\(\int \frac {a+b \sqrt {x}}{x^4} \, dx\) [2119]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \frac {a+b \sqrt {x}}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {2 b}{5 x^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \int \frac {a+b \sqrt {x}}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {2 b}{5 x^{5/2}} \]

[In]

Int[(a + b*Sqrt[x])/x^4,x]

[Out]

-1/3*a/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*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \sqrt {x}}{x^4} \, dx=\frac {-5 a-6 b \sqrt {x}}{15 x^3} \]

[In]

Integrate[(a + b*Sqrt[x])/x^4,x]

[Out]

(-5*a - 6*b*Sqrt[x])/(15*x^3)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(-\frac {a}{3 x^{3}}-\frac {2 b}{5 x^{\frac {5}{2}}}\)	\(14\)
default	\(-\frac {a}{3 x^{3}}-\frac {2 b}{5 x^{\frac {5}{2}}}\)	\(14\)
trager	\(\frac {a \left (x^{2}+x +1\right ) \left (-1+x \right )}{3 x^{3}}-\frac {2 b}{5 x^{\frac {5}{2}}}\)	\(23\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \sqrt {x}}{x^4} \, dx=-\frac {6 \, b \sqrt {x} + 5 \, a}{15 \, x^{3}} \]

[In]

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

[Out]

-1/15*(6*b*sqrt(x) + 5*a)/x^3

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \sqrt {x}}{x^4} \, dx=- \frac {a}{3 x^{3}} - \frac {2 b}{5 x^{\frac {5}{2}}} \]

[In]

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

[Out]

-a/(3*x**3) - 2*b/(5*x**(5/2))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \sqrt {x}}{x^4} \, dx=-\frac {6 \, b \sqrt {x} + 5 \, a}{15 \, x^{3}} \]

[In]

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

[Out]

-1/15*(6*b*sqrt(x) + 5*a)/x^3

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \sqrt {x}}{x^4} \, dx=-\frac {6 \, b \sqrt {x} + 5 \, a}{15 \, x^{3}} \]

[In]

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

[Out]

-1/15*(6*b*sqrt(x) + 5*a)/x^3

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \sqrt {x}}{x^4} \, dx=-\frac {5\,a+6\,b\,\sqrt {x}}{15\,x^3} \]

[In]

int((a + b*x^(1/2))/x^4,x)

[Out]

-(5*a + 6*b*x^(1/2))/(15*x^3)