Integrand size = 25, antiderivative size = 70 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5} \, dx=\frac {a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4}-\frac {x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3} \]
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Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {15, 375, 45} \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5} \, dx=\frac {a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4}-\frac {x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3} \]
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Rule 15
Rule 45
Rule 375
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{\frac {1}{n}} \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5} \, dx}{x} \\ & = \left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x}{(a+b x)^5} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^5}+\frac {1}{b (a+b x)^4}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4}-\frac {x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.69 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5} \, dx=-\frac {x \left (c x^n\right )^{-1/n} \left (a+4 b \left (c x^n\right )^{\frac {1}{n}}\right )}{12 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4} \]
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Time = 7.52 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.27
method | result | size |
parallelrisch | \(\frac {b^{6} \left (c \,x^{n}\right )^{\frac {3}{n}} x^{2} a +4 b^{5} \left (c \,x^{n}\right )^{\frac {2}{n}} x^{2} a^{2}+6 x^{2} \left (c \,x^{n}\right )^{\frac {1}{n}} a^{3} b^{4}}{12 a^{4} b^{4} x {\left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )}^{4}}\) | \(89\) |
risch | \(\frac {x \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} \left (c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} b^{2} {\mathrm e}^{\frac {3 i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+4 \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} a b \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}+6 a^{2} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}\right )}{12 a^{3} {\left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}^{4}}\) | \(279\) |
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Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.31 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5} \, dx=-\frac {4 \, b c^{\left (\frac {1}{n}\right )} x + a}{12 \, {\left (b^{6} c^{\frac {5}{n}} x^{4} + 4 \, a b^{5} c^{\frac {4}{n}} x^{3} + 6 \, a^{2} b^{4} c^{\frac {3}{n}} x^{2} + 4 \, a^{3} b^{3} c^{\frac {2}{n}} x + a^{4} b^{2} c^{\left (\frac {1}{n}\right )}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (58) = 116\).
Time = 8.62 (sec) , antiderivative size = 306, normalized size of antiderivative = 4.37 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5} \, dx=\begin {cases} \tilde {\infty } x \left (c x^{n}\right )^{- \frac {4}{n}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {x \left (c x^{n}\right )^{- \frac {4}{n}}}{3 b^{5}} & \text {for}\: a = 0 \\\tilde {\infty } x \left (c x^{n}\right )^{\frac {1}{n}} & \text {for}\: b = - a \left (c x^{n}\right )^{- \frac {1}{n}} \\\frac {6 a^{2} x \left (c x^{n}\right )^{\frac {1}{n}}}{12 a^{7} + 48 a^{6} b \left (c x^{n}\right )^{\frac {1}{n}} + 72 a^{5} b^{2} \left (c x^{n}\right )^{\frac {2}{n}} + 48 a^{4} b^{3} \left (c x^{n}\right )^{\frac {3}{n}} + 12 a^{3} b^{4} \left (c x^{n}\right )^{\frac {4}{n}}} + \frac {4 a b x \left (c x^{n}\right )^{\frac {2}{n}}}{12 a^{7} + 48 a^{6} b \left (c x^{n}\right )^{\frac {1}{n}} + 72 a^{5} b^{2} \left (c x^{n}\right )^{\frac {2}{n}} + 48 a^{4} b^{3} \left (c x^{n}\right )^{\frac {3}{n}} + 12 a^{3} b^{4} \left (c x^{n}\right )^{\frac {4}{n}}} + \frac {b^{2} x \left (c x^{n}\right )^{\frac {3}{n}}}{12 a^{7} + 48 a^{6} b \left (c x^{n}\right )^{\frac {1}{n}} + 72 a^{5} b^{2} \left (c x^{n}\right )^{\frac {2}{n}} + 48 a^{4} b^{3} \left (c x^{n}\right )^{\frac {3}{n}} + 12 a^{3} b^{4} \left (c x^{n}\right )^{\frac {4}{n}}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (66) = 132\).
Time = 0.25 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.26 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5} \, dx=\frac {b^{2} c^{\frac {3}{n}} x {\left (x^{n}\right )}^{\frac {3}{n}} + 4 \, a b c^{\frac {2}{n}} x {\left (x^{n}\right )}^{\frac {2}{n}} + 6 \, a^{2} c^{\left (\frac {1}{n}\right )} x {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}}{12 \, {\left (a^{3} b^{4} c^{\frac {4}{n}} {\left (x^{n}\right )}^{\frac {4}{n}} + 4 \, a^{4} b^{3} c^{\frac {3}{n}} {\left (x^{n}\right )}^{\frac {3}{n}} + 6 \, a^{5} b^{2} c^{\frac {2}{n}} {\left (x^{n}\right )}^{\frac {2}{n}} + 4 \, a^{6} b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{7}\right )}} \]
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\[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5} \, dx=\int { \frac {\left (c x^{n}\right )^{\left (\frac {1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{5}} \,d x } \]
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Time = 5.35 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.97 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5} \, dx=\frac {x}{12\,a\,b\,\left (b^3\,{\left (c\,x^n\right )}^{3/n}+a^3+3\,a\,b^2\,{\left (c\,x^n\right )}^{2/n}+3\,a^2\,b\,{\left (c\,x^n\right )}^{1/n}\right )}-\frac {x}{4\,b\,\left (b^4\,{\left (c\,x^n\right )}^{4/n}+a^4+4\,a\,b^3\,{\left (c\,x^n\right )}^{3/n}+6\,a^2\,b^2\,{\left (c\,x^n\right )}^{2/n}+4\,a^3\,b\,{\left (c\,x^n\right )}^{1/n}\right )}+\frac {x}{12\,a^3\,b\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}+\frac {x}{12\,a^2\,b\,\left (b^2\,{\left (c\,x^n\right )}^{2/n}+a^2+2\,a\,b\,{\left (c\,x^n\right )}^{1/n}\right )} \]
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