Optimal. Leaf size=252 \[ -\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}+\frac {\sqrt {2} c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}-\frac {\sqrt {2} c^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}+\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n} \]
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Rubi [A]
time = 0.16, 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 = {1598, 369,
352, 199, 327, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\sqrt {2} c^{5/4} \text {ArcTan}\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} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}+1\right )}{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 {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 {4 c x^{-n/4}}{b^2 n}-\frac {4 x^{-5 n/4}}{5 b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 210
Rule 217
Rule 327
Rule 352
Rule 369
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {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} \text {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} \text {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} \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 34, normalized size = 0.13 \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.22, size = 73, normalized size = 0.29
method | result | size |
risch | \(\frac {4 c \,x^{-\frac {n}{4}}}{b^{2} n}-\frac {4 x^{-\frac {5 n}{4}}}{5 b n}+\left (\munderset {\textit {\_R} =\RootOf \left (b^{9} n^{4} \textit {\_Z}^{4}+c^{5}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{4}}+\frac {b^{7} n^{3} \textit {\_R}^{3}}{c^{4}}\right )\right )\) | \(73\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 259, normalized size = 1.03 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{\frac {n}{4}+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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