Optimal. Leaf size=70 \[ \frac{a x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}-\frac{x \left (c x^n\right )^{-1/n}}{2 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]
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Rubi [A] time = 0.0347402, 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} \[ \frac{a x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}-\frac{x \left (c x^n\right )^{-1/n}}{2 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]
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 )^4} \, 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 )^4} \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x}{(a+b x)^4} \, 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)^4}+\frac{1}{b (a+b x)^3}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{a x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}-\frac{x \left (c x^n\right )^{-1/n}}{2 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0236064, size = 48, normalized size = 0.69 \[ -\frac{x \left (c x^n\right )^{-1/n} \left (a+3 b \left (c x^n\right )^{\frac{1}{n}}\right )}{6 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.079, size = 242, normalized size = 3.5 \begin{align*}{\frac{x}{6\,{a}^{2}} \left ( b \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}}}}+3\,a\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 ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0759, size = 147, normalized size = 2.1 \begin{align*} \frac{b c^{\frac{2}{n}} x{\left (x^{n}\right )}^{\frac{2}{n}} + 3 \, a c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}}{6 \,{\left (a^{2} b^{3} c^{\frac{3}{n}}{\left (x^{n}\right )}^{\frac{3}{n}} + 3 \, a^{3} b^{2} c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + 3 \, a^{4} b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65935, size = 143, normalized size = 2.04 \begin{align*} -\frac{3 \, b c^{\left (\frac{1}{n}\right )} x + a}{6 \,{\left (b^{5} c^{\frac{4}{n}} x^{3} + 3 \, a b^{4} c^{\frac{3}{n}} x^{2} + 3 \, a^{2} b^{3} c^{\frac{2}{n}} x + a^{3} 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 )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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