Optimal. Leaf size=234 \[ -\frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{\sqrt {2} a^{5/4} n}+\frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{\sqrt {2} a^{5/4} n}-\frac {\sqrt {2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} n}+\frac {\sqrt {2} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{5/4} n}-\frac {4 x^{-n/4}}{a n} \]
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Rubi [A] time = 0.18, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {345, 193, 321, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{\sqrt {2} a^{5/4} n}+\frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{\sqrt {2} a^{5/4} n}-\frac {\sqrt {2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} n}+\frac {\sqrt {2} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{5/4} n}-\frac {4 x^{-n/4}}{a n} \]
Antiderivative was successfully verified.
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Rule 193
Rule 204
Rule 211
Rule 321
Rule 345
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{-1-\frac {n}{4}}}{a+b x^n} \, dx &=-\frac {4 \operatorname {Subst}\left (\int \frac {1}{a+\frac {b}{x^4}} \, dx,x,x^{-n/4}\right )}{n}\\ &=-\frac {4 \operatorname {Subst}\left (\int \frac {x^4}{b+a x^4} \, dx,x,x^{-n/4}\right )}{n}\\ &=-\frac {4 x^{-n/4}}{a n}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a n}\\ &=-\frac {4 x^{-n/4}}{a n}+\frac {\left (2 \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {a} x^2}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a n}+\frac {\left (2 \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {a} x^2}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a n}\\ &=-\frac {4 x^{-n/4}}{a n}-\frac {\sqrt [4]{b} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{a}}+2 x}{-\frac {\sqrt {b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx,x,x^{-n/4}\right )}{\sqrt {2} a^{5/4} n}-\frac {\sqrt [4]{b} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{a}}-2 x}{-\frac {\sqrt {b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx,x,x^{-n/4}\right )}{\sqrt {2} a^{5/4} n}+\frac {\sqrt {b} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx,x,x^{-n/4}\right )}{a^{3/2} n}+\frac {\sqrt {b} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx,x,x^{-n/4}\right )}{a^{3/2} n}\\ &=-\frac {4 x^{-n/4}}{a n}-\frac {\sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{5/4} n}+\frac {\sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{5/4} n}+\frac {\left (\sqrt {2} \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} n}-\frac {\left (\sqrt {2} \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} n}\\ &=-\frac {4 x^{-n/4}}{a n}-\frac {\sqrt {2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} n}+\frac {\sqrt {2} \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} n}-\frac {\sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{5/4} n}+\frac {\sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{5/4} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 32, normalized size = 0.14 \[ -\frac {4 x^{-n/4} \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\frac {b x^n}{a}\right )}{a n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 211, normalized size = 0.90 \[ \frac {4 \, a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {a^{4} n^{3} x x^{-\frac {1}{4} \, n - 1} \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {3}{4}} - a^{4} n^{3} x \sqrt {\frac {a^{2} n^{2} \sqrt {-\frac {b}{a^{5} n^{4}}} + x^{2} x^{-\frac {1}{2} \, n - 2}}{x^{2}}} \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {3}{4}}}{b}\right ) + a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} + x x^{-\frac {1}{4} \, n - 1}}{x}\right ) - a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} - x x^{-\frac {1}{4} \, n - 1}}{x}\right ) - 4 \, x x^{-\frac {1}{4} \, n - 1}}{a n} \]
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 \[ \int \frac {x^{-\frac {1}{4} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 56, normalized size = 0.24 \[ \RootOf \left (a^{5} n^{4} \textit {\_Z}^{4}+b \right ) \ln \left (-\frac {\RootOf \left (a^{5} n^{4} \textit {\_Z}^{4}+b \right )^{3} a^{4} n^{3}}{b}+x^{\frac {n}{4}}\right )-\frac {4 x^{-\frac {n}{4}}}{a n} \]
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 \[ -b \int \frac {x^{\frac {3}{4} \, n}}{a b x x^{n} + a^{2} x}\,{d x} - \frac {4}{a n x^{\frac {1}{4} \, n}} \]
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 \[ \int \frac {1}{x^{\frac {n}{4}+1}\,\left (a+b\,x^n\right )} \,d x \]
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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