Optimal. Leaf size=54 \[ \frac {2}{a c n \sqrt {a+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2} c n} \]
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Rubi [A] time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {12, 266, 51, 63, 208} \[ \frac {2}{a c n \sqrt {a+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2} c n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{c x \left (a+b x^n\right )^{3/2}} \, dx &=\frac {\int \frac {1}{x \left (a+b x^n\right )^{3/2}} \, dx}{c}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^n\right )}{c n}\\ &=\frac {2}{a c n \sqrt {a+b x^n}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{a c n}\\ &=\frac {2}{a c n \sqrt {a+b x^n}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{a b c n}\\ &=\frac {2}{a c n \sqrt {a+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2} c n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 40, normalized size = 0.74 \[ \frac {2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x^n}{a}+1\right )}{a c n \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 148, normalized size = 2.74 \[ \left [\frac {{\left (\sqrt {a} b x^{n} + a^{\frac {3}{2}}\right )} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, \sqrt {b x^{n} + a} a}{a^{2} b c n x^{n} + a^{3} c n}, \frac {2 \, {\left ({\left (\sqrt {-a} b x^{n} + \sqrt {-a} a\right )} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + \sqrt {b x^{n} + a} a\right )}}{a^{2} b c n x^{n} + a^{3} c n}\right ] \]
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 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{\frac {3}{2}} c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 0.78 \[ \frac {-\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{n}+a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}+\frac {2}{\sqrt {b \,x^{n}+a}\, a}}{c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 61, normalized size = 1.13 \[ \frac {\frac {\log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}} n} + \frac {2}{\sqrt {b x^{n} + a} a n}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{c\,x\,{\left (a+b\,x^n\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.72, size = 185, normalized size = 3.43 \[ \frac {\frac {2 a^{3} \sqrt {1 + \frac {b x^{n}}{a}}}{a^{\frac {9}{2}} n + a^{\frac {7}{2}} b n x^{n}} + \frac {a^{3} \log {\left (\frac {b x^{n}}{a} \right )}}{a^{\frac {9}{2}} n + a^{\frac {7}{2}} b n x^{n}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{n}}{a}} + 1 \right )}}{a^{\frac {9}{2}} n + a^{\frac {7}{2}} b n x^{n}} + \frac {a^{2} b x^{n} \log {\left (\frac {b x^{n}}{a} \right )}}{a^{\frac {9}{2}} n + a^{\frac {7}{2}} b n x^{n}} - \frac {2 a^{2} b x^{n} \log {\left (\sqrt {1 + \frac {b x^{n}}{a}} + 1 \right )}}{a^{\frac {9}{2}} n + a^{\frac {7}{2}} b n x^{n}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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