Optimal. Leaf size=52 \[ \frac {2}{a n \sqrt {a+b (c x)^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{a^{3/2} n} \]
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Rubi [A]
time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {374, 12, 272,
53, 65, 214} \begin {gather*} \frac {2}{a n \sqrt {a+b (c x)^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{a^{3/2} n} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 53
Rule 65
Rule 214
Rule 272
Rule 374
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b (c x)^n\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {c}{x \left (a+b x^n\right )^{3/2}} \, dx,x,c x\right )}{c}\\ &=\text {Subst}\left (\int \frac {1}{x \left (a+b x^n\right )^{3/2}} \, dx,x,c x\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,(c x)^n\right )}{n}\\ &=\frac {2}{a n \sqrt {a+b (c x)^n}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,(c x)^n\right )}{a n}\\ &=\frac {2}{a n \sqrt {a+b (c x)^n}}+\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b (c x)^n}\right )}{a b n}\\ &=\frac {2}{a n \sqrt {a+b (c x)^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{a^{3/2} n}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 52, normalized size = 1.00 \begin {gather*} \frac {2}{a n \sqrt {a+b (c x)^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{a^{3/2} n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 43, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\frac {2}{a \sqrt {a +b \left (c x \right )^{n}}}-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}}{n}\) | \(43\) |
default | \(\frac {\frac {2}{a \sqrt {a +b \left (c x \right )^{n}}}-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}}{n}\) | \(43\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 164, normalized size = 3.15 \begin {gather*} \left [\frac {{\left (\left (c x\right )^{n} \sqrt {a} b + a^{\frac {3}{2}}\right )} \log \left (\frac {\left (c x\right )^{n} b - 2 \, \sqrt {\left (c x\right )^{n} b + a} \sqrt {a} + 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt {\left (c x\right )^{n} b + a} a}{\left (c x\right )^{n} a^{2} b n + a^{3} n}, \frac {2 \, {\left ({\left (\left (c x\right )^{n} \sqrt {-a} b + \sqrt {-a} a\right )} \arctan \left (\frac {\sqrt {\left (c x\right )^{n} b + a} \sqrt {-a}}{a}\right ) + \sqrt {\left (c x\right )^{n} b + a} a\right )}}{\left (c x\right )^{n} a^{2} b n + a^{3} n}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.55, size = 48, normalized size = 0.92 \begin {gather*} \frac {2}{a n \sqrt {a + b \left (c x\right )^{n}}} + \frac {2 \operatorname {atan}{\left (\frac {\sqrt {a + b \left (c x\right )^{n}}}{\sqrt {- a}} \right )}}{a n \sqrt {- a}} \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\,{\left (c\,x\right )}^n\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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