Optimal. Leaf size=34 \[ -\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {260, 32}
\begin {gather*} -\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 260
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{(a+b x)^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=-\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 34, normalized size = 1.00 \begin {gather*} -\frac {x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.27, size = 143, normalized size = 4.21
method | result | size |
risch | \(\frac {x \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (\mathrm {csgn}\left (i c \,x^{n}\right )-\mathrm {csgn}\left (i x^{n}\right )\right )}{2 n}}+2 a \right )}{2 a^{2} \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (\mathrm {csgn}\left (i c \,x^{n}\right )-\mathrm {csgn}\left (i x^{n}\right )\right )}{2 n}}+a \right )^{2}}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs.
\(2 (32) = 64\).
time = 0.35, size = 69, normalized size = 2.03 \begin {gather*} \frac {b c^{\left (\frac {1}{n}\right )} x {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + 2 \, a x}{2 \, {\left (a^{2} b^{2} c^{\frac {2}{n}} {\left (x^{n}\right )}^{\frac {2}{n}} + 2 \, a^{3} b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 43, normalized size = 1.26 \begin {gather*} -\frac {1}{2 \, {\left (b^{3} c^{\frac {3}{n}} x^{2} + 2 \, a b^{2} c^{\frac {2}{n}} x + a^{2} b c^{\left (\frac {1}{n}\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs.
\(2 (27) = 54\).
time = 4.52, size = 107, normalized size = 3.15 \begin {gather*} \begin {cases} \frac {2 a x}{2 a^{4} + 4 a^{3} b \left (c x^{n}\right )^{\frac {1}{n}} + 2 a^{2} b^{2} \left (c x^{n}\right )^{\frac {2}{n}}} + \frac {b x \left (c x^{n}\right )^{\frac {1}{n}}}{2 a^{4} + 4 a^{3} b \left (c x^{n}\right )^{\frac {1}{n}} + 2 a^{2} b^{2} \left (c x^{n}\right )^{\frac {2}{n}}} & \text {for}\: a \neq 0 \\- \frac {x \left (c x^{n}\right )^{- \frac {3}{n}}}{2 b^{3}} & \text {otherwise} \end {cases} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 36, normalized size = 1.06 \begin {gather*} \frac {x\,\left (2\,a+b\,{\left (c\,x^n\right )}^{1/n}\right )}{2\,a^2\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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