Optimal. Leaf size=22 \[ -\frac {2 (a-b x)^{3/2}}{3 a x^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {37}
\begin {gather*} -\frac {2 (a-b x)^{3/2}}{3 a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 22, 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]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs.
\(2(16)=32\).
time = 0.11, size = 52, normalized size = 2.36
method | result | size |
gosper | \(-\frac {2 \left (-b x +a \right )^{\frac {3}{2}}}{3 a \,x^{\frac {3}{2}}}\) | \(17\) |
risch | \(-\frac {2 \left (-b x +a \right )^{\frac {3}{2}}}{3 a \,x^{\frac {3}{2}}}\) | \(17\) |
default | \(-\frac {\sqrt {-b x +a}}{x^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \sqrt {-b x +a}}{3 a \,x^{\frac {3}{2}}}-\frac {4 b \sqrt {-b x +a}}{3 a^{2} \sqrt {x}}\right )}{2}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 16, normalized size = 0.73 \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]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.46, size = 23, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (b x - a\right )} \sqrt {-b x + a}}{3 \, a x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.86, size = 88, normalized size = 4.00 \begin {gather*} \begin {cases} - \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} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i \sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{3 x} + \frac {2 i b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{3 a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs.
\(2 (16) = 32\).
time = 1.24, size = 42, normalized size = 1.91 \begin {gather*} \frac {2 \, {\left (b x - a\right )} \sqrt {-b x + a} 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]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.24, size = 21, normalized size = 0.95 \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]
[Out]
________________________________________________________________________________________