Optimal. Leaf size=47 \[ -\frac {(c x)^{-1+m} \, _2F_1\left (1,\frac {1}{2} (-1+m);\frac {1+m}{2};-\frac {b x^2}{a}\right )}{a c (1-m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {371}
\begin {gather*} -\frac {(c x)^{m-1} \, _2F_1\left (1,\frac {m-1}{2};\frac {m+1}{2};-\frac {b x^2}{a}\right )}{a c (1-m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rubi steps
\begin {align*} \int \frac {(c x)^{-2+m}}{a+b x^2} \, dx &=-\frac {(c x)^{-1+m} \, _2F_1\left (1,\frac {1}{2} (-1+m);\frac {1+m}{2};-\frac {b x^2}{a}\right )}{a c (1-m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 44, normalized size = 0.94 \begin {gather*} \frac {x (c x)^{-2+m} \, _2F_1\left (1,\frac {1}{2} (-1+m);1+\frac {1}{2} (-1+m);-\frac {b x^2}{a}\right )}{a (-1+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (c x \right )^{-2+m}}{b \,x^{2}+a}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 4.96, size = 102, normalized size = 2.17 \begin {gather*} \frac {c^{m} m x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} - \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} - \frac {1}{2}\right )}{4 a c^{2} x \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )} - \frac {c^{m} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} - \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} - \frac {1}{2}\right )}{4 a c^{2} x \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (c\,x\right )}^{m-2}}{b\,x^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________