Optimal. Leaf size=79 \[ \frac {b x^8 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac {a x^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1111, 646, 43} \[ \frac {b x^8 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac {a x^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 646
Rule 1111
Rubi steps
\begin {align*} \int x^5 \sqrt {a^2+2 a b x^2+b^2 x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int x^2 \left (a b+b^2 x\right ) \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \left (a b x^2+b^2 x^3\right ) \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )}\\ &=\frac {a x^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac {b x^8 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 39, normalized size = 0.49 \[ \frac {\sqrt {\left (a+b x^2\right )^2} \left (4 a x^6+3 b x^8\right )}{24 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.04, size = 13, normalized size = 0.16 \[ \frac {1}{8} \, b x^{8} + \frac {1}{6} \, a x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 29, normalized size = 0.37 \[ \frac {1}{8} \, b x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{6} \, a x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 36, normalized size = 0.46 \[ \frac {\left (3 b \,x^{2}+4 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x^{6}}{24 b \,x^{2}+24 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 13, normalized size = 0.16 \[ \frac {1}{8} \, b x^{8} + \frac {1}{6} \, a x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.45, size = 71, normalized size = 0.90 \[ \frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (a^3-4\,a^2\,b\,x^2-5\,a\,b^2\,x^4+3\,b\,x^2\,\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )\right )}{24\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 12, normalized size = 0.15 \[ \frac {a x^{6}}{6} + \frac {b x^{8}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________