Optimal. Leaf size=32 \[ \frac {x \left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]
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Rubi [A] time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {15, 368, 37} \[ \frac {x \left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
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 )^3} \, 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 )^3} \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x}{(a+b x)^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {x \left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ \frac {x \left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 56, normalized size = 1.75 \[ -\frac {2 \, b c^{\left (\frac {1}{n}\right )} x + a}{2 \, {\left (b^{4} c^{\frac {3}{n}} x^{2} + 2 \, a b^{3} c^{\frac {2}{n}} x + a^{2} b^{2} c^{\left (\frac {1}{n}\right )}\right )}} \]
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 \[ \int \frac {\left (c x^{n}\right )^{\left (\frac {1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 137, normalized size = 4.28 \[ \frac {x \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}}{2 \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 60, normalized size = 1.88 \[ \frac {c^{\left (\frac {1}{n}\right )} x {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}}{2 \, {\left (a b^{2} c^{\frac {2}{n}} {\left (x^{n}\right )}^{\frac {2}{n}} + 2 \, a^{2} b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 30, normalized size = 0.94 \[ \frac {x\,{\left (c\,x^n\right )}^{1/n}}{2\,a\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 178.51, size = 160, normalized size = 5.00 \[ \begin {cases} \tilde {\infty } c^{- \frac {2}{n}} x \left (x^{n}\right )^{- \frac {2}{n}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {x}{b^{3} \left (2 \cdot 0^{n} \tilde {\infty }^{n} \left (0^{n}\right )^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} - \left (0^{n}\right )^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}}\right )} & \text {for}\: a = 0 \wedge c = 0^{n} \\- \frac {c^{- \frac {2}{n}} x \left (x^{n}\right )^{- \frac {2}{n}}}{b^{3}} & \text {for}\: a = 0 \\\tilde {\infty } c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}} & \text {for}\: a = - b c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} \\\frac {c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}}}{2 a^{3} + 4 a^{2} b c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} + 2 a b^{2} c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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