3.2151 \(\int \frac{(a+b \sqrt{x})^5}{x^6} \, dx\)

Optimal. Leaf size=75 \[ -\frac{5 a^3 b^2}{2 x^4}-\frac{20 a^2 b^3}{7 x^{7/2}}-\frac{10 a^4 b}{9 x^{9/2}}-\frac{a^5}{5 x^5}-\frac{5 a b^4}{3 x^3}-\frac{2 b^5}{5 x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0319221, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{5 a^3 b^2}{2 x^4}-\frac{20 a^2 b^3}{7 x^{7/2}}-\frac{10 a^4 b}{9 x^{9/2}}-\frac{a^5}{5 x^5}-\frac{5 a b^4}{3 x^3}-\frac{2 b^5}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5/x^6,x]

[Out]

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

Rule 266

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]]

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

Mathematica [A]  time = 0.0260976, size = 65, normalized size = 0.87 \[ -\frac{1800 a^2 b^3 x^{3/2}+1575 a^3 b^2 x+700 a^4 b \sqrt{x}+126 a^5+1050 a b^4 x^2+252 b^5 x^{5/2}}{630 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5/x^6,x]

[Out]

-(126*a^5 + 700*a^4*b*Sqrt[x] + 1575*a^3*b^2*x + 1800*a^2*b^3*x^(3/2) + 1050*a*b^4*x^2 + 252*b^5*x^(5/2))/(630
*x^5)

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Maple [A]  time = 0.003, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{5\,{x}^{5}}}-{\frac{10\,{a}^{4}b}{9}{x}^{-{\frac{9}{2}}}}-{\frac{5\,{a}^{3}{b}^{2}}{2\,{x}^{4}}}-{\frac{20\,{a}^{2}{b}^{3}}{7}{x}^{-{\frac{7}{2}}}}-{\frac{5\,a{b}^{4}}{3\,{x}^{3}}}-{\frac{2\,{b}^{5}}{5}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^5/x^6,x)

[Out]

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

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Maxima [A]  time = 0.978925, size = 77, normalized size = 1.03 \begin{align*} -\frac{252 \, b^{5} x^{\frac{5}{2}} + 1050 \, a b^{4} x^{2} + 1800 \, a^{2} b^{3} x^{\frac{3}{2}} + 1575 \, a^{3} b^{2} x + 700 \, a^{4} b \sqrt{x} + 126 \, a^{5}}{630 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x^6,x, algorithm="maxima")

[Out]

-1/630*(252*b^5*x^(5/2) + 1050*a*b^4*x^2 + 1800*a^2*b^3*x^(3/2) + 1575*a^3*b^2*x + 700*a^4*b*sqrt(x) + 126*a^5
)/x^5

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Fricas [A]  time = 1.46752, size = 147, normalized size = 1.96 \begin{align*} -\frac{1050 \, a b^{4} x^{2} + 1575 \, a^{3} b^{2} x + 126 \, a^{5} + 4 \,{\left (63 \, b^{5} x^{2} + 450 \, a^{2} b^{3} x + 175 \, a^{4} b\right )} \sqrt{x}}{630 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x^6,x, algorithm="fricas")

[Out]

-1/630*(1050*a*b^4*x^2 + 1575*a^3*b^2*x + 126*a^5 + 4*(63*b^5*x^2 + 450*a^2*b^3*x + 175*a^4*b)*sqrt(x))/x^5

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Sympy [A]  time = 2.78584, size = 75, normalized size = 1. \begin{align*} - \frac{a^{5}}{5 x^{5}} - \frac{10 a^{4} b}{9 x^{\frac{9}{2}}} - \frac{5 a^{3} b^{2}}{2 x^{4}} - \frac{20 a^{2} b^{3}}{7 x^{\frac{7}{2}}} - \frac{5 a b^{4}}{3 x^{3}} - \frac{2 b^{5}}{5 x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**5/x**6,x)

[Out]

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

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Giac [A]  time = 1.10437, size = 77, normalized size = 1.03 \begin{align*} -\frac{252 \, b^{5} x^{\frac{5}{2}} + 1050 \, a b^{4} x^{2} + 1800 \, a^{2} b^{3} x^{\frac{3}{2}} + 1575 \, a^{3} b^{2} x + 700 \, a^{4} b \sqrt{x} + 126 \, a^{5}}{630 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x^6,x, algorithm="giac")

[Out]

-1/630*(252*b^5*x^(5/2) + 1050*a*b^4*x^2 + 1800*a^2*b^3*x^(3/2) + 1575*a^3*b^2*x + 700*a^4*b*sqrt(x) + 126*a^5
)/x^5