Optimal. Leaf size=31 \[ -\frac {2 (c x)^{-n/2} \sqrt {a+b x^n}}{a c n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {264} \[ -\frac {2 (c x)^{-n/2} \sqrt {a+b x^n}}{a c n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rubi steps
\begin {align*} \int \frac {(c x)^{-1-\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx &=-\frac {2 (c x)^{-n/2} \sqrt {a+b x^n}}{a c n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ -\frac {2 x (c x)^{-\frac {n}{2}-1} \sqrt {a+b x^n}}{a n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{-\frac {1}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{-\frac {n}{2}-1}}{\sqrt {b \,x^{n}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{-\frac {1}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.21, size = 29, normalized size = 0.94 \[ -\frac {2\,x\,\sqrt {a+b\,x^n}}{a\,n\,{\left (c\,x\right )}^{\frac {n}{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.45, size = 29, normalized size = 0.94 \[ - \frac {2 \sqrt {b} c^{- \frac {n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{a c n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________