Optimal. Leaf size=100 \[ -\frac {2 b^{3/2} x^{3 n/2} (c x)^{-3 n/2} \tan ^{-1}\left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} c n}+\frac {2 b x^n (c x)^{-3 n/2}}{a^2 c n}-\frac {2 (c x)^{-3 n/2}}{3 a c n} \]
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Rubi [A] time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {363, 362, 345, 193, 321, 205} \[ -\frac {2 b^{3/2} x^{3 n/2} (c x)^{-3 n/2} \tan ^{-1}\left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} c n}+\frac {2 b x^n (c x)^{-3 n/2}}{a^2 c n}-\frac {2 (c x)^{-3 n/2}}{3 a c n} \]
Antiderivative was successfully verified.
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Rule 193
Rule 205
Rule 321
Rule 345
Rule 362
Rule 363
Rubi steps
\begin {align*} \int \frac {(c x)^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx &=\frac {\left (x^{3 n/2} (c x)^{-3 n/2}\right ) \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx}{c}\\ &=-\frac {2 (c x)^{-3 n/2}}{3 a c n}-\frac {\left (b x^{3 n/2} (c x)^{-3 n/2}\right ) \int \frac {x^{-1-\frac {n}{2}}}{a+b x^n} \, dx}{a c}\\ &=-\frac {2 (c x)^{-3 n/2}}{3 a c n}+\frac {\left (2 b x^{3 n/2} (c x)^{-3 n/2}\right ) \operatorname {Subst}\left (\int \frac {1}{a+\frac {b}{x^2}} \, dx,x,x^{-n/2}\right )}{a c n}\\ &=-\frac {2 (c x)^{-3 n/2}}{3 a c n}+\frac {\left (2 b x^{3 n/2} (c x)^{-3 n/2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a c n}\\ &=-\frac {2 (c x)^{-3 n/2}}{3 a c n}+\frac {2 b x^n (c x)^{-3 n/2}}{a^2 c n}-\frac {\left (2 b^2 x^{3 n/2} (c x)^{-3 n/2}\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a^2 c n}\\ &=-\frac {2 (c x)^{-3 n/2}}{3 a c n}+\frac {2 b x^n (c x)^{-3 n/2}}{a^2 c n}-\frac {2 b^{3/2} x^{3 n/2} (c x)^{-3 n/2} \tan ^{-1}\left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} c n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.39 \[ -\frac {2 x (c x)^{-\frac {3 n}{2}-1} \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {b x^n}{a}\right )}{3 a n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 492, normalized size = 4.92 \[ \left [\frac {3 \, b c^{-n - \frac {2}{3}} \sqrt {-\frac {b c^{-n - \frac {2}{3}}}{a}} \log \left (-\frac {2 \, a^{2} b c^{-n - \frac {2}{3}} x^{\frac {4}{3}} e^{\left (-\frac {2}{3} \, {\left (3 \, n + 2\right )} \log \relax (c) - \frac {2}{3} \, {\left (3 \, n + 2\right )} \log \relax (x)\right )} - a^{3} x^{2} e^{\left (-{\left (3 \, n + 2\right )} \log \relax (c) - {\left (3 \, n + 2\right )} \log \relax (x)\right )} - 2 \, a b^{2} c^{-2 \, n - \frac {4}{3}} x^{\frac {2}{3}} e^{\left (-\frac {1}{3} \, {\left (3 \, n + 2\right )} \log \relax (c) - \frac {1}{3} \, {\left (3 \, n + 2\right )} \log \relax (x)\right )} + b^{3} c^{-3 \, n - 2} - 2 \, {\left (a^{2} b c^{-n - \frac {2}{3}} x e^{\left (-\frac {1}{2} \, {\left (3 \, n + 2\right )} \log \relax (c) - \frac {1}{2} \, {\left (3 \, n + 2\right )} \log \relax (x)\right )} - a^{3} x^{\frac {5}{3}} e^{\left (-\frac {5}{6} \, {\left (3 \, n + 2\right )} \log \relax (c) - \frac {5}{6} \, {\left (3 \, n + 2\right )} \log \relax (x)\right )} - a b^{2} c^{-2 \, n - \frac {4}{3}} x^{\frac {1}{3}} e^{\left (-\frac {1}{6} \, {\left (3 \, n + 2\right )} \log \relax (c) - \frac {1}{6} \, {\left (3 \, n + 2\right )} \log \relax (x)\right )}\right )} \sqrt {-\frac {b c^{-n - \frac {2}{3}}}{a}}}{a^{3} x^{2} e^{\left (-{\left (3 \, n + 2\right )} \log \relax (c) - {\left (3 \, n + 2\right )} \log \relax (x)\right )} + b^{3} c^{-3 \, n - 2}}\right ) + 6 \, b c^{-n - \frac {2}{3}} x^{\frac {1}{3}} e^{\left (-\frac {1}{6} \, {\left (3 \, n + 2\right )} \log \relax (c) - \frac {1}{6} \, {\left (3 \, n + 2\right )} \log \relax (x)\right )} - 2 \, a x e^{\left (-\frac {1}{2} \, {\left (3 \, n + 2\right )} \log \relax (c) - \frac {1}{2} \, {\left (3 \, n + 2\right )} \log \relax (x)\right )}}{3 \, a^{2} n}, \frac {2 \, {\left (3 \, b c^{-n - \frac {2}{3}} \sqrt {\frac {b c^{-n - \frac {2}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {b c^{-n - \frac {2}{3}}}{a}} e^{\left (\frac {1}{6} \, {\left (3 \, n + 2\right )} \log \relax (c) + \frac {1}{6} \, {\left (3 \, n + 2\right )} \log \relax (x)\right )}}{x^{\frac {1}{3}}}\right ) + 3 \, b c^{-n - \frac {2}{3}} x^{\frac {1}{3}} e^{\left (-\frac {1}{6} \, {\left (3 \, n + 2\right )} \log \relax (c) - \frac {1}{6} \, {\left (3 \, n + 2\right )} \log \relax (x)\right )} - a x e^{\left (-\frac {1}{2} \, {\left (3 \, n + 2\right )} \log \relax (c) - \frac {1}{2} \, {\left (3 \, n + 2\right )} \log \relax (x)\right )}\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 \[ \int \frac {\left (c x\right )^{-\frac {3}{2} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{-\frac {3 n}{2}-1}}{b \,x^{n}+a}\, dx \]
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 \[ b^{2} \int \frac {x^{\frac {1}{2} \, n}}{a^{2} b c^{\frac {3}{2} \, n + 1} x x^{n} + a^{3} c^{\frac {3}{2} \, n + 1} x}\,{d x} + \frac {2 \, {\left (3 \, b x^{n} - a\right )} c^{-\frac {3}{2} \, n - 1}}{3 \, a^{2} n x^{\frac {3}{2} \, n}} \]
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 \[ \int \frac {1}{{\left (c\,x\right )}^{\frac {3\,n}{2}+1}\,\left (a+b\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.65, size = 82, normalized size = 0.82 \[ - \frac {2 c^{- \frac {3 n}{2}} x^{- \frac {3 n}{2}}}{3 a c n} + \frac {2 b c^{- \frac {3 n}{2}} x^{- \frac {n}{2}}}{a^{2} c n} + \frac {2 b^{\frac {3}{2}} c^{- \frac {3 n}{2}} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{a^{\frac {5}{2}} c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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