Optimal. Leaf size=20 \[ \frac{d x}{b c \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.0046504, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {21, 191} \[ \frac{d x}{b c \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (\frac{b c}{d}+b x^2\right ) \sqrt{c+d x^2}} \, dx &=\frac{d \int \frac{1}{\left (c+d x^2\right )^{3/2}} \, dx}{b}\\ &=\frac{d x}{b c \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0071051, size = 20, normalized size = 1. \[ \frac{d x}{b c \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 1. \begin{align*}{\frac{dx}{bc}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50166, size = 55, normalized size = 2.75 \begin{align*} \frac{\sqrt{d x^{2} + c} d x}{b c d x^{2} + b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{d \int \frac{1}{c \sqrt{c + d x^{2}} + d x^{2} \sqrt{c + d x^{2}}}\, dx}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11679, size = 24, normalized size = 1.2 \begin{align*} \frac{d x}{\sqrt{d x^{2} + c} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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