Optimal. Leaf size=74 \[ \frac {1}{b x^2 \sqrt {b x^2+c x^4}}-\frac {4 \sqrt {b x^2+c x^4}}{3 b^2 x^4}+\frac {8 c \sqrt {b x^2+c x^4}}{3 b^3 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2040, 2041,
2039} \begin {gather*} \frac {8 c \sqrt {b x^2+c x^4}}{3 b^3 x^2}-\frac {4 \sqrt {b x^2+c x^4}}{3 b^2 x^4}+\frac {1}{b x^2 \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2039
Rule 2040
Rule 2041
Rubi steps
\begin {align*} \int \frac {1}{x \left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{b x^2 \sqrt {b x^2+c x^4}}+\frac {4 \int \frac {1}{x^3 \sqrt {b x^2+c x^4}} \, dx}{b}\\ &=\frac {1}{b x^2 \sqrt {b x^2+c x^4}}-\frac {4 \sqrt {b x^2+c x^4}}{3 b^2 x^4}-\frac {(8 c) \int \frac {1}{x \sqrt {b x^2+c x^4}} \, dx}{3 b^2}\\ &=\frac {1}{b x^2 \sqrt {b x^2+c x^4}}-\frac {4 \sqrt {b x^2+c x^4}}{3 b^2 x^4}+\frac {8 c \sqrt {b x^2+c x^4}}{3 b^3 x^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 46, normalized size = 0.62 \begin {gather*} \frac {-b^2+4 b c x^2+8 c^2 x^4}{3 b^3 x^2 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 45, normalized size = 0.61
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (-8 c^{2} x^{4}-4 b c \,x^{2}+b^{2}\right )}{3 b^{3} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}\) | \(45\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (-8 c^{2} x^{4}-4 b c \,x^{2}+b^{2}\right )}{3 b^{3} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}\) | \(45\) |
trager | \(-\frac {\left (-8 c^{2} x^{4}-4 b c \,x^{2}+b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3 \left (c \,x^{2}+b \right ) b^{3} x^{4}}\) | \(50\) |
risch | \(-\frac {\left (c \,x^{2}+b \right ) \left (-5 c \,x^{2}+b \right )}{3 b^{3} x^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {x^{2} c^{2}}{b^{3} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 65, normalized size = 0.88 \begin {gather*} \frac {8 \, c^{2} x^{2}}{3 \, \sqrt {c x^{4} + b x^{2}} b^{3}} + \frac {4 \, c}{3 \, \sqrt {c x^{4} + b x^{2}} b^{2}} - \frac {1}{3 \, \sqrt {c x^{4} + b x^{2}} b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 54, normalized size = 0.73 \begin {gather*} \frac {{\left (8 \, c^{2} x^{4} + 4 \, b c x^{2} - b^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{3 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} \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 {1}{x \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 = 5.83, size = 114, normalized size = 1.54 \begin {gather*} \frac {c^{2} x}{\sqrt {c x^{2} + b} b^{3} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} c^{\frac {3}{2}} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b c^{\frac {3}{2}} + 5 \, b^{2} c^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{3} b^{2} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.24, size = 51, normalized size = 0.69 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}\,\left (-b^2+4\,b\,c\,x^2+8\,c^2\,x^4\right )}{3\,b^3\,x^4\,\left (c\,x^2+b\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________