Optimal. Leaf size=98 \[ \frac {3 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}}+\frac {3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2} \]
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Rubi [A] time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {254, 199, 205} \[ \frac {3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {3 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}}+\frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 254
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {\left (3 x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{4 a}\\ &=\frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {\left (3 x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{8 a^2}\\ &=\frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {3 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 91, normalized size = 0.93 \[ \frac {x \left (\frac {\sqrt {a} \left (5 a+3 b \left (c x^n\right )^{2/n}\right )}{\left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {3 \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{\sqrt {b}}\right )}{8 a^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 328, normalized size = 3.35 \[ \left [\frac {6 \, a b^{2} c^{\frac {4}{n}} x^{3} + 10 \, a^{2} b c^{\frac {2}{n}} x - 3 \, {\left (b^{2} c^{\frac {4}{n}} x^{4} + 2 \, a b c^{\frac {2}{n}} x^{2} + a^{2}\right )} \sqrt {-a b c^{\frac {2}{n}}} \log \left (\frac {b c^{\frac {2}{n}} x^{2} - 2 \, \sqrt {-a b c^{\frac {2}{n}}} x - a}{b c^{\frac {2}{n}} x^{2} + a}\right )}{16 \, {\left (a^{3} b^{3} c^{\frac {6}{n}} x^{4} + 2 \, a^{4} b^{2} c^{\frac {4}{n}} x^{2} + a^{5} b c^{\frac {2}{n}}\right )}}, \frac {3 \, a b^{2} c^{\frac {4}{n}} x^{3} + 5 \, a^{2} b c^{\frac {2}{n}} x + 3 \, {\left (b^{2} c^{\frac {4}{n}} x^{4} + 2 \, a b c^{\frac {2}{n}} x^{2} + a^{2}\right )} \sqrt {a b c^{\frac {2}{n}}} \arctan \left (\frac {\sqrt {a b c^{\frac {2}{n}}} x}{a}\right )}{8 \, {\left (a^{3} b^{3} c^{\frac {6}{n}} x^{4} + 2 \, a^{4} b^{2} c^{\frac {4}{n}} x^{2} + a^{5} b c^{\frac {2}{n}}\right )}}\right ] \]
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 \[ \int \frac {1}{{\left (\left (c x^{n}\right )^{\frac {2}{n}} b + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.43, size = 378, normalized size = 3.86 \[ \frac {\left (3 b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}}+5 a \right ) x}{8 \left (b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}}+a \right )^{2} a^{2}}+\frac {3 \arctan \left (\frac {b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}}}{\sqrt {\frac {a b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}}}{x^{2}}}\, x}\right )}{8 \sqrt {\frac {a b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}}}{x^{2}}}\, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3 \, b c^{\frac {2}{n}} x {\left (x^{n}\right )}^{\frac {2}{n}} + 5 \, a x}{8 \, {\left (a^{2} b^{2} c^{\frac {4}{n}} {\left (x^{n}\right )}^{\frac {4}{n}} + 2 \, a^{3} b c^{\frac {2}{n}} {\left (x^{n}\right )}^{\frac {2}{n}} + a^{4}\right )}} + 3 \, \int \frac {1}{8 \, {\left (a^{2} b c^{\frac {2}{n}} {\left (x^{n}\right )}^{\frac {2}{n}} + a^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+b\,{\left (c\,x^n\right )}^{2/n}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \left (c x^{n}\right )^{\frac {2}{n}}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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