Optimal. Leaf size=27 \[ \frac {2 \log \left (c \sqrt {a+b x^n}+d\right )}{b c n} \]
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Rubi [A] time = 0.11, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2155, 31} \[ \frac {2 \log \left (c \sqrt {a+b x^n}+d\right )}{b c n} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2155
Rubi steps
\begin {align*} \int \frac {x^{-1+n}}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a c+b c x+d \sqrt {a+b x}} \, dx,x,x^n\right )}{n}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{d+c x} \, dx,x,\sqrt {a+b x^n}\right )}{b n}\\ &=\frac {2 \log \left (d+c \sqrt {a+b x^n}\right )}{b c n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 27, normalized size = 1.00 \[ \frac {2 \log \left (c \sqrt {a+b x^n}+d\right )}{b c n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 25, normalized size = 0.93 \[ \frac {2 \, \log \left (\sqrt {b x^{n} + a} c + d\right )}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 26, normalized size = 0.96 \[ \frac {2 \, \log \left ({\left | \sqrt {b x^{n} + a} c + d \right |}\right )}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {x^{n -1}}{b c \,x^{n}+a c +\sqrt {b \,x^{n}+a}\, d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 61, normalized size = 2.26 \[ -\frac {\log \left (\frac {b x^{n} + a}{b}\right )}{b c n} + \frac {2 \, \log \left (\frac {b c x^{n} + a c + \sqrt {b x^{n} + a} d}{d}\right )}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.35, size = 25, normalized size = 0.93 \[ \frac {2\,\ln \left (d+c\,\sqrt {a+b\,x^n}\right )}{b\,c\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 33.63, size = 32, normalized size = 1.19 \[ \frac {2 \left (\begin {cases} \frac {\sqrt {a + b x^{n}}}{d} & \text {for}\: c = 0 \\\frac {\log {\left (c \sqrt {a + b x^{n}} + d \right )}}{c} & \text {otherwise} \end {cases}\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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