Optimal. Leaf size=91 \[ \frac{3 (c+d x)}{8 a^2 d \left (a+b (c+d x)^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} d}+\frac{c+d x}{4 a d \left (a+b (c+d x)^2\right )^2} \]
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Rubi [A] time = 0.0482007, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 199, 205} \[ \frac{3 (c+d x)}{8 a^2 d \left (a+b (c+d x)^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} d}+\frac{c+d x}{4 a d \left (a+b (c+d x)^2\right )^2} \]
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 )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{c+d x}{4 a d \left (a+b (c+d x)^2\right )^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^2} \, dx,x,c+d x\right )}{4 a d}\\ &=\frac{c+d x}{4 a d \left (a+b (c+d x)^2\right )^2}+\frac{3 (c+d x)}{8 a^2 d \left (a+b (c+d x)^2\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,c+d x\right )}{8 a^2 d}\\ &=\frac{c+d x}{4 a d \left (a+b (c+d x)^2\right )^2}+\frac{3 (c+d x)}{8 a^2 d \left (a+b (c+d x)^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} d}\\ \end{align*}
Mathematica [A] time = 0.0601837, size = 75, normalized size = 0.82 \[ \frac{\frac{\sqrt{a} (c+d x) \left (5 a+3 b (c+d x)^2\right )}{\left (a+b (c+d x)^2\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{\sqrt{b}}}{8 a^{5/2} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 147, normalized size = 1.6 \begin{align*}{\frac{2\,b{d}^{2}x+2\,bcd}{8\,ab{d}^{2} \left ( b{d}^{2}{x}^{2}+2\,bcdx+{c}^{2}b+a \right ) ^{2}}}+{\frac{3\,x}{8\,{a}^{2} \left ( b{d}^{2}{x}^{2}+2\,bcdx+{c}^{2}b+a \right ) }}+{\frac{3\,c}{8\,{a}^{2}d \left ( b{d}^{2}{x}^{2}+2\,bcdx+{c}^{2}b+a \right ) }}+{\frac{3}{8\,{a}^{2}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 [B] time = 1.94832, size = 1231, normalized size = 13.53 \begin{align*} \left [\frac{6 \, a b^{2} d^{3} x^{3} + 18 \, a b^{2} c d^{2} x^{2} + 6 \, a b^{2} c^{3} + 10 \, a^{2} b c + 2 \,{\left (9 \, a b^{2} c^{2} + 5 \, a^{2} b\right )} d x - 3 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + b^{2} c^{4} + 2 \,{\left (3 \, b^{2} c^{2} + a b\right )} d^{2} x^{2} + 2 \, a b c^{2} + 4 \,{\left (b^{2} c^{3} + a b c\right )} d x + a^{2}\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 )}{16 \,{\left (a^{3} b^{3} d^{5} x^{4} + 4 \, a^{3} b^{3} c d^{4} x^{3} + 2 \,{\left (3 \, a^{3} b^{3} c^{2} + a^{4} b^{2}\right )} d^{3} x^{2} + 4 \,{\left (a^{3} b^{3} c^{3} + a^{4} b^{2} c\right )} d^{2} x +{\left (a^{3} b^{3} c^{4} + 2 \, a^{4} b^{2} c^{2} + a^{5} b\right )} d\right )}}, \frac{3 \, a b^{2} d^{3} x^{3} + 9 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{3} + 5 \, a^{2} b c +{\left (9 \, a b^{2} c^{2} + 5 \, a^{2} b\right )} d x + 3 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + b^{2} c^{4} + 2 \,{\left (3 \, b^{2} c^{2} + a b\right )} d^{2} x^{2} + 2 \, a b c^{2} + 4 \,{\left (b^{2} c^{3} + a b c\right )} d x + a^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}{\left (d x + c\right )}}{a}\right )}{8 \,{\left (a^{3} b^{3} d^{5} x^{4} + 4 \, a^{3} b^{3} c d^{4} x^{3} + 2 \,{\left (3 \, a^{3} b^{3} c^{2} + a^{4} b^{2}\right )} d^{3} x^{2} + 4 \,{\left (a^{3} b^{3} c^{3} + a^{4} b^{2} c\right )} d^{2} x +{\left (a^{3} b^{3} c^{4} + 2 \, a^{4} b^{2} c^{2} + a^{5} b\right )} d\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.65176, size = 257, normalized size = 2.82 \begin{align*} \frac{5 a c + 3 b c^{3} + 9 b c d^{2} x^{2} + 3 b d^{3} x^{3} + x \left (5 a d + 9 b c^{2} d\right )}{8 a^{4} d + 16 a^{3} b c^{2} d + 8 a^{2} b^{2} c^{4} d + 32 a^{2} b^{2} c d^{4} x^{3} + 8 a^{2} b^{2} d^{5} x^{4} + x^{2} \left (16 a^{3} b d^{3} + 48 a^{2} b^{2} c^{2} d^{3}\right ) + x \left (32 a^{3} b c d^{2} + 32 a^{2} b^{2} c^{3} d^{2}\right )} + \frac{- \frac{3 \sqrt{- \frac{1}{a^{5} b}} \log{\left (x + \frac{- 3 a^{3} \sqrt{- \frac{1}{a^{5} b}} + 3 c}{3 d} \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} b}} \log{\left (x + \frac{3 a^{3} \sqrt{- \frac{1}{a^{5} b}} + 3 c}{3 d} \right )}}{16}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09979, size = 139, normalized size = 1.53 \begin{align*} \frac{3 \, \arctan \left (\frac{b d x + b c}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} d} + \frac{3 \, b d^{3} x^{3} + 9 \, b c d^{2} x^{2} + 9 \, b c^{2} d x + 3 \, b c^{3} + 5 \, a d x + 5 \, a c}{8 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{2} a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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