3.2152 \(\int \frac{(a+b \sqrt{x})^5}{x^7} \, dx\)

Optimal. Leaf size=73 \[ -\frac{2 a^3 b^2}{x^5}-\frac{20 a^2 b^3}{9 x^{9/2}}-\frac{10 a^4 b}{11 x^{11/2}}-\frac{a^5}{6 x^6}-\frac{5 a b^4}{4 x^4}-\frac{2 b^5}{7 x^{7/2}} \]

[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))

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Rubi [A]  time = 0.032088, antiderivative size = 73, 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} \[ -\frac{2 a^3 b^2}{x^5}-\frac{20 a^2 b^3}{9 x^{9/2}}-\frac{10 a^4 b}{11 x^{11/2}}-\frac{a^5}{6 x^6}-\frac{5 a b^4}{4 x^4}-\frac{2 b^5}{7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^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))

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^7} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{13}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{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]  time = 0.0252444, size = 65, normalized size = 0.89 \[ -\frac{6160 a^2 b^3 x^{3/2}+5544 a^3 b^2 x+2520 a^4 b \sqrt{x}+462 a^5+3465 a b^4 x^2+792 b^5 x^{5/2}}{2772 x^6} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.003, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{6\,{x}^{6}}}-{\frac{10\,{a}^{4}b}{11}{x}^{-{\frac{11}{2}}}}-2\,{\frac{{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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]  time = 0.956876, size = 77, normalized size = 1.05 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.51563, size = 149, normalized size = 2.04 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 4.14868, size = 73, normalized size = 1. \begin{align*} - \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.0916, size = 77, normalized size = 1.05 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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