Optimal. Leaf size=26 \[ \frac {\log (x)}{a}-\frac {\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {368, 36, 29, 31} \begin {gather*} \frac {\log (x)}{a}-\frac {\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 368
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{a}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{a}\\ &=\frac {\log (x)}{a}-\frac {\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 23, normalized size = 0.88 \begin {gather*} \frac {\log (x)-\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.16, size = 37, normalized size = 1.42 \begin {gather*} \frac {\log \left (\left (c x^n\right )^{\frac {1}{n}}\right )}{a}-\frac {\log \left (a^2+a b \left (c x^n\right )^{\frac {1}{n}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.53, size = 21, normalized size = 0.81 \begin {gather*} -\frac {\log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) - \log \relax (x)}{a} \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*} \int \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 35, normalized size = 1.35 \begin {gather*} \frac {\ln \left (\left (c \,x^{n}\right )^{\frac {1}{n}}\right )}{a}-\frac {\ln \left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.65, size = 40, normalized size = 1.54 \begin {gather*} \frac {\log \relax (x)}{a} - \frac {\log \left (\frac {b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a}{b c^{\left (\frac {1}{n}\right )}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.22, size = 24, normalized size = 0.92 \begin {gather*} -\frac {\ln \left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )-\ln \relax (x)}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.55, size = 56, normalized size = 2.15 \begin {gather*} \begin {cases} \tilde {\infty } c^{- \frac {1}{n}} \left (x^{n}\right )^{- \frac {1}{n}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {c^{- \frac {1}{n}} \left (x^{n}\right )^{- \frac {1}{n}}}{b} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{a} - \frac {\log {\left (\frac {a}{b} + c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} \right )}}{a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________