\(\int \frac {(a+b \sqrt {x})^5}{x^7} \, dx\) [2152]

Optimal result

Integrand size = 15, antiderivative size = 73 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^7} \, dx=-\frac {a^5}{6 x^6}-\frac {10 a^4 b}{11 x^{11/2}}-\frac {2 a^3 b^2}{x^5}-\frac {20 a^2 b^3}{9 x^{9/2}}-\frac {5 a b^4}{4 x^4}-\frac {2 b^5}{7 x^{7/2}} \]

[Out]

-1/6*a^5/x^6-10/11*a^4*b/x^(11/2)-2*a^3*b^2/x^5-20/9*a^2*b^3/x^(9/2)-5/4*a*b^4/x^4-2/7*b^5/x^(7/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, 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} \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^7} \, dx=-\frac {a^5}{6 x^6}-\frac {10 a^4 b}{11 x^{11/2}}-\frac {2 a^3 b^2}{x^5}-\frac {20 a^2 b^3}{9 x^{9/2}}-\frac {5 a b^4}{4 x^4}-\frac {2 b^5}{7 x^{7/2}} \]

[In]

Int[(a + b*Sqrt[x])^5/x^7,x]

[Out]

-1/6*a^5/x^6 - (10*a^4*b)/(11*x^(11/2)) - (2*a^3*b^2)/x^5 - (20*a^2*b^3)/(9*x^(9/2)) - (5*a*b^4)/(4*x^4) - (2*
b^5)/(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)^5}{x^{13}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {a^5}{x^{13}}+\frac {5 a^4 b}{x^{12}}+\frac {10 a^3 b^2}{x^{11}}+\frac {10 a^2 b^3}{x^{10}}+\frac {5 a b^4}{x^9}+\frac {b^5}{x^8}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^5}{6 x^6}-\frac {10 a^4 b}{11 x^{11/2}}-\frac {2 a^3 b^2}{x^5}-\frac {20 a^2 b^3}{9 x^{9/2}}-\frac {5 a b^4}{4 x^4}-\frac {2 b^5}{7 x^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^7} \, dx=\frac {-462 a^5-2520 a^4 b \sqrt {x}-5544 a^3 b^2 x-6160 a^2 b^3 x^{3/2}-3465 a b^4 x^2-792 b^5 x^{5/2}}{2772 x^6} \]

[In]

Integrate[(a + b*Sqrt[x])^5/x^7,x]

[Out]

(-462*a^5 - 2520*a^4*b*Sqrt[x] - 5544*a^3*b^2*x - 6160*a^2*b^3*x^(3/2) - 3465*a*b^4*x^2 - 792*b^5*x^(5/2))/(27
72*x^6)

Maple [A] (verified)

Time = 3.58 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(-\frac {a^{5}}{6 x^{6}}-\frac {10 a^{4} b}{11 x^{\frac {11}{2}}}-\frac {2 a^{3} b^{2}}{x^{5}}-\frac {20 a^{2} b^{3}}{9 x^{\frac {9}{2}}}-\frac {5 a \,b^{4}}{4 x^{4}}-\frac {2 b^{5}}{7 x^{\frac {7}{2}}}\)	\(58\)
default	\(-\frac {a^{5}}{6 x^{6}}-\frac {10 a^{4} b}{11 x^{\frac {11}{2}}}-\frac {2 a^{3} b^{2}}{x^{5}}-\frac {20 a^{2} b^{3}}{9 x^{\frac {9}{2}}}-\frac {5 a \,b^{4}}{4 x^{4}}-\frac {2 b^{5}}{7 x^{\frac {7}{2}}}\)	\(58\)
trager	\(\frac {\left (-1+x \right ) \left (2 a^{4} x^{5}+24 a^{2} b^{2} x^{5}+15 b^{4} x^{5}+2 a^{4} x^{4}+24 a^{2} x^{4} b^{2}+15 b^{4} x^{4}+2 a^{4} x^{3}+24 a^{2} b^{2} x^{3}+15 b^{4} x^{3}+2 a^{4} x^{2}+24 a^{2} b^{2} x^{2}+15 b^{4} x^{2}+2 a^{4} x +24 a^{2} b^{2} x +2 a^{4}\right ) a}{12 x^{6}}-\frac {2 \left (99 b^{4} x^{2}+770 a^{2} b^{2} x +315 a^{4}\right ) b}{693 x^{\frac {11}{2}}}\)	\(169\)

[In]

int((a+b*x^(1/2))^5/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*a^5/x^6-10/11*a^4*b/x^(11/2)-2*a^3*b^2/x^5-20/9*a^2*b^3/x^(9/2)-5/4*a*b^4/x^4-2/7*b^5/x^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^7} \, dx=-\frac {3465 \, a b^{4} x^{2} + 5544 \, a^{3} b^{2} x + 462 \, a^{5} + 8 \, {\left (99 \, b^{5} x^{2} + 770 \, a^{2} b^{3} x + 315 \, a^{4} b\right )} \sqrt {x}}{2772 \, x^{6}} \]

[In]

integrate((a+b*x^(1/2))^5/x^7,x, algorithm="fricas")

[Out]

-1/2772*(3465*a*b^4*x^2 + 5544*a^3*b^2*x + 462*a^5 + 8*(99*b^5*x^2 + 770*a^2*b^3*x + 315*a^4*b)*sqrt(x))/x^6

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^7} \, dx=- \frac {a^{5}}{6 x^{6}} - \frac {10 a^{4} b}{11 x^{\frac {11}{2}}} - \frac {2 a^{3} b^{2}}{x^{5}} - \frac {20 a^{2} b^{3}}{9 x^{\frac {9}{2}}} - \frac {5 a b^{4}}{4 x^{4}} - \frac {2 b^{5}}{7 x^{\frac {7}{2}}} \]

[In]

integrate((a+b*x**(1/2))**5/x**7,x)

[Out]

-a**5/(6*x**6) - 10*a**4*b/(11*x**(11/2)) - 2*a**3*b**2/x**5 - 20*a**2*b**3/(9*x**(9/2)) - 5*a*b**4/(4*x**4) -
2*b**5/(7*x**(7/2))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^7} \, dx=-\frac {792 \, b^{5} x^{\frac {5}{2}} + 3465 \, a b^{4} x^{2} + 6160 \, a^{2} b^{3} x^{\frac {3}{2}} + 5544 \, a^{3} b^{2} x + 2520 \, a^{4} b \sqrt {x} + 462 \, a^{5}}{2772 \, x^{6}} \]

[In]

integrate((a+b*x^(1/2))^5/x^7,x, algorithm="maxima")

[Out]

-1/2772*(792*b^5*x^(5/2) + 3465*a*b^4*x^2 + 6160*a^2*b^3*x^(3/2) + 5544*a^3*b^2*x + 2520*a^4*b*sqrt(x) + 462*a
^5)/x^6

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^7} \, dx=-\frac {792 \, b^{5} x^{\frac {5}{2}} + 3465 \, a b^{4} x^{2} + 6160 \, a^{2} b^{3} x^{\frac {3}{2}} + 5544 \, a^{3} b^{2} x + 2520 \, a^{4} b \sqrt {x} + 462 \, a^{5}}{2772 \, x^{6}} \]

[In]

integrate((a+b*x^(1/2))^5/x^7,x, algorithm="giac")

[Out]

-1/2772*(792*b^5*x^(5/2) + 3465*a*b^4*x^2 + 6160*a^2*b^3*x^(3/2) + 5544*a^3*b^2*x + 2520*a^4*b*sqrt(x) + 462*a
^5)/x^6

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^7} \, dx=-\frac {\frac {a^5}{6}+\frac {2\,b^5\,x^{5/2}}{7}+2\,a^3\,b^2\,x+\frac {5\,a\,b^4\,x^2}{4}+\frac {10\,a^4\,b\,\sqrt {x}}{11}+\frac {20\,a^2\,b^3\,x^{3/2}}{9}}{x^6} \]

[In]

int((a + b*x^(1/2))^5/x^7,x)

[Out]

-(a^5/6 + (2*b^5*x^(5/2))/7 + 2*a^3*b^2*x + (5*a*b^4*x^2)/4 + (10*a^4*b*x^(1/2))/11 + (20*a^2*b^3*x^(3/2))/9)/
x^6