3.2150 \(\int \frac{(a+b \sqrt{x})^5}{x^5} \, dx\)

Optimal. Leaf size=70 \[ -\frac{b^2 \left (a+b \sqrt{x}\right )^6}{84 a^3 x^3}+\frac{b \left (a+b \sqrt{x}\right )^6}{14 a^2 x^{7/2}}-\frac{\left (a+b \sqrt{x}\right )^6}{4 a x^4} \]

[Out]

-(a + b*Sqrt[x])^6/(4*a*x^4) + (b*(a + b*Sqrt[x])^6)/(14*a^2*x^(7/2)) - (b^2*(a + b*Sqrt[x])^6)/(84*a^3*x^3)

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Rubi [A]  time = 0.0217816, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ -\frac{b^2 \left (a+b \sqrt{x}\right )^6}{84 a^3 x^3}+\frac{b \left (a+b \sqrt{x}\right )^6}{14 a^2 x^{7/2}}-\frac{\left (a+b \sqrt{x}\right )^6}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5/x^5,x]

[Out]

-(a + b*Sqrt[x])^6/(4*a*x^4) + (b*(a + b*Sqrt[x])^6)/(14*a^2*x^(7/2)) - (b^2*(a + b*Sqrt[x])^6)/(84*a^3*x^3)

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^5}{x^5} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^9} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \sqrt{x}\right )^6}{4 a x^4}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^8} \, dx,x,\sqrt{x}\right )}{2 a}\\ &=-\frac{\left (a+b \sqrt{x}\right )^6}{4 a x^4}+\frac{b \left (a+b \sqrt{x}\right )^6}{14 a^2 x^{7/2}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^7} \, dx,x,\sqrt{x}\right )}{14 a^2}\\ &=-\frac{\left (a+b \sqrt{x}\right )^6}{4 a x^4}+\frac{b \left (a+b \sqrt{x}\right )^6}{14 a^2 x^{7/2}}-\frac{b^2 \left (a+b \sqrt{x}\right )^6}{84 a^3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0261548, size = 65, normalized size = 0.93 \[ -\frac{336 a^2 b^3 x^{3/2}+280 a^3 b^2 x+120 a^4 b \sqrt{x}+21 a^5+210 a b^4 x^2+56 b^5 x^{5/2}}{84 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5/x^5,x]

[Out]

-(21*a^5 + 120*a^4*b*Sqrt[x] + 280*a^3*b^2*x + 336*a^2*b^3*x^(3/2) + 210*a*b^4*x^2 + 56*b^5*x^(5/2))/(84*x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.962597, size = 77, normalized size = 1.1 \begin{align*} -\frac{56 \, b^{5} x^{\frac{5}{2}} + 210 \, a b^{4} x^{2} + 336 \, a^{2} b^{3} x^{\frac{3}{2}} + 280 \, a^{3} b^{2} x + 120 \, a^{4} b \sqrt{x} + 21 \, a^{5}}{84 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/84*(56*b^5*x^(5/2) + 210*a*b^4*x^2 + 336*a^2*b^3*x^(3/2) + 280*a^3*b^2*x + 120*a^4*b*sqrt(x) + 21*a^5)/x^4

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Fricas [A]  time = 1.69642, size = 138, normalized size = 1.97 \begin{align*} -\frac{210 \, a b^{4} x^{2} + 280 \, a^{3} b^{2} x + 21 \, a^{5} + 8 \,{\left (7 \, b^{5} x^{2} + 42 \, a^{2} b^{3} x + 15 \, a^{4} b\right )} \sqrt{x}}{84 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/84*(210*a*b^4*x^2 + 280*a^3*b^2*x + 21*a^5 + 8*(7*b^5*x^2 + 42*a^2*b^3*x + 15*a^4*b)*sqrt(x))/x^4

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Sympy [A]  time = 1.88796, size = 73, normalized size = 1.04 \begin{align*} - \frac{a^{5}}{4 x^{4}} - \frac{10 a^{4} b}{7 x^{\frac{7}{2}}} - \frac{10 a^{3} b^{2}}{3 x^{3}} - \frac{4 a^{2} b^{3}}{x^{\frac{5}{2}}} - \frac{5 a b^{4}}{2 x^{2}} - \frac{2 b^{5}}{3 x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11061, size = 77, normalized size = 1.1 \begin{align*} -\frac{56 \, b^{5} x^{\frac{5}{2}} + 210 \, a b^{4} x^{2} + 336 \, a^{2} b^{3} x^{\frac{3}{2}} + 280 \, a^{3} b^{2} x + 120 \, a^{4} b \sqrt{x} + 21 \, a^{5}}{84 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/84*(56*b^5*x^(5/2) + 210*a*b^4*x^2 + 336*a^2*b^3*x^(3/2) + 280*a^3*b^2*x + 120*a^4*b*sqrt(x) + 21*a^5)/x^4