3.2119 \(\int \frac{a+b \sqrt{x}}{x^4} \, dx\)

Optimal. Leaf size=19 \[ -\frac{a}{3 x^3}-\frac{2 b}{5 x^{5/2}} \]

[Out]

-a/(3*x^3) - (2*b)/(5*x^(5/2))

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Rubi [A]  time = 0.004937, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ -\frac{a}{3 x^3}-\frac{2 b}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])/x^4,x]

[Out]

-a/(3*x^3) - (2*b)/(5*x^(5/2))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0068429, size = 19, normalized size = 1. \[ -\frac{a}{3 x^3}-\frac{2 b}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])/x^4,x]

[Out]

-a/(3*x^3) - (2*b)/(5*x^(5/2))

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Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{2\,b}{5}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.961646, size = 20, normalized size = 1.05 \begin{align*} -\frac{6 \, b \sqrt{x} + 5 \, a}{15 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(6*b*sqrt(x) + 5*a)/x^3

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Fricas [A]  time = 1.51372, size = 42, normalized size = 2.21 \begin{align*} -\frac{6 \, b \sqrt{x} + 5 \, a}{15 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(6*b*sqrt(x) + 5*a)/x^3

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Sympy [A]  time = 0.884565, size = 17, normalized size = 0.89 \begin{align*} - \frac{a}{3 x^{3}} - \frac{2 b}{5 x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(3*x**3) - 2*b/(5*x**(5/2))

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Giac [A]  time = 1.08902, size = 20, normalized size = 1.05 \begin{align*} -\frac{6 \, b \sqrt{x} + 5 \, a}{15 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(6*b*sqrt(x) + 5*a)/x^3