Optimal. Leaf size=41 \[ \frac {x^4}{4 a \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.68, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1111, 640, 607} \begin {gather*} \frac {a}{4 b^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 607
Rule 640
Rule 1111
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {1}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right )}{2 b}\\ &=-\frac {1}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a}{4 b^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 39, normalized size = 0.95 \begin {gather*} \frac {-a-2 b x^2}{4 b^2 \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 0.57, size = 157, normalized size = 3.83 \begin {gather*} \frac {-a^3 b+\sqrt {b^2} \left (-a^2+a b x^2-2 b^2 x^4\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}+a b^3 x^4+2 b^4 x^6}{2 x^4 \left (-2 a b^5-2 b^6 x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}+2 \sqrt {b^2} x^4 \left (2 a^2 b^4+4 a b^5 x^2+2 b^6 x^4\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 36, normalized size = 0.88 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 32, normalized size = 0.78 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b x^{2} + a\right )}^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 32, normalized size = 0.78 \begin {gather*} -\frac {\left (b \,x^{2}+a \right ) \left (2 b \,x^{2}+a \right )}{4 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.38, size = 36, normalized size = 0.88 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.24, size = 42, normalized size = 1.02 \begin {gather*} -\frac {\left (2\,b\,x^2+a\right )\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,b^2\,{\left (b\,x^2+a\right )}^3} \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}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________