Optimal. Leaf size=45 \[ \frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}}+\frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37}
\begin {gather*} \frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}}+\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (a-b x)^{5/2}} \, dx &=\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}}+\frac {2 \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx}{3 a}\\ &=\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}}+\frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 30, normalized size = 0.67 \begin {gather*} \frac {2 \sqrt {x} (3 a-2 b x)}{3 a^2 (a-b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 3.60, size = 157, normalized size = 3.49 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \sqrt {b} x \left (3 a-2 b x\right ) \sqrt {\frac {a-b x}{b x}}}{3 a^2 \left (a-b x\right )^2},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-6 I a b}{3 a^3 b^{\frac {3}{2}} \sqrt {1-\frac {a}{b x}}-3 a^2 b^{\frac {5}{2}} x \sqrt {1-\frac {a}{b x}}}+\frac {I 4 b^2 x}{3 a^3 b^{\frac {3}{2}} \sqrt {1-\frac {a}{b x}}-3 a^2 b^{\frac {5}{2}} x \sqrt {1-\frac {a}{b x}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 34, normalized size = 0.76
method | result | size |
gosper | \(\frac {2 \sqrt {x}\, \left (-2 b x +3 a \right )}{3 \left (-b x +a \right )^{\frac {3}{2}} a^{2}}\) | \(25\) |
default | \(\frac {2 \sqrt {x}}{3 a \left (-b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {-b x +a}}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 30, normalized size = 0.67 \begin {gather*} \frac {2 \, {\left (b - \frac {3 \, {\left (b x - a\right )}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.31, size = 44, normalized size = 0.98 \begin {gather*} -\frac {2 \, {\left (2 \, b x - 3 \, a\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.12, size = 197, normalized size = 4.38 \begin {gather*} \begin {cases} - \frac {6 a}{- 3 a^{3} \sqrt {b} \sqrt {\frac {a}{b x} - 1} + 3 a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} - 1}} + \frac {4 b x}{- 3 a^{3} \sqrt {b} \sqrt {\frac {a}{b x} - 1} + 3 a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {6 i a b}{- 3 a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1} + 3 a^{2} b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}} - \frac {4 i b^{2} x}{- 3 a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1} + 3 a^{2} b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}} & \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. 96 vs.
\(2 (33) = 66\).
time = 0.01, size = 109, normalized size = 2.42 \begin {gather*} -\frac {32 b \sqrt {-b} b \left (-3 \left (\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right )^{2}+a b\right )}{2\cdot 6 \left |b\right | \left (\left (\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right )^{2}-a b\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.41, size = 56, normalized size = 1.24 \begin {gather*} \frac {6\,a\,\sqrt {x}\,\sqrt {a-b\,x}-4\,b\,x^{3/2}\,\sqrt {a-b\,x}}{3\,a^4-6\,a^3\,b\,x+3\,a^2\,b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________