\(\int \frac {(a+b \sqrt {x})^5}{x^2} \, dx\) [2147]

Optimal result

Integrand size = 15, antiderivative size = 62 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=-\frac {a^5}{x}-\frac {10 a^4 b}{\sqrt {x}}+20 a^2 b^3 \sqrt {x}+5 a b^4 x+\frac {2}{3} b^5 x^{3/2}+10 a^3 b^2 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 62, 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} \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=-\frac {a^5}{x}-\frac {10 a^4 b}{\sqrt {x}}+10 a^3 b^2 \log (x)+20 a^2 b^3 \sqrt {x}+5 a b^4 x+\frac {2}{3} b^5 x^{3/2} \]

[In]

Int[(a + b*Sqrt[x])^5/x^2,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=-\frac {a^5}{x}-\frac {10 a^4 b}{\sqrt {x}}+20 a^2 b^3 \sqrt {x}+5 a b^4 x+\frac {2}{3} b^5 x^{3/2}+10 a^3 b^2 \log (x) \]

[In]

Integrate[(a + b*Sqrt[x])^5/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 3.59 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(-\frac {a^{5}}{x}+5 b^{4} x a +\frac {2 b^{5} x^{\frac {3}{2}}}{3}+10 a^{3} b^{2} \ln \left (x \right )-\frac {10 a^{4} b}{\sqrt {x}}+20 a^{2} b^{3} \sqrt {x}\)	\(55\)
default	\(-\frac {a^{5}}{x}+5 b^{4} x a +\frac {2 b^{5} x^{\frac {3}{2}}}{3}+10 a^{3} b^{2} \ln \left (x \right )-\frac {10 a^{4} b}{\sqrt {x}}+20 a^{2} b^{3} \sqrt {x}\)	\(55\)
trager	\(\frac {\left (-1+x \right ) \left (5 b^{4} x +a^{4}\right ) a}{x}-\frac {2 \left (-b^{4} x^{2}-30 a^{2} b^{2} x +15 a^{4}\right ) b}{3 \sqrt {x}}+10 a^{3} b^{2} \ln \left (x \right )\)	\(59\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=\frac {15 \, a b^{4} x^{2} + 60 \, a^{3} b^{2} x \log \left (\sqrt {x}\right ) - 3 \, a^{5} + 2 \, {\left (b^{5} x^{2} + 30 \, a^{2} b^{3} x - 15 \, a^{4} b\right )} \sqrt {x}}{3 \, x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=- \frac {a^{5}}{x} - \frac {10 a^{4} b}{\sqrt {x}} + 10 a^{3} b^{2} \log {\left (x \right )} + 20 a^{2} b^{3} \sqrt {x} + 5 a b^{4} x + \frac {2 b^{5} x^{\frac {3}{2}}}{3} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=\frac {2}{3} \, b^{5} x^{\frac {3}{2}} + 5 \, a b^{4} x + 10 \, a^{3} b^{2} \log \left (x\right ) + 20 \, a^{2} b^{3} \sqrt {x} - \frac {10 \, a^{4} b \sqrt {x} + a^{5}}{x} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=\frac {2}{3} \, b^{5} x^{\frac {3}{2}} + 5 \, a b^{4} x + 10 \, a^{3} b^{2} \log \left ({\left | x \right |}\right ) + 20 \, a^{2} b^{3} \sqrt {x} - \frac {10 \, a^{4} b \sqrt {x} + a^{5}}{x} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=\frac {2\,b^5\,x^{3/2}}{3}-\frac {a^5+10\,a^4\,b\,\sqrt {x}}{x}+20\,a^3\,b^2\,\ln \left (\sqrt {x}\right )+20\,a^2\,b^3\,\sqrt {x}+5\,a\,b^4\,x \]

[In]

int((a + b*x^(1/2))^5/x^2,x)

[Out]

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