Optimal. Leaf size=93 \[ \frac{5 d x}{16 a^3 \left (a+c x^2\right )}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \sqrt{c}}-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3} \]
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Rubi [A] time = 0.0280046, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {639, 199, 205} \[ \frac{5 d x}{16 a^3 \left (a+c x^2\right )}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \sqrt{c}}-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 639
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a+c x^2\right )^4} \, dx &=-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac{(5 d) \int \frac{1}{\left (a+c x^2\right )^3} \, dx}{6 a}\\ &=-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac{(5 d) \int \frac{1}{\left (a+c x^2\right )^2} \, dx}{8 a^2}\\ &=-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac{5 d x}{16 a^3 \left (a+c x^2\right )}+\frac{(5 d) \int \frac{1}{a+c x^2} \, dx}{16 a^3}\\ &=-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}+\frac{5 d x}{16 a^3 \left (a+c x^2\right )}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0463111, size = 83, normalized size = 0.89 \[ \frac{\frac{\sqrt{a} \left (33 a^2 c d x-8 a^3 e+40 a c^2 d x^3+15 c^3 d x^5\right )}{\left (a+c x^2\right )^3}+15 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{48 a^{7/2} c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 81, normalized size = 0.9 \begin{align*}{\frac{2\,cdx-2\,ae}{12\,ac \left ( c{x}^{2}+a \right ) ^{3}}}+{\frac{5\,dx}{24\,{a}^{2} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{5\,dx}{16\,{a}^{3} \left ( c{x}^{2}+a \right ) }}+{\frac{5\,d}{16\,{a}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 2.09654, size = 608, normalized size = 6.54 \begin{align*} \left [\frac{30 \, a c^{3} d x^{5} + 80 \, a^{2} c^{2} d x^{3} + 66 \, a^{3} c d x - 16 \, a^{4} e - 15 \,{\left (c^{3} d x^{6} + 3 \, a c^{2} d x^{4} + 3 \, a^{2} c d x^{2} + a^{3} d\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{96 \,{\left (a^{4} c^{4} x^{6} + 3 \, a^{5} c^{3} x^{4} + 3 \, a^{6} c^{2} x^{2} + a^{7} c\right )}}, \frac{15 \, a c^{3} d x^{5} + 40 \, a^{2} c^{2} d x^{3} + 33 \, a^{3} c d x - 8 \, a^{4} e + 15 \,{\left (c^{3} d x^{6} + 3 \, a c^{2} d x^{4} + 3 \, a^{2} c d x^{2} + a^{3} d\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{48 \,{\left (a^{4} c^{4} x^{6} + 3 \, a^{5} c^{3} x^{4} + 3 \, a^{6} c^{2} x^{2} + a^{7} c\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.838742, size = 150, normalized size = 1.61 \begin{align*} d \left (- \frac{5 \sqrt{- \frac{1}{a^{7} c}} \log{\left (- a^{4} \sqrt{- \frac{1}{a^{7} c}} + x \right )}}{32} + \frac{5 \sqrt{- \frac{1}{a^{7} c}} \log{\left (a^{4} \sqrt{- \frac{1}{a^{7} c}} + x \right )}}{32}\right ) + \frac{- 8 a^{3} e + 33 a^{2} c d x + 40 a c^{2} d x^{3} + 15 c^{3} d x^{5}}{48 a^{6} c + 144 a^{5} c^{2} x^{2} + 144 a^{4} c^{3} x^{4} + 48 a^{3} c^{4} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3043, size = 99, normalized size = 1.06 \begin{align*} \frac{5 \, d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3}} + \frac{15 \, c^{3} d x^{5} + 40 \, a c^{2} d x^{3} + 33 \, a^{2} c d x - 8 \, a^{3} e}{48 \,{\left (c x^{2} + a\right )}^{3} a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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