Integrand size = 19, antiderivative size = 101 \[ \int \frac {x^3}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac {a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac {x^4 \left (c x^n\right )^{-1/n}}{3 b}-\frac {a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^4} \]
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Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {375, 45} \[ \int \frac {x^3}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=-\frac {a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^4}+\frac {a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac {a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac {x^4 \left (c x^n\right )^{-1/n}}{3 b} \]
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Rule 45
Rule 375
Rubi steps \begin{align*} \text {integral}& = \left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {x^3}{a+b x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac {a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac {x^4 \left (c x^n\right )^{-1/n}}{3 b}-\frac {a^3 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*}
Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.86 \[ \int \frac {x^3}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {x^4 \left (c x^n\right )^{-4/n} \left (b \left (c x^n\right )^{\frac {1}{n}} \left (6 a^2-3 a b \left (c x^n\right )^{\frac {1}{n}}+2 b^2 \left (c x^n\right )^{2/n}\right )-6 a^3 \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{6 b^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.42 (sec) , antiderivative size = 380, normalized size of antiderivative = 3.76
method | result | size |
risch | \(\frac {c^{-\frac {3}{n}} \left (x^{n}\right )^{-\frac {3}{n}} x^{3} \left (\frac {b^{2} \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} x \,{\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{3}-\frac {b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} a x \,{\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{2}+a^{2} x \,{\mathrm e}^{-\frac {3 i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}\right )}{b^{3}}-\frac {\ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) \left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} a^{3} c^{-\frac {3}{n}} \left (x^{n}\right )^{-\frac {3}{n}} x^{4} {\mathrm e}^{-\frac {2 i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{b^{4}}\) | \(380\) |
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none
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {2 \, b^{3} c^{\frac {3}{n}} x^{3} - 3 \, a b^{2} c^{\frac {2}{n}} x^{2} + 6 \, a^{2} b c^{\left (\frac {1}{n}\right )} x - 6 \, a^{3} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{6 \, b^{4} c^{\frac {4}{n}}} \]
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\[ \int \frac {x^3}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int \frac {x^{3}}{a + b \left (c x^{n}\right )^{\frac {1}{n}}}\, dx \]
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\[ \int \frac {x^3}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int { \frac {x^{3}}{\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a} \,d x } \]
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\[ \int \frac {x^3}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int { \frac {x^{3}}{\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a} \,d x } \]
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Timed out. \[ \int \frac {x^3}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\int \frac {x^3}{a+b\,{\left (c\,x^n\right )}^{1/n}} \,d x \]
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