Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d} \]
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Rubi [A] time = 0.0235124, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {247, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d} \]
Antiderivative was successfully verified.
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Rule 247
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{a+b (c+d x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d}\\ \end{align*}
Mathematica [A] time = 0.0090773, size = 31, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 34, normalized size = 1.1 \begin{align*}{\frac{1}{d}\arctan \left ({\frac{2\,b{d}^{2}x+2\,bcd}{2\,d}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.67146, size = 243, normalized size = 7.84 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{b d^{2} x^{2} + 2 \, b c d x + b c^{2} - 2 \, \sqrt{-a b}{\left (d x + c\right )} - a}{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}\right )}{2 \, a b d}, \frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b}{\left (d x + c\right )}}{a}\right )}{a b d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.201686, size = 61, normalized size = 1.97 \begin{align*} \frac{- \frac{\sqrt{- \frac{1}{a b}} \log{\left (x + \frac{- a \sqrt{- \frac{1}{a b}} + c}{d} \right )}}{2} + \frac{\sqrt{- \frac{1}{a b}} \log{\left (x + \frac{a \sqrt{- \frac{1}{a b}} + c}{d} \right )}}{2}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12971, size = 32, normalized size = 1.03 \begin{align*} \frac{\arctan \left (\frac{b d x + b c}{\sqrt{a b}}\right )}{\sqrt{a b} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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