Optimal. Leaf size=65 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{3/2}}+\frac {x}{8 b c \left (b+c x^2\right )}-\frac {x}{4 c \left (b+c x^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1584, 288, 199, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{3/2}}+\frac {x}{8 b c \left (b+c x^2\right )}-\frac {x}{4 c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 199
Rule 205
Rule 288
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^8}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^2}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac {x}{4 c \left (b+c x^2\right )^2}+\frac {\int \frac {1}{\left (b+c x^2\right )^2} \, dx}{4 c}\\ &=-\frac {x}{4 c \left (b+c x^2\right )^2}+\frac {x}{8 b c \left (b+c x^2\right )}+\frac {\int \frac {1}{b+c x^2} \, dx}{8 b c}\\ &=-\frac {x}{4 c \left (b+c x^2\right )^2}+\frac {x}{8 b c \left (b+c x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 58, normalized size = 0.89 \[ \frac {\frac {\sqrt {b} \sqrt {c} x \left (c x^2-b\right )}{\left (b+c x^2\right )^2}+\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 190, normalized size = 2.92 \[ \left [\frac {2 \, b c^{2} x^{3} - 2 \, b^{2} c x - {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right )}{16 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{3} c^{3} x^{2} + b^{4} c^{2}\right )}}, \frac {b c^{2} x^{3} - b^{2} c x + {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right )}{8 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{3} c^{3} x^{2} + b^{4} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 50, normalized size = 0.77 \[ \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b c} + \frac {c x^{3} - b x}{8 \, {\left (c x^{2} + b\right )}^{2} b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 49, normalized size = 0.75 \[ \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}\, b c}+\frac {\frac {x^{3}}{8 b}-\frac {x}{8 c}}{\left (c \,x^{2}+b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.86, size = 62, normalized size = 0.95 \[ \frac {c x^{3} - b x}{8 \, {\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )}} + \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.23, size = 55, normalized size = 0.85 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{8\,b^{3/2}\,c^{3/2}}-\frac {\frac {x}{8\,c}-\frac {x^3}{8\,b}}{b^2+2\,b\,c\,x^2+c^2\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.35, size = 110, normalized size = 1.69 \[ - \frac {\sqrt {- \frac {1}{b^{3} c^{3}}} \log {\left (- b^{2} c \sqrt {- \frac {1}{b^{3} c^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{b^{3} c^{3}}} \log {\left (b^{2} c \sqrt {- \frac {1}{b^{3} c^{3}}} + x \right )}}{16} + \frac {- b x + c x^{3}}{8 b^{3} c + 16 b^{2} c^{2} x^{2} + 8 b c^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________