Optimal. Leaf size=45 \[ -\frac {(c x)^m \, _2F_1\left (3,\frac {m-5}{2};\frac {m-3}{2};-\frac {c x^2}{b}\right )}{b^3 (5-m) x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 45, 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 = {1142, 1584, 364} \[ -\frac {(c x)^m \, _2F_1\left (3,\frac {m-5}{2};\frac {m-3}{2};-\frac {c x^2}{b}\right )}{b^3 (5-m) x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 1142
Rule 1584
Rubi steps
\begin {align*} \int \frac {(c x)^m}{\left (b x^2+c x^4\right )^3} \, dx &=\left (x^{-m} (c x)^m\right ) \operatorname {Subst}\left (\int \frac {x^m}{\left (b x^2+c x^4\right )^3} \, dx,x,x\right )\\ &=\left (x^{-m} (c x)^m\right ) \operatorname {Subst}\left (\int \frac {x^{-6+m}}{\left (b+c x^2\right )^3} \, dx,x,x\right )\\ &=-\frac {(c x)^m \, _2F_1\left (3,\frac {1}{2} (-5+m);\frac {1}{2} (-3+m);-\frac {c x^2}{b}\right )}{b^3 (5-m) x^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 44, normalized size = 0.98 \[ \frac {(c x)^m \, _2F_1\left (3,\frac {m-5}{2};\frac {m-5}{2}+1;-\frac {c x^2}{b}\right )}{b^3 (m-5) x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c x\right )^{m}}{c^{3} x^{12} + 3 \, b c^{2} x^{10} + 3 \, b^{2} c x^{8} + b^{3} x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{m}}{{\left (c x^{4} + b x^{2}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{m}}{\left (c \,x^{4}+b \,x^{2}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{m}}{{\left (c x^{4} + b x^{2}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x\right )}^m}{{\left (c\,x^4+b\,x^2\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{m}}{x^{6} \left (b + c x^{2}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________