Optimal. Leaf size=70 \[ \frac{a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}-\frac{x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \]
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Rubi [A] time = 0.034735, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {15, 368, 43} \[ \frac{a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}-\frac{x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \]
Antiderivative was successfully verified.
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Rule 15
Rule 368
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c x^n\right )^{\frac{1}{n}}}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \int \frac{x}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5} \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x}{(a+b x)^5} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^5}+\frac{1}{b (a+b x)^4}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}-\frac{x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0248681, size = 48, normalized size = 0.69 \[ -\frac{x \left (c x^n\right )^{-1/n} \left (a+4 b \left (c x^n\right )^{\frac{1}{n}}\right )}{12 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.059, size = 316, normalized size = 4.5 \begin{align*}{\frac{x}{12\,{a}^{3}} \left ({b}^{2} \left ( \sqrt [n]{{x}^{n}} \right ) ^{3} \left ( \sqrt [n]{c} \right ) ^{3}{{\rm e}^{{\frac{-{\frac{3\,i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ({\it csgn} \left ( ic \right ) -{\it csgn} \left ( ic{x}^{n} \right ) \right ) \left ({\it csgn} \left ( i{x}^{n} \right ) -{\it csgn} \left ( ic{x}^{n} \right ) \right ) }{n}}}}+4\,ba \left ( \sqrt [n]{{x}^{n}} \right ) ^{2} \left ( \sqrt [n]{c} \right ) ^{2}{{\rm e}^{{\frac{-i{\it csgn} \left ( ic{x}^{n} \right ) \pi \, \left ({\it csgn} \left ( ic \right ) -{\it csgn} \left ( ic{x}^{n} \right ) \right ) \left ({\it csgn} \left ( i{x}^{n} \right ) -{\it csgn} \left ( ic{x}^{n} \right ) \right ) }{n}}}}+6\,{a}^{2}\sqrt [n]{{x}^{n}}\sqrt [n]{c}{{\rm e}^{{\frac{-i/2{\it csgn} \left ( ic{x}^{n} \right ) \pi \, \left ({\it csgn} \left ( ic \right ) -{\it csgn} \left ( ic{x}^{n} \right ) \right ) \left ({\it csgn} \left ( i{x}^{n} \right ) -{\it csgn} \left ( ic{x}^{n} \right ) \right ) }{n}}}} \right ) \left ( a+b{{\rm e}^{-{\frac{i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) +i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08316, size = 213, normalized size = 3.04 \begin{align*} \frac{b^{2} c^{\frac{3}{n}} x{\left (x^{n}\right )}^{\frac{3}{n}} + 4 \, a b c^{\frac{2}{n}} x{\left (x^{n}\right )}^{\frac{2}{n}} + 6 \, a^{2} c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}}{12 \,{\left (a^{3} b^{4} c^{\frac{4}{n}}{\left (x^{n}\right )}^{\frac{4}{n}} + 4 \, a^{4} b^{3} c^{\frac{3}{n}}{\left (x^{n}\right )}^{\frac{3}{n}} + 6 \, a^{5} b^{2} c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + 4 \, a^{6} b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66837, size = 177, normalized size = 2.53 \begin{align*} -\frac{4 \, b c^{\left (\frac{1}{n}\right )} x + a}{12 \,{\left (b^{6} c^{\frac{5}{n}} x^{4} + 4 \, a b^{5} c^{\frac{4}{n}} x^{3} + 6 \, a^{2} b^{4} c^{\frac{3}{n}} x^{2} + 4 \, a^{3} b^{3} c^{\frac{2}{n}} x + a^{4} b^{2} c^{\left (\frac{1}{n}\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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