Optimal. Leaf size=252 \[ \frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{\sqrt {2} a^{9/4} n}-\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{\sqrt {2} a^{9/4} n}+\frac {\sqrt {2} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}-\frac {\sqrt {2} b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{9/4} n}+\frac {4 b x^{-n/4}}{a^2 n}-\frac {4 x^{-5 n/4}}{5 a n} \]
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Rubi [A] time = 0.20, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {362, 345, 193, 321, 211, 1165, 628, 1162, 617, 204} \[ \frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{\sqrt {2} a^{9/4} n}-\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt {a} x^{-n/2}+\sqrt {b}\right )}{\sqrt {2} a^{9/4} n}+\frac {\sqrt {2} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}-\frac {\sqrt {2} b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{9/4} n}+\frac {4 b x^{-n/4}}{a^2 n}-\frac {4 x^{-5 n/4}}{5 a n} \]
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
Rubi steps
\begin {align*} \int \frac {x^{-1-\frac {5 n}{4}}}{a+b x^n} \, dx &=-\frac {4 x^{-5 n/4}}{5 a n}-\frac {b \int \frac {x^{-1-\frac {n}{4}}}{a+b x^n} \, dx}{a}\\ &=-\frac {4 x^{-5 n/4}}{5 a n}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{a+\frac {b}{x^4}} \, dx,x,x^{-n/4}\right )}{a n}\\ &=-\frac {4 x^{-5 n/4}}{5 a n}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {x^4}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a n}\\ &=-\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}-\frac {\left (4 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a^2 n}\\ &=-\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}-\frac {\left (2 b^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {a} x^2}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a^2 n}-\frac {\left (2 b^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {a} x^2}{b+a x^4} \, dx,x,x^{-n/4}\right )}{a^2 n}\\ &=-\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}+\frac {b^{5/4} \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^{9/4} n}+\frac {b^{5/4} \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^{9/4} n}-\frac {b^{3/2} \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^{5/2} n}-\frac {b^{3/2} \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^{5/2} n}\\ &=-\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}+\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{9/4} n}-\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{9/4} n}-\frac {\left (\sqrt {2} b^{5/4}\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^{9/4} n}+\frac {\left (\sqrt {2} b^{5/4}\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^{9/4} n}\\ &=-\frac {4 x^{-5 n/4}}{5 a n}+\frac {4 b x^{-n/4}}{a^2 n}+\frac {\sqrt {2} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}-\frac {\sqrt {2} b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}+\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{9/4} n}-\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {a} x^{-n/2}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}\right )}{\sqrt {2} a^{9/4} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.13 \[ -\frac {4 x^{-5 n/4} \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\frac {b x^n}{a}\right )}{5 a n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 270, normalized size = 1.07 \[ -\frac {20 \, a^{2} n \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {a^{7} b n^{3} x^{\frac {1}{5}} x^{-\frac {1}{4} \, n - \frac {1}{5}} \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {3}{4}} - a^{7} n^{3} x^{\frac {1}{5}} \sqrt {\frac {a^{4} n^{2} x^{\frac {3}{5}} \sqrt {-\frac {b^{5}}{a^{9} n^{4}}} + b^{2} x x^{-\frac {1}{2} \, n - \frac {2}{5}}}{x}} \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {3}{4}}}{b^{5}}\right ) + 5 \, a^{2} n \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} n x^{\frac {4}{5}} \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} + b x x^{-\frac {1}{4} \, n - \frac {1}{5}}}{x}\right ) - 5 \, a^{2} n \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{2} n x^{\frac {4}{5}} \left (-\frac {b^{5}}{a^{9} n^{4}}\right )^{\frac {1}{4}} - b x x^{-\frac {1}{4} \, n - \frac {1}{5}}}{x}\right ) + 4 \, a x x^{-\frac {5}{4} \, n - 1} - 20 \, b x^{\frac {1}{5}} x^{-\frac {1}{4} \, n - \frac {1}{5}}}{5 \, a^{2} n} \]
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 \[ \int \frac {x^{-\frac {5}{4} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 73, normalized size = 0.29 \[ \RootOf \left (a^{9} n^{4} \textit {\_Z}^{4}+b^{5}\right ) \ln \left (\frac {\RootOf \left (a^{9} n^{4} \textit {\_Z}^{4}+b^{5}\right )^{3} a^{7} n^{3}}{b^{4}}+x^{\frac {n}{4}}\right )-\frac {4 x^{-\frac {5 n}{4}}}{5 a n}+\frac {4 b \,x^{-\frac {n}{4}}}{a^{2} n} \]
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 \[ b^{2} \int \frac {x^{\frac {3}{4} \, n}}{a^{2} b x x^{n} + a^{3} x}\,{d x} + \frac {4 \, {\left (5 \, b x^{n} - a\right )}}{5 \, a^{2} n x^{\frac {5}{4} \, n}} \]
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 \[ \int \frac {1}{x^{\frac {5\,n}{4}+1}\,\left (a+b\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.75, size = 309, normalized size = 1.23 \[ \frac {x^{- \frac {5 n}{4}} \Gamma \left (- \frac {5}{4}\right )}{a n \Gamma \left (- \frac {1}{4}\right )} - \frac {5 b x^{- \frac {n}{4}} \Gamma \left (- \frac {5}{4}\right )}{a^{2} n \Gamma \left (- \frac {1}{4}\right )} + \frac {5 b^{\frac {5}{4}} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} + \frac {5 i b^{\frac {5}{4}} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {3 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} - \frac {5 b^{\frac {5}{4}} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {5 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} - \frac {5 i b^{\frac {5}{4}} e^{- \frac {3 i \pi }{4}} \log {\left (1 - \frac {\sqrt [4]{b} x^{\frac {n}{4}} e^{\frac {7 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac {5}{4}\right )}{4 a^{\frac {9}{4}} n \Gamma \left (- \frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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