Optimal. Leaf size=79 \[ -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )} \]
[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} \[ -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 646
Rule 1111
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \frac {a b+b^2 x}{x^5} \, 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 (\frac {a b}{x^5}+\frac {b^2}{x^4}\right ) \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )}\\ &=-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 39, normalized size = 0.49 \[ -\frac {\sqrt {\left (a+b x^2\right )^2} \left (3 a+4 b x^2\right )}{24 x^8 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.80, size = 15, normalized size = 0.19 \[ -\frac {4 \, b x^{2} + 3 \, a}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 31, normalized size = 0.39 \[ -\frac {4 \, b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, a \mathrm {sgn}\left (b x^{2} + a\right )}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 36, normalized size = 0.46 \[ -\frac {\left (4 b \,x^{2}+3 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{24 \left (b \,x^{2}+a \right ) x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 15, normalized size = 0.19 \[ -\frac {4 \, b x^{2} + 3 \, a}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.24, size = 35, normalized size = 0.44 \[ -\frac {\left (4\,b\,x^2+3\,a\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{24\,x^8\,\left (b\,x^2+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.20, size = 15, normalized size = 0.19 \[ \frac {- 3 a - 4 b x^{2}}{24 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________