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 = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {15, 368, 43} \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rule 368
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*}
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Mathematica [A] time = 0.05, size = 48, normalized size = 0.69 \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.86, size = 74, normalized size = 1.06 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 205, normalized size = 2.93 \begin {gather*} \frac {\left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}}+3 a \,{\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}\right ) x \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}}}{6 \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )^{3} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 109, normalized size = 1.56 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 127, normalized size = 1.81 \begin {gather*} \frac {x}{6\,a^2\,b\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}-\frac {x}{3\,b\,\left (b^3\,{\left (c\,x^n\right )}^{3/n}+a^3+3\,a\,b^2\,{\left (c\,x^n\right )}^{2/n}+3\,a^2\,b\,{\left (c\,x^n\right )}^{1/n}\right )}+\frac {x}{6\,a\,b\,\left (b^2\,{\left (c\,x^n\right )}^{2/n}+a^2+2\,a\,b\,{\left (c\,x^n\right )}^{1/n}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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