3.2149 \(\int \frac{(a+b \sqrt{x})^5}{x^4} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b \sqrt{x}\right )^6}{3 a x^3} \]

[Out]

-(a + b*Sqrt[x])^6/(3*a*x^3)

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Rubi [A]  time = 0.0032821, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\left (a+b \sqrt{x}\right )^6}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5/x^4,x]

[Out]

-(a + b*Sqrt[x])^6/(3*a*x^3)

Rule 264

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^5}{x^4} \, dx &=-\frac{\left (a+b \sqrt{x}\right )^6}{3 a x^3}\\ \end{align*}

Mathematica [A]  time = 0.0046224, size = 21, normalized size = 1. \[ -\frac{\left (a+b \sqrt{x}\right )^6}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5/x^4,x]

[Out]

-(a + b*Sqrt[x])^6/(3*a*x^3)

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Maple [B]  time = 0.001, size = 58, normalized size = 2.8 \begin{align*} -2\,{\frac{{b}^{5}}{\sqrt{x}}}-5\,{\frac{a{b}^{4}}{x}}-{\frac{20\,{a}^{2}{b}^{3}}{3}{x}^{-{\frac{3}{2}}}}-5\,{\frac{{a}^{3}{b}^{2}}{{x}^{2}}}-2\,{\frac{{a}^{4}b}{{x}^{5/2}}}-{\frac{{a}^{5}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^5/x^4,x)

[Out]

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

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Maxima [B]  time = 1.51152, size = 74, normalized size = 3.52 \begin{align*} -\frac{6 \, b^{5} x^{\frac{5}{2}} + 15 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac{3}{2}} + 15 \, a^{3} b^{2} x + 6 \, a^{4} b \sqrt{x} + a^{5}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x^4,x, algorithm="maxima")

[Out]

-1/3*(6*b^5*x^(5/2) + 15*a*b^4*x^2 + 20*a^2*b^3*x^(3/2) + 15*a^3*b^2*x + 6*a^4*b*sqrt(x) + a^5)/x^3

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Fricas [B]  time = 1.49889, size = 128, normalized size = 6.1 \begin{align*} -\frac{15 \, a b^{4} x^{2} + 15 \, a^{3} b^{2} x + a^{5} + 2 \,{\left (3 \, b^{5} x^{2} + 10 \, a^{2} b^{3} x + 3 \, a^{4} b\right )} \sqrt{x}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x^4,x, algorithm="fricas")

[Out]

-1/3*(15*a*b^4*x^2 + 15*a^3*b^2*x + a^5 + 2*(3*b^5*x^2 + 10*a^2*b^3*x + 3*a^4*b)*sqrt(x))/x^3

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Sympy [B]  time = 1.27584, size = 66, normalized size = 3.14 \begin{align*} - \frac{a^{5}}{3 x^{3}} - \frac{2 a^{4} b}{x^{\frac{5}{2}}} - \frac{5 a^{3} b^{2}}{x^{2}} - \frac{20 a^{2} b^{3}}{3 x^{\frac{3}{2}}} - \frac{5 a b^{4}}{x} - \frac{2 b^{5}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**5/x**4,x)

[Out]

-a**5/(3*x**3) - 2*a**4*b/x**(5/2) - 5*a**3*b**2/x**2 - 20*a**2*b**3/(3*x**(3/2)) - 5*a*b**4/x - 2*b**5/sqrt(x
)

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Giac [B]  time = 1.09051, size = 74, normalized size = 3.52 \begin{align*} -\frac{6 \, b^{5} x^{\frac{5}{2}} + 15 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac{3}{2}} + 15 \, a^{3} b^{2} x + 6 \, a^{4} b \sqrt{x} + a^{5}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x^4,x, algorithm="giac")

[Out]

-1/3*(6*b^5*x^(5/2) + 15*a*b^4*x^2 + 20*a^2*b^3*x^(3/2) + 15*a^3*b^2*x + 6*a^4*b*sqrt(x) + a^5)/x^3