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

Optimal. Leaf size=21 \[ -\frac {2 (a+b x)^{3/2}}{3 a x^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {37} \begin {gather*} -\frac {2 (a+b x)^{3/2}}{3 a x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2 (a+b x)^{3/2}}{3 a x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x]/x^(5/2),x]

[Out]

(-2*(a + b*x)^(3/2))/(3*a*x^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(15)=30\).
time = 0.12, size = 49, normalized size = 2.33

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	\(-\frac {\sqrt {b x +a}}{x^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \sqrt {b x +a}}{3 a \,x^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 a^{2} \sqrt {x}}\right )}{2}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 15, normalized size = 0.71 \begin {gather*} -\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, a x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [A]
time = 0.43, size = 15, normalized size = 0.71 \begin {gather*} -\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, a x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
time = 0.79, size = 41, normalized size = 1.95 \begin {gather*} - \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).
time = 0.93, size = 33, normalized size = 1.57 \begin {gather*} -\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 |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Mupad [B]
time = 0.24, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\left (\frac {2\,b\,x}{3\,a}+\frac {2}{3}\right )\,\sqrt {a+b\,x}}{x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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