Optimal. Leaf size=142 \[ \frac{d^2 x^3 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{3 b^3}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^3 x^5 (4 b c-a d)}{5 b^2}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{9/2}}+\frac{d^4 x^7}{7 b} \]
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Rubi [A] time = 0.093036, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {390, 205} \[ \frac{d^2 x^3 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{3 b^3}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^3 x^5 (4 b c-a d)}{5 b^2}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{9/2}}+\frac{d^4 x^7}{7 b} \]
Antiderivative was successfully verified.
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Rule 390
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^4}{a+b x^2} \, dx &=\int \left (\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right )}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^2}{b^3}+\frac{d^3 (4 b c-a d) x^4}{b^2}+\frac{d^4 x^6}{b}+\frac{b^4 c^4-4 a b^3 c^3 d+6 a^2 b^2 c^2 d^2-4 a^3 b c d^3+a^4 d^4}{b^4 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac{d^3 (4 b c-a d) x^5}{5 b^2}+\frac{d^4 x^7}{7 b}+\frac{(b c-a d)^4 \int \frac{1}{a+b x^2} \, dx}{b^4}\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac{d^3 (4 b c-a d) x^5}{5 b^2}+\frac{d^4 x^7}{7 b}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0855929, size = 136, normalized size = 0.96 \[ \frac{d x \left (35 a^2 b d^2 \left (12 c+d x^2\right )-105 a^3 d^3-7 a b^2 d \left (90 c^2+20 c d x^2+3 d^2 x^4\right )+3 b^3 \left (70 c^2 d x^2+140 c^3+28 c d^2 x^4+5 d^3 x^6\right )\right )}{105 b^4}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 246, normalized size = 1.7 \begin{align*}{\frac{{d}^{4}{x}^{7}}{7\,b}}-{\frac{{d}^{4}{x}^{5}a}{5\,{b}^{2}}}+{\frac{4\,{d}^{3}{x}^{5}c}{5\,b}}+{\frac{{d}^{4}{x}^{3}{a}^{2}}{3\,{b}^{3}}}-{\frac{4\,{d}^{3}{x}^{3}ac}{3\,{b}^{2}}}+2\,{\frac{{d}^{2}{x}^{3}{c}^{2}}{b}}-{\frac{{d}^{4}{a}^{3}x}{{b}^{4}}}+4\,{\frac{{a}^{2}{d}^{3}cx}{{b}^{3}}}-6\,{\frac{a{c}^{2}{d}^{2}x}{{b}^{2}}}+4\,{\frac{d{c}^{3}x}{b}}+{\frac{{a}^{4}{d}^{4}}{{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-4\,{\frac{{a}^{3}c{d}^{3}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+6\,{\frac{{a}^{2}{c}^{2}{d}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-4\,{\frac{a{c}^{3}d}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{{c}^{4}\arctan \left ({bx{\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 = 1.85122, size = 892, normalized size = 6.28 \begin{align*} \left [\frac{30 \, a b^{4} d^{4} x^{7} + 42 \,{\left (4 \, a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x^{5} + 70 \,{\left (6 \, a b^{4} c^{2} d^{2} - 4 \, a^{2} b^{3} c d^{3} + a^{3} b^{2} d^{4}\right )} x^{3} - 105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 210 \,{\left (4 \, a b^{4} c^{3} d - 6 \, a^{2} b^{3} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x}{210 \, a b^{5}}, \frac{15 \, a b^{4} d^{4} x^{7} + 21 \,{\left (4 \, a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x^{5} + 35 \,{\left (6 \, a b^{4} c^{2} d^{2} - 4 \, a^{2} b^{3} c d^{3} + a^{3} b^{2} d^{4}\right )} x^{3} + 105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 105 \,{\left (4 \, a b^{4} c^{3} d - 6 \, a^{2} b^{3} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x}{105 \, a b^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.881547, size = 323, normalized size = 2.27 \begin{align*} - \frac{\sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4} \log{\left (- \frac{a b^{4} \sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4} \log{\left (\frac{a b^{4} \sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac{d^{4} x^{7}}{7 b} - \frac{x^{5} \left (a d^{4} - 4 b c d^{3}\right )}{5 b^{2}} + \frac{x^{3} \left (a^{2} d^{4} - 4 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{3 b^{3}} - \frac{x \left (a^{3} d^{4} - 4 a^{2} b c d^{3} + 6 a b^{2} c^{2} d^{2} - 4 b^{3} c^{3} d\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11394, size = 267, normalized size = 1.88 \begin{align*} \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{15 \, b^{6} d^{4} x^{7} + 84 \, b^{6} c d^{3} x^{5} - 21 \, a b^{5} d^{4} x^{5} + 210 \, b^{6} c^{2} d^{2} x^{3} - 140 \, a b^{5} c d^{3} x^{3} + 35 \, a^{2} b^{4} d^{4} x^{3} + 420 \, b^{6} c^{3} d x - 630 \, a b^{5} c^{2} d^{2} x + 420 \, a^{2} b^{4} c d^{3} x - 105 \, a^{3} b^{3} d^{4} x}{105 \, b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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