Optimal. Leaf size=40 \[ -\frac {2}{a \sqrt {x}}-\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {53, 65, 211}
\begin {gather*} -\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 53
Rule 65
Rule 211
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} (a+b x)} \, dx &=-\frac {2}{a \sqrt {x}}-\frac {b \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{a}\\ &=-\frac {2}{a \sqrt {x}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a}\\ &=-\frac {2}{a \sqrt {x}}-\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 40, normalized size = 1.00 \begin {gather*} -\frac {2}{a \sqrt {x}}-\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \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 = 2.53, size = 100, normalized size = 2.50 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{x^{\frac {3}{2}}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-2}{3 b x^{\frac {3}{2}}},a\text {==}0\right \},\left \{\frac {-2}{a \sqrt {x}},b\text {==}0\right \}\right \},-\frac {\text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{a \sqrt {-\frac {a}{b}}}+\frac {\text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{a \sqrt {-\frac {a}{b}}}-\frac {2}{a \sqrt {x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 32, normalized size = 0.80
method | result | size |
derivativedivides | \(-\frac {2 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}-\frac {2}{a \sqrt {x}}\) | \(32\) |
default | \(-\frac {2 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}-\frac {2}{a \sqrt {x}}\) | \(32\) |
risch | \(-\frac {2 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}-\frac {2}{a \sqrt {x}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.34, size = 31, normalized size = 0.78 \begin {gather*} -\frac {2 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {2}{a \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.32, size = 93, normalized size = 2.32 \begin {gather*} \left [\frac {x \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) - 2 \, \sqrt {x}}{a x}, \frac {2 \, {\left (x \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - \sqrt {x}\right )}}{a x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.03, size = 85, normalized size = 2.12 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 b x^{\frac {3}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a \sqrt {x}} & \text {for}\: b = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{a \sqrt {- \frac {a}{b}}} + \frac {\log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{a \sqrt {- \frac {a}{b}}} - \frac {2}{a \sqrt {x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 45, normalized size = 1.12 \begin {gather*} 2 \left (-\frac {1}{a \sqrt {x}}-\frac {2 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a\cdot 2 \sqrt {a b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 28, normalized size = 0.70 \begin {gather*} -\frac {2}{a\,\sqrt {x}}-\frac {2\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________