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 = 0.10991, antiderivative size = 27, normalized size of antiderivative = 1., 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 2155
Rule 31
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*}
Mathematica [A] time = 0.0609164, size = 27, normalized size = 1. \[ \frac{2 \log \left (c \sqrt{a+b x^n}+d\right )}{b c n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+n} \left ( ac+bc{x}^{n}+d\sqrt{a+b{x}^{n}} \right ) ^{-1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.32554, size = 82, normalized size = 3.04 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86726, size = 51, normalized size = 1.89 \begin{align*} \frac{2 \, \log \left (\sqrt{b x^{n} + a} c + d\right )}{b c n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 18.4952, size = 32, normalized size = 1.19 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12408, size = 55, normalized size = 2.04 \begin{align*} \frac{2 \, \log \left ({\left | \sqrt{b x^{n} + a} c + d \right |}\right )}{b c n} - \frac{2 \, \log \left ({\left | d \right |}\right )}{b c n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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