Optimal. Leaf size=18 \[ \frac {x \left (c+d x^n\right )^{-1/n}}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {191} \[ \frac {x \left (c+d x^n\right )^{-1/n}}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rubi steps
\begin {align*} \int \left (c+d x^n\right )^{-1-\frac {1}{n}} \, dx &=\frac {x \left (c+d x^n\right )^{-1/n}}{c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 18, normalized size = 1.00 \[ \frac {x \left (c+d x^n\right )^{-1/n}}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.00, size = 31, normalized size = 1.72 \[ \frac {d x x^{n} + c x}{{\left (d x^{n} + c\right )}^{\frac {n + 1}{n}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 53, normalized size = 2.94 \[ \frac {d x \,{\mathrm e}^{n \ln \relax (x )} {\mathrm e}^{\left (-\frac {1}{n}-1\right ) \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )}}{c}+x \,{\mathrm e}^{\left (-\frac {1}{n}-1\right ) \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.76, size = 75, normalized size = 4.17 \[ \frac {d\,x^{n+1}\,\left (\frac {c}{d\,x^n}-{\left (\frac {c}{d\,x^n}+1\right )}^{\frac {n+1}{n}}+1\right )}{c\,n\,\left (\frac {n+1}{n}-1\right )\,{\left (c+d\,x^n\right )}^{\frac {n+1}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 33.05, size = 211, normalized size = 11.72 \[ \begin {cases} - \frac {d^{- \frac {1}{n}} x x^{- n} \left (x^{n}\right )^{- \frac {1}{n}}}{d n} & \text {for}\: c = 0 \\0^{-1 - \frac {1}{n}} x & \text {for}\: c = - d x^{n} \\x \left (0^{n}\right )^{-1 - \frac {1}{n}} & \text {for}\: c = 0^{n} - d x^{n} \\\frac {c^{2} x}{c^{3} \left (c + d x^{n}\right )^{\frac {1}{n}} + 2 c^{2} d x^{n} \left (c + d x^{n}\right )^{\frac {1}{n}} + c d^{2} x^{2 n} \left (c + d x^{n}\right )^{\frac {1}{n}}} + \frac {c d x x^{n}}{c^{3} \left (c + d x^{n}\right )^{\frac {1}{n}} + 2 c^{2} d x^{n} \left (c + d x^{n}\right )^{\frac {1}{n}} + c d^{2} x^{2 n} \left (c + d x^{n}\right )^{\frac {1}{n}}} + \frac {d x x^{n}}{c^{2} \left (c + d x^{n}\right )^{\frac {1}{n}} + c d x^{n} \left (c + d x^{n}\right )^{\frac {1}{n}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________