Optimal. Leaf size=210 \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0948812, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {254, 199, 200, 31, 634, 617, 204, 628} \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 254
Rule 199
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^3\right )^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{\left (2 x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{3 a}\\ &=\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{\left (2 x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3}}+\frac{\left (2 x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3}}\\ &=\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{3 a^{4/3}}-\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}\\ &=\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac{2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{\left (2 x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt [3]{a}}\right )}{3 a^{5/3} \sqrt [3]{b}}\\ &=\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )}-\frac{2 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.223498, size = 158, normalized size = 0.75 \[ \frac{x \left (\frac{1}{a+b \left (c x^n\right )^{3/n}}-\frac{\left (c x^n\right )^{-1/n} \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 2.388, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{-2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x}{3 \,{\left (a b c^{\frac{3}{n}}{\left (x^{n}\right )}^{\frac{3}{n}} + a^{2}\right )}} + 2 \, \int \frac{1}{3 \,{\left (a b c^{\frac{3}{n}}{\left (x^{n}\right )}^{\frac{3}{n}} + a^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.39486, size = 1413, normalized size = 6.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \left (c x^{n}\right )^{\frac{3}{n}}\right )^{2}}\, dx \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 )^{\frac{3}{n}} b + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]