Optimal. Leaf size=48 \[ \frac {2}{a n \sqrt {a+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2} n} \]
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Rubi [A]
time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 53, 65,
214} \begin {gather*} \frac {2}{a n \sqrt {a+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2} n} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^n\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^n\right )}{n}\\ &=\frac {2}{a n \sqrt {a+b x^n}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{a n}\\ &=\frac {2}{a n \sqrt {a+b x^n}}+\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{a b n}\\ &=\frac {2}{a n \sqrt {a+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2} n}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 48, normalized size = 1.00 \begin {gather*} \frac {2}{a n \sqrt {a+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2} n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 39, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {-\frac {2 \arctanh \left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}+\frac {2}{a \sqrt {a +b \,x^{n}}}}{n}\) | \(39\) |
default | \(\frac {-\frac {2 \arctanh \left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}+\frac {2}{a \sqrt {a +b \,x^{n}}}}{n}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 57, normalized size = 1.19 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 144, normalized size = 3.00 \begin {gather*} \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 n x^{n} + a^{3} 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 n x^{n} + a^{3} n}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 184 vs.
\(2 (39) = 78\).
time = 1.10, size = 184, normalized size = 3.83 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x\,{\left (a+b\,x^n\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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