Optimal. Leaf size=54 \[ \frac {\log \left (a x^4+2 a x^2+a+b\right )}{4 a}-\frac {\tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1114, 634, 618, 204, 628} \[ \frac {\log \left (a x^4+2 a x^2+a+b\right )}{4 a}-\frac {\tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 628
Rule 634
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{a+b+2 a x+a x^2} \, dx,x,x^2\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a+b+2 a x+a x^2} \, dx,x,x^2\right )\right )+\frac {\operatorname {Subst}\left (\int \frac {2 a+2 a x}{a+b+2 a x+a x^2} \, dx,x,x^2\right )}{4 a}\\ &=\frac {\log \left (a+b+2 a x^2+a x^4\right )}{4 a}+\operatorname {Subst}\left (\int \frac {1}{-4 a b-x^2} \, dx,x,2 a \left (1+x^2\right )\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}}+\frac {\log \left (a+b+2 a x^2+a x^4\right )}{4 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 49, normalized size = 0.91 \[ \frac {\log \left (a \left (x^2+1\right )^2+b\right )-\frac {2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{\sqrt {b}}}{4 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.87, size = 131, normalized size = 2.43 \[ \left [\frac {b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) - \sqrt {-a b} \log \left (\frac {a x^{4} + 2 \, a x^{2} + 2 \, \sqrt {-a b} {\left (x^{2} + 1\right )} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right )}{4 \, a b}, \frac {b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 2 \, \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{a x^{2} + a}\right )}{4 \, a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 42, normalized size = 0.78 \[ -\frac {\arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} + \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 47, normalized size = 0.87 \[ -\frac {\arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{2 \sqrt {a b}}+\frac {\ln \left (a \,x^{4}+2 a \,x^{2}+a +b \right )}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.87, size = 42, normalized size = 0.78 \[ -\frac {\arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} + \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 85, normalized size = 1.57 \[ \frac {\ln \left (a\,x^4+2\,a\,x^2+a+b\right )}{4\,a}-\frac {\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}}{a+b}+\frac {a^{3/2}}{\sqrt {b}\,\left (a+b\right )}+\frac {\sqrt {a}\,\sqrt {b}\,x^2}{a+b}+\frac {a^{3/2}\,x^2}{\sqrt {b}\,\left (a+b\right )}\right )}{2\,\sqrt {a}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.60, size = 117, normalized size = 2.17 \[ \left (\frac {1}{4 a} - \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) \log {\left (x^{2} + \frac {- 4 a b \left (\frac {1}{4 a} - \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) + a + b}{a} \right )} + \left (\frac {1}{4 a} + \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) \log {\left (x^{2} + \frac {- 4 a b \left (\frac {1}{4 a} + \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) + a + b}{a} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________