^3} \, dx\) [2148]

Optimal result

Integrand size = 15, antiderivative size = 66 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^3} \, dx=-\frac {a^5}{2 x^2}-\frac {10 a^4 b}{3 x^{3/2}}-\frac {10 a^3 b^2}{x}-\frac {20 a^2 b^3}{\sqrt {x}}+2 b^5 \sqrt {x}+5 a b^4 \log (x) \]

[Out]

-1/2*a^5/x^2-10/3*a^4*b/x^(3/2)-10*a^3*b^2/x+5*a*b^4*ln(x)-20*a^2*b^3/x^(1/2)+2*b^5*x^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^3} \, dx=-\frac {a^5}{2 x^2}-\frac {10 a^4 b}{3 x^{3/2}}-\frac {10 a^3 b^2}{x}-\frac {20 a^2 b^3}{\sqrt {x}}+5 a b^4 \log (x)+2 b^5 \sqrt {x} \]

[In]

Int[(a + b*Sqrt[x])^5/x^3,x]

[Out]

-1/2*a^5/x^2 - (10*a^4*b)/(3*x^(3/2)) - (10*a^3*b^2)/x - (20*a^2*b^3)/Sqrt[x] + 2*b^5*Sqrt[x] + 5*a*b^4*Log[x]

Rule 45

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])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {(a+b x)^5}{x^5} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (b^5+\frac {a^5}{x^5}+\frac {5 a^4 b}{x^4}+\frac {10 a^3 b^2}{x^3}+\frac {10 a^2 b^3}{x^2}+\frac {5 a b^4}{x}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^5}{2 x^2}-\frac {10 a^4 b}{3 x^{3/2}}-\frac {10 a^3 b^2}{x}-\frac {20 a^2 b^3}{\sqrt {x}}+2 b^5 \sqrt {x}+5 a b^4 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^3} \, dx=\frac {-3 a^5-20 a^4 b \sqrt {x}-60 a^3 b^2 x-120 a^2 b^3 x^{3/2}+12 b^5 x^{5/2}}{6 x^2}+10 a b^4 \log \left (\sqrt {x}\right ) \]

[In]

Integrate[(a + b*Sqrt[x])^5/x^3,x]

[Out]

(-3*a^5 - 20*a^4*b*Sqrt[x] - 60*a^3*b^2*x - 120*a^2*b^3*x^(3/2) + 12*b^5*x^(5/2))/(6*x^2) + 10*a*b^4*Log[Sqrt[
x]]

Maple [A] (verified)

Time = 3.61 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(-\frac {a^{5}}{2 x^{2}}-\frac {10 a^{4} b}{3 x^{\frac {3}{2}}}-\frac {10 a^{3} b^{2}}{x}+5 a \,b^{4} \ln \left (x \right )-\frac {20 a^{2} b^{3}}{\sqrt {x}}+2 b^{5} \sqrt {x}\)	\(57\)
default	\(-\frac {a^{5}}{2 x^{2}}-\frac {10 a^{4} b}{3 x^{\frac {3}{2}}}-\frac {10 a^{3} b^{2}}{x}+5 a \,b^{4} \ln \left (x \right )-\frac {20 a^{2} b^{3}}{\sqrt {x}}+2 b^{5} \sqrt {x}\)	\(57\)
trager	\(\frac {\left (-1+x \right ) \left (a^{2} x +20 b^{2} x +a^{2}\right ) a^{3}}{2 x^{2}}-\frac {2 \left (-3 b^{4} x^{2}+30 a^{2} b^{2} x +5 a^{4}\right ) b}{3 x^{\frac {3}{2}}}+5 a \,b^{4} \ln \left (x \right )\)	\(65\)

[In]

int((a+b*x^(1/2))^5/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*a^5/x^2-10/3*a^4*b/x^(3/2)-10*a^3*b^2/x+5*a*b^4*ln(x)-20*a^2*b^3/x^(1/2)+2*b^5*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^3} \, dx=\frac {60 \, a b^{4} x^{2} \log \left (\sqrt {x}\right ) - 60 \, a^{3} b^{2} x - 3 \, a^{5} + 4 \, {\left (3 \, b^{5} x^{2} - 30 \, a^{2} b^{3} x - 5 \, a^{4} b\right )} \sqrt {x}}{6 \, x^{2}} \]

[In]

integrate((a+b*x^(1/2))^5/x^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^3} \, dx=- \frac {a^{5}}{2 x^{2}} - \frac {10 a^{4} b}{3 x^{\frac {3}{2}}} - \frac {10 a^{3} b^{2}}{x} - \frac {20 a^{2} b^{3}}{\sqrt {x}} + 5 a b^{4} \log {\left (x \right )} + 2 b^{5} \sqrt {x} \]

[In]

integrate((a+b*x**(1/2))**5/x**3,x)

[Out]

-a**5/(2*x**2) - 10*a**4*b/(3*x**(3/2)) - 10*a**3*b**2/x - 20*a**2*b**3/sqrt(x) + 5*a*b**4*log(x) + 2*b**5*sqr
t(x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^3} \, dx=5 \, a b^{4} \log \left (x\right ) + 2 \, b^{5} \sqrt {x} - \frac {120 \, a^{2} b^{3} x^{\frac {3}{2}} + 60 \, a^{3} b^{2} x + 20 \, a^{4} b \sqrt {x} + 3 \, a^{5}}{6 \, x^{2}} \]

[In]

integrate((a+b*x^(1/2))^5/x^3,x, algorithm="maxima")

[Out]

5*a*b^4*log(x) + 2*b^5*sqrt(x) - 1/6*(120*a^2*b^3*x^(3/2) + 60*a^3*b^2*x + 20*a^4*b*sqrt(x) + 3*a^5)/x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^3} \, dx=5 \, a b^{4} \log \left ({\left | x \right |}\right ) + 2 \, b^{5} \sqrt {x} - \frac {120 \, a^{2} b^{3} x^{\frac {3}{2}} + 60 \, a^{3} b^{2} x + 20 \, a^{4} b \sqrt {x} + 3 \, a^{5}}{6 \, x^{2}} \]

[In]

integrate((a+b*x^(1/2))^5/x^3,x, algorithm="giac")

[Out]

5*a*b^4*log(abs(x)) + 2*b^5*sqrt(x) - 1/6*(120*a^2*b^3*x^(3/2) + 60*a^3*b^2*x + 20*a^4*b*sqrt(x) + 3*a^5)/x^2

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^3} \, dx=2\,b^5\,\sqrt {x}-\frac {\frac {a^5}{2}+10\,a^3\,b^2\,x+\frac {10\,a^4\,b\,\sqrt {x}}{3}+20\,a^2\,b^3\,x^{3/2}}{x^2}+10\,a\,b^4\,\ln \left (\sqrt {x}\right ) \]

[In]

int((a + b*x^(1/2))^5/x^3,x)

[Out]

2*b^5*x^(1/2) - (a^5/2 + 10*a^3*b^2*x + (10*a^4*b*x^(1/2))/3 + 20*a^2*b^3*x^(3/2))/x^2 + 10*a*b^4*log(x^(1/2))