Optimal. Leaf size=68 \[ \frac {b}{a^2 \sqrt {d x^2}}-\frac {1}{3 a x^2 \sqrt {d x^2}}+\frac {b^{3/2} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 331, 211}
\begin {gather*} \frac {b^{3/2} x \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {d x^2}}+\frac {b}{a^2 \sqrt {d x^2}}-\frac {1}{3 a x^2 \sqrt {d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 211
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {d x^2} \left (a+b x^2\right )} \, dx &=\frac {x \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{\sqrt {d x^2}}\\ &=-\frac {1}{3 a x^2 \sqrt {d x^2}}-\frac {(b x) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{a \sqrt {d x^2}}\\ &=\frac {b}{a^2 \sqrt {d x^2}}-\frac {1}{3 a x^2 \sqrt {d x^2}}+\frac {\left (b^2 x\right ) \int \frac {1}{a+b x^2} \, dx}{a^2 \sqrt {d x^2}}\\ &=\frac {b}{a^2 \sqrt {d x^2}}-\frac {1}{3 a x^2 \sqrt {d x^2}}+\frac {b^{3/2} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 58, normalized size = 0.85 \begin {gather*} \frac {d \left (-\sqrt {a} \left (a-3 b x^2\right )+3 b^{3/2} x^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}{3 a^{5/2} \left (d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 57, normalized size = 0.84
method | result | size |
default | \(-\frac {-3 b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right ) x^{3}-3 b \sqrt {a b}\, x^{2}+a \sqrt {a b}}{3 x^{2} \sqrt {d \,x^{2}}\, a^{2} \sqrt {a b}}\) | \(57\) |
risch | \(\frac {\frac {b \,x^{2}}{a^{2}}-\frac {1}{3 a}}{\sqrt {d \,x^{2}}\, x^{2}}+\frac {x \left (\munderset {\textit {\_R} =\RootOf \left (a^{5} \textit {\_Z}^{2}+b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{5}+2 b^{3}\right ) x -a^{3} b \textit {\_R} \right )\right )}{2 \sqrt {d \,x^{2}}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 52, normalized size = 0.76 \begin {gather*} \frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2} \sqrt {d}} + \frac {3 \, b \sqrt {d} x^{2} - a \sqrt {d}}{3 \, a^{2} d x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.64, size = 157, normalized size = 2.31 \begin {gather*} \left [\frac {3 \, b d x^{4} \sqrt {-\frac {b}{a d}} \log \left (\frac {b x^{2} + 2 \, \sqrt {d x^{2}} a \sqrt {-\frac {b}{a d}} - a}{b x^{2} + a}\right ) + 2 \, {\left (3 \, b x^{2} - a\right )} \sqrt {d x^{2}}}{6 \, a^{2} d x^{4}}, \frac {3 \, b d x^{4} \sqrt {\frac {b}{a d}} \arctan \left (\sqrt {d x^{2}} \sqrt {\frac {b}{a d}}\right ) + {\left (3 \, b x^{2} - a\right )} \sqrt {d x^{2}}}{3 \, a^{2} d x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \sqrt {d x^{2}} \left (a + b x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.93, size = 54, normalized size = 0.79 \begin {gather*} \frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2} \sqrt {d} \mathrm {sgn}\left (x\right )} + \frac {3 \, b x^{2} - a}{3 \, a^{2} \sqrt {d} x^{3} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.35, size = 53, normalized size = 0.78 \begin {gather*} \frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x^2}}{\sqrt {a}}\right )}{a^{5/2}\,\sqrt {d}}-\frac {1}{3\,a\,\sqrt {d}\,{\left (x^2\right )}^{3/2}}+\frac {b\,x^2}{a^2\,\sqrt {d}\,{\left (x^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________