Optimal. Leaf size=26 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0089402, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {368, 36, 29, 31} \[ \frac{\log (x)}{a}-\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 368
Rule 36
Rule 29
Rule 31
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 = 0.0082195, size = 23, normalized size = 0.88 \[ \frac{\log (x)-\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 35, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( \sqrt [n]{c{x}^{n}} \right ) }{a}}-{\frac{\ln \left ( a+b\sqrt [n]{c{x}^{n}} \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02334, size = 54, normalized size = 2.08 \begin{align*} \frac{\log \left (x\right )}{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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.53546, size = 49, normalized size = 1.88 \begin{align*} -\frac{\log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) - \log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.53923, size = 56, normalized size = 2.15 \begin{align*} \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{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]