Optimal. Leaf size=58 \[ \frac {\sqrt {a x^2+b x^4}}{2 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^2+b x^4}}\right )}{2 b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2043, 654, 634,
212} \begin {gather*} \frac {\sqrt {a x^2+b x^4}}{2 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^2+b x^4}}\right )}{2 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 634
Rule 654
Rule 2043
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a x^2+b x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{\sqrt {a x+b x^2}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a x^2+b x^4}}{2 b}-\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {a x+b x^2}} \, dx,x,x^2\right )}{4 b}\\ &=\frac {\sqrt {a x^2+b x^4}}{2 b}-\frac {a \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a x^2+b x^4}}\right )}{2 b}\\ &=\frac {\sqrt {a x^2+b x^4}}{2 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^2+b x^4}}\right )}{2 b^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 74, normalized size = 1.28 \begin {gather*} \frac {x \left (\sqrt {b} x \left (a+b x^2\right )+a \sqrt {a+b x^2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{2 b^{3/2} \sqrt {x^2 \left (a+b x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 64, normalized size = 1.10
method | result | size |
default | \(\frac {x \sqrt {b \,x^{2}+a}\, \left (x \sqrt {b \,x^{2}+a}\, b^{\frac {3}{2}}-a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) b \right )}{2 \sqrt {b \,x^{4}+a \,x^{2}}\, b^{\frac {5}{2}}}\) | \(64\) |
risch | \(\frac {x^{2} \left (b \,x^{2}+a \right )}{2 b \sqrt {x^{2} \left (b \,x^{2}+a \right )}}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) x \sqrt {b \,x^{2}+a}}{2 b^{\frac {3}{2}} \sqrt {x^{2} \left (b \,x^{2}+a \right )}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 52, normalized size = 0.90 \begin {gather*} -\frac {a \log \left (2 \, b x^{2} + a + 2 \, \sqrt {b x^{4} + a x^{2}} \sqrt {b}\right )}{4 \, b^{\frac {3}{2}}} + \frac {\sqrt {b x^{4} + a x^{2}}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 114, normalized size = 1.97 \begin {gather*} \left [\frac {a \sqrt {b} \log \left (-2 \, b x^{2} - a + 2 \, \sqrt {b x^{4} + a x^{2}} \sqrt {b}\right ) + 2 \, \sqrt {b x^{4} + a x^{2}} b}{4 \, b^{2}}, \frac {a \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{4} + a x^{2}} \sqrt {-b}}{b x^{2} + a}\right ) + \sqrt {b x^{4} + a x^{2}} b}{2 \, b^{2}}\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 {x^{3}}{\sqrt {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 = 3.85, size = 59, normalized size = 1.02 \begin {gather*} -\frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{4 \, b^{\frac {3}{2}}} + \frac {\sqrt {b x^{2} + a} x}{2 \, b \mathrm {sgn}\left (x\right )} + \frac {a \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.71, size = 53, normalized size = 0.91 \begin {gather*} \frac {\sqrt {b\,x^4+a\,x^2}}{2\,b}-\frac {a\,\ln \left (\frac {b\,x^2+\frac {a}{2}}{\sqrt {b}}+\sqrt {b\,x^4+a\,x^2}\right )}{4\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________