Optimal. Leaf size=74 \[ \frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {44, 65, 211}
\begin {gather*} \frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{5/2}}+\frac {3 b \sqrt {b x-a}}{4 a^2 x}+\frac {\sqrt {b x-a}}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 65
Rule 211
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx &=\frac {\sqrt {-a+b x}}{2 a x^2}+\frac {(3 b) \int \frac {1}{x^2 \sqrt {-a+b x}} \, dx}{4 a}\\ &=\frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {\left (3 b^2\right ) \int \frac {1}{x \sqrt {-a+b x}} \, dx}{8 a^2}\\ &=\frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{4 a^2}\\ &=\frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 60, normalized size = 0.81 \begin {gather*} \frac {\sqrt {-a+b x} (2 a+3 b x)}{4 a^2 x^2}+\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 95.10, size = 210, normalized size = 2.84 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (2 a^{\frac {11}{2}} x \left (a-b x\right )+a^{\frac {9}{2}} b x^2 \left (a-b x\right )-3 a^{\frac {7}{2}} b^2 x^3 \left (a-b x\right )+3 a^3 b^{\frac {7}{2}} x^{\frac {9}{2}} \text {ArcCosh}\left [\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}}\right ] \left (\frac {a-b x}{b x}\right )^{\frac {3}{2}}\right )}{4 a^{\frac {11}{2}} b^{\frac {3}{2}} x^{\frac {9}{2}} \left (\frac {a-b x}{b x}\right )^{\frac {3}{2}}},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-3 b^2 \text {ArcSin}\left [\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}}\right ]}{4 a^{\frac {5}{2}}}-\frac {1}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {1-\frac {a}{b x}}}-\frac {\sqrt {b}}{4 a x^{\frac {3}{2}} \sqrt {1-\frac {a}{b x}}}+\frac {3 b^{\frac {3}{2}}}{4 a^2 \sqrt {x} \sqrt {1-\frac {a}{b x}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 72, normalized size = 0.97
method | result | size |
risch | \(-\frac {\left (-b x +a \right ) \left (3 b x +2 a \right )}{4 a^{2} x^{2} \sqrt {b x -a}}+\frac {3 b^{2} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{4 a^{\frac {5}{2}}}\) | \(55\) |
derivativedivides | \(2 b^{2} \left (\frac {\sqrt {b x -a}}{4 a \,b^{2} x^{2}}+\frac {\frac {3 \sqrt {b x -a}}{8 a b x}+\frac {3 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}}{a}\right )\) | \(72\) |
default | \(2 b^{2} \left (\frac {\sqrt {b x -a}}{4 a \,b^{2} x^{2}}+\frac {\frac {3 \sqrt {b x -a}}{8 a b x}+\frac {3 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}}{a}\right )\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.35, size = 86, normalized size = 1.16 \begin {gather*} \frac {3 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{2} + 5 \, \sqrt {b x - a} a b^{2}}{4 \, {\left ({\left (b x - a\right )}^{2} a^{2} + 2 \, {\left (b x - a\right )} a^{3} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.32, size = 128, normalized size = 1.73 \begin {gather*} \left [-\frac {3 \, \sqrt {-a} b^{2} x^{2} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) - 2 \, {\left (3 \, a b x + 2 \, a^{2}\right )} \sqrt {b x - a}}{8 \, a^{3} x^{2}}, \frac {3 \, \sqrt {a} b^{2} x^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (3 \, a b x + 2 \, a^{2}\right )} \sqrt {b x - a}}{4 \, a^{3} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 2.57, size = 216, normalized size = 2.92 \begin {gather*} \begin {cases} \frac {i}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} - 1}} + \frac {i \sqrt {b}}{4 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}} - \frac {3 i b^{\frac {3}{2}}}{4 a^{2} \sqrt {x} \sqrt {\frac {a}{b x} - 1}} + \frac {3 i b^{2} \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {5}{2}}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {1}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {- \frac {a}{b x} + 1}} - \frac {\sqrt {b}}{4 a x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} + \frac {3 b^{\frac {3}{2}}}{4 a^{2} \sqrt {x} \sqrt {- \frac {a}{b x} + 1}} - \frac {3 b^{2} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 97, normalized size = 1.31 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {-a+b x} \left (-a+b x\right ) b^{3}+5 \sqrt {-a+b x} a b^{3}}{8 a^{2} \left (-a+b x+a\right )^{2}}+\frac {3 b^{3} \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{2}\cdot 2 \sqrt {a}}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 57, normalized size = 0.77 \begin {gather*} \frac {3\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{4\,a^{5/2}}+\frac {5\,\sqrt {b\,x-a}}{4\,a\,x^2}+\frac {3\,{\left (b\,x-a\right )}^{3/2}}{4\,a^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________