Optimal. Leaf size=25 \[ -\frac {\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2014} \begin {gather*} -\frac {\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2014
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^9} \, dx &=-\frac {\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -\frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2}}{5 b x^{10}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.22, size = 46, normalized size = 1.84 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-b^2-2 b c x^2-c^2 x^4\right )}{5 b x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 2.06, size = 39, normalized size = 1.56 \begin {gather*} -\frac {{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{5 \, b x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.30, size = 92, normalized size = 3.68 \begin {gather*} \frac {2 \, {\left (5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b^{2} c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + b^{4} c^{\frac {5}{2}} \mathrm {sgn}\relax (x)\right )}}{5 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 29, normalized size = 1.16 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{5 b \,x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.41, size = 81, normalized size = 3.24 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}} c^{2}}{5 \, b x^{2}} + \frac {\sqrt {c x^{4} + b x^{2}} c}{10 \, x^{4}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}} b}{10 \, x^{6}} - \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{2 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.38, size = 30, normalized size = 1.20 \begin {gather*} -\frac {{\left (c\,x^2+b\right )}^2\,\sqrt {c\,x^4+b\,x^2}}{5\,b\,x^6} \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 {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________