Optimal. Leaf size=68 \[ -\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} n}+\frac {2 b x^{-n/2}}{a^2 n}-\frac {2 x^{-3 n/2}}{3 a n} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {362, 345, 193, 321, 205} \[ -\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} n}+\frac {2 b x^{-n/2}}{a^2 n}-\frac {2 x^{-3 n/2}}{3 a n} \]
Antiderivative was successfully verified.
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Rule 193
Rule 205
Rule 321
Rule 345
Rule 362
Rubi steps
\begin {align*} \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx &=-\frac {2 x^{-3 n/2}}{3 a n}-\frac {b \int \frac {x^{-1-\frac {n}{2}}}{a+b x^n} \, dx}{a}\\ &=-\frac {2 x^{-3 n/2}}{3 a n}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{a+\frac {b}{x^2}} \, dx,x,x^{-n/2}\right )}{a n}\\ &=-\frac {2 x^{-3 n/2}}{3 a n}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {x^2}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a n}\\ &=-\frac {2 x^{-3 n/2}}{3 a n}+\frac {2 b x^{-n/2}}{a^2 n}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a^2 n}\\ &=-\frac {2 x^{-3 n/2}}{3 a n}+\frac {2 b x^{-n/2}}{a^2 n}-\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.50 \[ -\frac {2 x^{-3 n/2} \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {b x^n}{a}\right )}{3 a n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 251, normalized size = 3.69 \[ \left [-\frac {2 \, a x x^{-\frac {3}{2} \, n - 1} - 3 \, b \sqrt {-\frac {b}{a}} \log \left (-\frac {2 \, a^{3} x^{\frac {5}{3}} x^{-\frac {5}{2} \, n - \frac {5}{3}} \sqrt {-\frac {b}{a}} - a^{3} x^{2} x^{-3 \, n - 2} - 2 \, a^{2} b x x^{-\frac {3}{2} \, n - 1} \sqrt {-\frac {b}{a}} + 2 \, a^{2} b x^{\frac {4}{3}} x^{-2 \, n - \frac {4}{3}} + 2 \, a b^{2} x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}} \sqrt {-\frac {b}{a}} - 2 \, a b^{2} x^{\frac {2}{3}} x^{-n - \frac {2}{3}} + b^{3}}{a^{3} x^{2} x^{-3 \, n - 2} + b^{3}}\right ) - 6 \, b x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}}}{3 \, a^{2} n}, -\frac {2 \, {\left (a x x^{-\frac {3}{2} \, n - 1} - 3 \, b \sqrt {\frac {b}{a}} \arctan \left (\frac {\sqrt {\frac {b}{a}}}{x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}}}\right ) - 3 \, b x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}}\right )}}{3 \, a^{2} n}\right ] \]
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 \[ \int \frac {x^{-\frac {3}{2} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 97, normalized size = 1.43 \[ -\frac {2 x^{-\frac {3 n}{2}}}{3 a n}+\frac {2 b \,x^{-\frac {n}{2}}}{a^{2} n}-\frac {\sqrt {-a b}\, b \ln \left (x^{\frac {n}{2}}-\frac {\sqrt {-a b}}{b}\right )}{a^{3} n}+\frac {\sqrt {-a b}\, b \ln \left (x^{\frac {n}{2}}+\frac {\sqrt {-a b}}{b}\right )}{a^{3} n} \]
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 \[ b^{2} \int \frac {x^{\frac {1}{2} \, n}}{a^{2} b x x^{n} + a^{3} x}\,{d x} + \frac {2 \, {\left (3 \, b x^{n} - a\right )}}{3 \, a^{2} n x^{\frac {3}{2} \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{\frac {3\,n}{2}+1}\,\left (a+b\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.66, size = 56, normalized size = 0.82 \[ - \frac {2 x^{- \frac {3 n}{2}}}{3 a n} + \frac {2 b x^{- \frac {n}{2}}}{a^{2} n} + \frac {2 b^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{a^{\frac {5}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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