Optimal. Leaf size=252 \[ \frac {c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}+\sqrt {b} x^{-n/2}+\sqrt {c}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{5/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}+\sqrt {b} x^{-n/2}+\sqrt {c}\right )}{\sqrt {2} b^{9/4} n}+\frac {\sqrt {2} c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}-\frac {\sqrt {2} c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}+1\right )}{b^{9/4} n}+\frac {4 c x^{-n/4}}{b^2 n}-\frac {4 x^{-5 n/4}}{5 b n} \]
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Rubi [A] time = 0.22, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1584, 362, 345, 193, 321, 211, 1165, 628, 1162, 617, 204} \[ \frac {c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}+\sqrt {b} x^{-n/2}+\sqrt {c}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{5/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}+\sqrt {b} x^{-n/2}+\sqrt {c}\right )}{\sqrt {2} b^{9/4} n}+\frac {\sqrt {2} c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}-\frac {\sqrt {2} c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}+1\right )}{b^{9/4} n}+\frac {4 c x^{-n/4}}{b^2 n}-\frac {4 x^{-5 n/4}}{5 b n} \]
Antiderivative was successfully verified.
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Rule 193
Rule 204
Rule 211
Rule 321
Rule 345
Rule 362
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1-\frac {5 n}{4}}}{b+c x^n} \, dx\\ &=-\frac {4 x^{-5 n/4}}{5 b n}-\frac {c \int \frac {x^{-1-\frac {n}{4}}}{b+c x^n} \, dx}{b}\\ &=-\frac {4 x^{-5 n/4}}{5 b n}+\frac {(4 c) \operatorname {Subst}\left (\int \frac {1}{b+\frac {c}{x^4}} \, dx,x,x^{-n/4}\right )}{b n}\\ &=-\frac {4 x^{-5 n/4}}{5 b n}+\frac {(4 c) \operatorname {Subst}\left (\int \frac {x^4}{c+b x^4} \, dx,x,x^{-n/4}\right )}{b n}\\ &=-\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}-\frac {\left (4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+b x^4} \, dx,x,x^{-n/4}\right )}{b^2 n}\\ &=-\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}-\frac {\left (2 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {b} x^2}{c+b x^4} \, dx,x,x^{-n/4}\right )}{b^2 n}-\frac {\left (2 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {b} x^2}{c+b x^4} \, dx,x,x^{-n/4}\right )}{b^2 n}\\ &=-\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}+\frac {c^{5/4} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {c}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}+\frac {c^{5/4} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {c}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{3/2} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^{-n/4}\right )}{b^{5/2} n}-\frac {c^{3/2} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^{-n/4}\right )}{b^{5/2} n}\\ &=-\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}+\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {\left (\sqrt {2} c^{5/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}+\frac {\left (\sqrt {2} c^{5/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}\\ &=-\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}+\frac {\sqrt {2} c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}-\frac {\sqrt {2} c^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}+\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.13 \[ -\frac {4 x^{-5 n/4} \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\frac {c x^n}{b}\right )}{5 b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 259, normalized size = 1.03 \[ -\frac {4 \, b x^{5} x^{-\frac {5}{4} \, n - 5} + 20 \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {b^{7} c n^{3} x x^{-\frac {1}{4} \, n - 1} \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {3}{4}} - b^{7} n^{3} x \sqrt {\frac {b^{4} n^{2} \sqrt {-\frac {c^{5}}{b^{9} n^{4}}} + c^{2} x^{2} x^{-\frac {1}{2} \, n - 2}}{x^{2}}} \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {3}{4}}}{c^{5}}\right ) + 5 \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} + c x x^{-\frac {1}{4} \, n - 1}}{x}\right ) - 5 \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} - c x x^{-\frac {1}{4} \, n - 1}}{x}\right ) - 20 \, c x x^{-\frac {1}{4} \, n - 1}}{5 \, b^{2} n} \]
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 \[ \int \frac {x^{-\frac {1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 73, normalized size = 0.29 \[ \RootOf \left (b^{9} n^{4} \textit {\_Z}^{4}+c^{5}\right ) \ln \left (\frac {\RootOf \left (b^{9} n^{4} \textit {\_Z}^{4}+c^{5}\right )^{3} b^{7} n^{3}}{c^{4}}+x^{\frac {n}{4}}\right )-\frac {4 x^{-\frac {5 n}{4}}}{5 b n}+\frac {4 c \,x^{-\frac {n}{4}}}{b^{2} n} \]
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 \[ c^{2} \int \frac {x^{\frac {3}{4} \, n}}{b^{2} c x x^{n} + b^{3} x}\,{d x} + \frac {4 \, {\left (5 \, c x^{n} - b\right )}}{5 \, b^{2} n x^{\frac {5}{4} \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^{\frac {n}{4}+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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