Optimal. Leaf size=114 \[ \frac {a^3 x^4 \left (c x^n\right )^{-4/n}}{b^4 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {3 a^2 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^4}-\frac {2 a x^4 \left (c x^n\right )^{-3/n}}{b^3}+\frac {x^4 \left (c x^n\right )^{-2/n}}{2 b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {368, 43} \[ \frac {a^3 x^4 \left (c x^n\right )^{-4/n}}{b^4 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {3 a^2 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^4}-\frac {2 a x^4 \left (c x^n\right )^{-3/n}}{b^3}+\frac {x^4 \left (c x^n\right )^{-2/n}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 368
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx &=\left (x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\left (x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname {Subst}\left (\int \left (-\frac {2 a}{b^3}+\frac {x}{b^2}-\frac {a^3}{b^3 (a+b x)^2}+\frac {3 a^2}{b^3 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=-\frac {2 a x^4 \left (c x^n\right )^{-3/n}}{b^3}+\frac {x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac {a^3 x^4 \left (c x^n\right )^{-4/n}}{b^4 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {3 a^2 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 89, normalized size = 0.78 \[ \frac {x^4 \left (c x^n\right )^{-4/n} \left (\frac {2 a^3}{a+b \left (c x^n\right )^{\frac {1}{n}}}+6 a^2 \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )-4 a b \left (c x^n\right )^{\frac {1}{n}}+b^2 \left (c x^n\right )^{2/n}\right )}{2 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 105, normalized size = 0.92 \[ \frac {b^{3} c^{\frac {3}{n}} x^{3} - 3 \, a b^{2} c^{\frac {2}{n}} x^{2} - 4 \, a^{2} b c^{\left (\frac {1}{n}\right )} x + 2 \, a^{3} + 6 \, {\left (a^{2} b c^{\left (\frac {1}{n}\right )} x + a^{3}\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{2 \, {\left (b^{5} c^{\frac {5}{n}} x + a b^{4} c^{\frac {4}{n}}\right )}} \]
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 \[ \int \frac {x^{3}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 463, normalized size = 4.06 \[ \frac {3 a^{2} x^{4} c^{-\frac {3}{n}} c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {3}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {2 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}} \ln \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 )}{b^{4}}-\frac {3 a \,x^{4} c^{-\frac {3}{n}} \left (x^{n}\right )^{-\frac {3}{n}} {\mathrm e}^{-\frac {3 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}}}{b^{3}}-\frac {x^{4} 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 b}+\frac {3 x^{4} 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}}}{2 b^{2}}+\frac {x^{4}}{\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 ) a} \]
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 \[ \frac {x^{4}}{a b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2}} - 3 \, \int \frac {x^{3}}{a b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2}}\,{d x} \]
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 \[ \int \frac {x^3}{{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \,d x \]
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 \[ \int \frac {x^{3}}{\left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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