Optimal. Leaf size=45 \[ \frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}}+\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, 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 = {45, 37} \[ \frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}}+\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 45
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 = 0.01, size = 30, normalized size = 0.67 \[ \frac {2 \sqrt {x} (3 a-2 b x)}{3 a^2 (a-b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 44, normalized size = 0.98 \[ -\frac {2 \, {\left (2 \, b x - 3 \, a\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.43, size = 96, normalized size = 2.13 \[ \frac {8 \, {\left (3 \, {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} \sqrt {-b} b^{2}}{3 \, {\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 25, normalized size = 0.56 \[ \frac {2 \left (-2 b x +3 a \right ) \sqrt {x}}{3 \left (-b x +a \right )^{\frac {3}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.34, size = 30, normalized size = 0.67 \[ \frac {2 \, {\left (b - \frac {3 \, {\left (b x - a\right )}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 56, normalized size = 1.24 \[ \frac {6\,a\,\sqrt {x}\,\sqrt {a-b\,x}-4\,b\,x^{3/2}\,\sqrt {a-b\,x}}{3\,a^4-6\,a^3\,b\,x+3\,a^2\,b^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 2.00, size = 211, normalized size = 4.69 \[ \begin {cases} \frac {6 i a}{3 i a^{3} \sqrt {b} \sqrt {\frac {a}{b x} - 1} - 3 i a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} - 1}} - \frac {4 i b x}{3 i a^{3} \sqrt {b} \sqrt {\frac {a}{b x} - 1} - 3 i a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {6 a b}{3 i a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1} - 3 i a^{2} b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}} - \frac {4 b^{2} x}{3 i a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1} - 3 i a^{2} b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________