Optimal. Leaf size=63 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} d}+\frac{c+d x}{2 a d \left (a+b (c+d x)^2\right )} \]
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Rubi [A] time = 0.0329842, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} d}+\frac{c+d x}{2 a d \left (a+b (c+d x)^2\right )} \]
Antiderivative was successfully verified.
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Rule 247
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b (c+d x)^2\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{c+d x}{2 a d \left (a+b (c+d x)^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,c+d x\right )}{2 a d}\\ &=\frac{c+d x}{2 a d \left (a+b (c+d x)^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} d}\\ \end{align*}
Mathematica [A] time = 0.0247364, size = 60, normalized size = 0.95 \[ \frac{\frac{\sqrt{a} (c+d x)}{a+b (c+d x)^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{\sqrt{b}}}{2 a^{3/2} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 86, normalized size = 1.4 \begin{align*}{\frac{2\,b{d}^{2}x+2\,bcd}{4\,ab{d}^{2} \left ( b{d}^{2}{x}^{2}+2\,bcdx+{c}^{2}b+a \right ) }}+{\frac{1}{2\,ad}\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 = 1.8116, size = 555, normalized size = 8.81 \begin{align*} \left [\frac{2 \, a b d x + 2 \, a b c -{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )} \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 )}{4 \,{\left (a^{2} b^{2} d^{3} x^{2} + 2 \, a^{2} b^{2} c d^{2} x +{\left (a^{2} b^{2} c^{2} + a^{3} b\right )} d\right )}}, \frac{a b d x + a b c +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}{\left (d x + c\right )}}{a}\right )}{2 \,{\left (a^{2} b^{2} d^{3} x^{2} + 2 \, a^{2} b^{2} c d^{2} x +{\left (a^{2} b^{2} c^{2} + a^{3} b\right )} d\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.727365, size = 117, normalized size = 1.86 \begin{align*} \frac{c + d x}{2 a^{2} d + 2 a b c^{2} d + 4 a b c d^{2} x + 2 a b d^{3} x^{2}} + \frac{- \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (x + \frac{- a^{2} \sqrt{- \frac{1}{a^{3} b}} + c}{d} \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (x + \frac{a^{2} \sqrt{- \frac{1}{a^{3} b}} + c}{d} \right )}}{4}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11284, size = 88, normalized size = 1.4 \begin{align*} \frac{\arctan \left (\frac{b d x + b c}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a d} + \frac{d x + c}{2 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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