Optimal. Leaf size=21 \[ -\frac {x}{c \sqrt {b x^2+c x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 21, 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 = {1602}
\begin {gather*} -\frac {x}{c \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1602
Rubi steps
\begin {align*} \int \frac {x^4}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {x}{c \sqrt {b x^2+c x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -\frac {x}{c \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 29, normalized size = 1.38
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) x^{3}}{c \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}\) | \(29\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) x^{3}}{c \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}\) | \(29\) |
trager | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}}}{\left (c \,x^{2}+b \right ) c x}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 14, normalized size = 0.67 \begin {gather*} -\frac {1}{\sqrt {c x^{2} + b} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 29, normalized size = 1.38 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}}}{c^{2} x^{3} + b c x} \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^{4}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 6.46, size = 28, normalized size = 1.33 \begin {gather*} \frac {\mathrm {sgn}\left (x\right )}{\sqrt {b} c} - \frac {1}{\sqrt {c x^{2} + b} c \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.15, size = 30, normalized size = 1.43 \begin {gather*} -\frac {\sqrt {c\,x^4+b\,x^2}}{c\,x\,\left (c\,x^2+b\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________