\(\int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx\) [494]

Optimal result

Integrand size = 15, antiderivative size = 21 \[ \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx=-\frac {2 (a+b x)^{3/2}}{3 a x^{3/2}} \]

[Out]

-2/3*(b*x+a)^(3/2)/a/x^(3/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx=-\frac {2 (a+b x)^{3/2}}{3 a x^{3/2}} \]

[In]

Int[Sqrt[a + b*x]/x^(5/2),x]

[Out]

(-2*(a + b*x)^(3/2))/(3*a*x^(3/2))

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*} \text {integral}& = -\frac {2 (a+b x)^{3/2}}{3 a x^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx=-\frac {2 (a+b x)^{3/2}}{3 a x^{3/2}} \]

[In]

Integrate[Sqrt[a + b*x]/x^(5/2),x]

[Out]

(-2*(a + b*x)^(3/2))/(3*a*x^(3/2))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3 a \,x^{\frac {3}{2}}}\)	\(16\)
risch	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3 a \,x^{\frac {3}{2}}}\)	\(16\)
default	\(-\frac {\sqrt {b x +a}}{x^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \sqrt {b x +a}}{3 a \,x^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 a^{2} \sqrt {x}}\right )}{2}\)	\(49\)

[In]

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

[Out]

-2/3*(b*x+a)^(3/2)/a/x^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx=-\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, a x^{\frac {3}{2}}} \]

[In]

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

[Out]

-2/3*(b*x + a)^(3/2)/(a*x^(3/2))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).

Time = 1.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx=- \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a} \]

[In]

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

[Out]

-2*sqrt(b)*sqrt(a/(b*x) + 1)/(3*x) - 2*b**(3/2)*sqrt(a/(b*x) + 1)/(3*a)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx=-\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, a x^{\frac {3}{2}}} \]

[In]

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

[Out]

-2/3*(b*x + a)^(3/2)/(a*x^(3/2))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx=-\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{4}}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}} a {\left | b \right |}} \]

[In]

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

[Out]

-2/3*(b*x + a)^(3/2)*b^4/(((b*x + a)*b - a*b)^(3/2)*a*abs(b))

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx=-\frac {\left (\frac {2\,b\,x}{3\,a}+\frac {2}{3}\right )\,\sqrt {a+b\,x}}{x^{3/2}} \]

[In]

int((a + b*x)^(1/2)/x^(5/2),x)

[Out]

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