\(\int \frac {(a+b \sqrt {x})^3}{x^6} \, dx\) [2140]

Optimal result

Integrand size = 15, antiderivative size = 47 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {a^3}{5 x^5}-\frac {2 a^2 b}{3 x^{9/2}}-\frac {3 a b^2}{4 x^4}-\frac {2 b^3}{7 x^{7/2}} \]

[Out]

-1/5*a^3/x^5-2/3*a^2*b/x^(9/2)-3/4*a*b^2/x^4-2/7*b^3/x^(7/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, 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} \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {a^3}{5 x^5}-\frac {2 a^2 b}{3 x^{9/2}}-\frac {3 a b^2}{4 x^4}-\frac {2 b^3}{7 x^{7/2}} \]

[In]

Int[(a + b*Sqrt[x])^3/x^6,x]

[Out]

-1/5*a^3/x^5 - (2*a^2*b)/(3*x^(9/2)) - (3*a*b^2)/(4*x^4) - (2*b^3)/(7*x^(7/2))

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)^3}{x^{11}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {a^3}{x^{11}}+\frac {3 a^2 b}{x^{10}}+\frac {3 a b^2}{x^9}+\frac {b^3}{x^8}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^3}{5 x^5}-\frac {2 a^2 b}{3 x^{9/2}}-\frac {3 a b^2}{4 x^4}-\frac {2 b^3}{7 x^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=\frac {-84 a^3-280 a^2 b \sqrt {x}-315 a b^2 x-120 b^3 x^{3/2}}{420 x^5} \]

[In]

Integrate[(a + b*Sqrt[x])^3/x^6,x]

[Out]

(-84*a^3 - 280*a^2*b*Sqrt[x] - 315*a*b^2*x - 120*b^3*x^(3/2))/(420*x^5)

Maple [A] (verified)

Time = 3.74 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(-\frac {a^{3}}{5 x^{5}}-\frac {2 a^{2} b}{3 x^{\frac {9}{2}}}-\frac {3 a \,b^{2}}{4 x^{4}}-\frac {2 b^{3}}{7 x^{\frac {7}{2}}}\)	\(36\)
default	\(-\frac {a^{3}}{5 x^{5}}-\frac {2 a^{2} b}{3 x^{\frac {9}{2}}}-\frac {3 a \,b^{2}}{4 x^{4}}-\frac {2 b^{3}}{7 x^{\frac {7}{2}}}\)	\(36\)
trager	\(\frac {\left (-1+x \right ) \left (4 a^{2} x^{4}+15 b^{2} x^{4}+4 a^{2} x^{3}+15 b^{2} x^{3}+4 a^{2} x^{2}+15 b^{2} x^{2}+4 a^{2} x +15 b^{2} x +4 a^{2}\right ) a}{20 x^{5}}-\frac {2 \left (3 b^{2} x +7 a^{2}\right ) b}{21 x^{\frac {9}{2}}}\)	\(95\)

[In]

int((a+b*x^(1/2))^3/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*a^3/x^5-2/3*a^2*b/x^(9/2)-3/4*a*b^2/x^4-2/7*b^3/x^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {315 \, a b^{2} x + 84 \, a^{3} + 40 \, {\left (3 \, b^{3} x + 7 \, a^{2} b\right )} \sqrt {x}}{420 \, x^{5}} \]

[In]

integrate((a+b*x^(1/2))^3/x^6,x, algorithm="fricas")

[Out]

-1/420*(315*a*b^2*x + 84*a^3 + 40*(3*b^3*x + 7*a^2*b)*sqrt(x))/x^5

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=- \frac {a^{3}}{5 x^{5}} - \frac {2 a^{2} b}{3 x^{\frac {9}{2}}} - \frac {3 a b^{2}}{4 x^{4}} - \frac {2 b^{3}}{7 x^{\frac {7}{2}}} \]

[In]

integrate((a+b*x**(1/2))**3/x**6,x)

[Out]

-a**3/(5*x**5) - 2*a**2*b/(3*x**(9/2)) - 3*a*b**2/(4*x**4) - 2*b**3/(7*x**(7/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {120 \, b^{3} x^{\frac {3}{2}} + 315 \, a b^{2} x + 280 \, a^{2} b \sqrt {x} + 84 \, a^{3}}{420 \, x^{5}} \]

[In]

integrate((a+b*x^(1/2))^3/x^6,x, algorithm="maxima")

[Out]

-1/420*(120*b^3*x^(3/2) + 315*a*b^2*x + 280*a^2*b*sqrt(x) + 84*a^3)/x^5

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {120 \, b^{3} x^{\frac {3}{2}} + 315 \, a b^{2} x + 280 \, a^{2} b \sqrt {x} + 84 \, a^{3}}{420 \, x^{5}} \]

[In]

integrate((a+b*x^(1/2))^3/x^6,x, algorithm="giac")

[Out]

-1/420*(120*b^3*x^(3/2) + 315*a*b^2*x + 280*a^2*b*sqrt(x) + 84*a^3)/x^5

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {84\,a^3+120\,b^3\,x^{3/2}+280\,a^2\,b\,\sqrt {x}+315\,a\,b^2\,x}{420\,x^5} \]

[In]

int((a + b*x^(1/2))^3/x^6,x)

[Out]

-(84*a^3 + 120*b^3*x^(3/2) + 280*a^2*b*x^(1/2) + 315*a*b^2*x)/(420*x^5)