Optimal. Leaf size=155 \[ -\frac{(d+e x) \left (a e^2+5 c d^2\right ) (a e-c d x)}{16 a^3 c^2 \left (a+c x^2\right )}+\frac{\left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{5/2}}-\frac{(d+e x)^3 (a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac{x (d+e x)^4}{6 a \left (a+c x^2\right )^3} \]
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Rubi [A] time = 0.0845514, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {737, 805, 723, 205} \[ -\frac{(d+e x) \left (a e^2+5 c d^2\right ) (a e-c d x)}{16 a^3 c^2 \left (a+c x^2\right )}+\frac{\left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{5/2}}-\frac{(d+e x)^3 (a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac{x (d+e x)^4}{6 a \left (a+c x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 737
Rule 805
Rule 723
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\left (a+c x^2\right )^4} \, dx &=\frac{x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac{\int \frac{(-5 d-e x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx}{6 a}\\ &=\frac{x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac{(a e-5 c d x) (d+e x)^3}{24 a^2 c \left (a+c x^2\right )^2}+\frac{\left (5 c d^2+a e^2\right ) \int \frac{(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c}\\ &=\frac{x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac{(a e-5 c d x) (d+e x)^3}{24 a^2 c \left (a+c x^2\right )^2}-\frac{\left (5 c d^2+a e^2\right ) (a e-c d x) (d+e x)}{16 a^3 c^2 \left (a+c x^2\right )}+\frac{\left (\left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )\right ) \int \frac{1}{a+c x^2} \, dx}{16 a^3 c^2}\\ &=\frac{x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac{(a e-5 c d x) (d+e x)^3}{24 a^2 c \left (a+c x^2\right )^2}-\frac{\left (5 c d^2+a e^2\right ) (a e-c d x) (d+e x)}{16 a^3 c^2 \left (a+c x^2\right )}+\frac{\left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.146524, size = 197, normalized size = 1.27 \[ \frac{3 a^2 c^2 x \left (16 d^2 e^2 x^2+11 d^4+e^4 x^4\right )-2 a^3 c e \left (9 d^2 e x+16 d^3+24 d e^2 x^2+4 e^3 x^3\right )-a^4 e^3 (16 d+3 e x)+2 a c^3 d^2 x^3 \left (20 d^2+9 e^2 x^2\right )+15 c^4 d^4 x^5}{48 a^3 c^2 \left (a+c x^2\right )^3}+\frac{\left (a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 225, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}} \left ({\frac{ \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}+5\,{c}^{2}{d}^{4} \right ){x}^{5}}{16\,{a}^{3}}}-{\frac{ \left ({a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}-5\,{c}^{2}{d}^{4} \right ){x}^{3}}{6\,{a}^{2}c}}-{\frac{d{e}^{3}{x}^{2}}{c}}-{\frac{ \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}-11\,{c}^{2}{d}^{4} \right ) x}{16\,a{c}^{2}}}-{\frac{de \left ( a{e}^{2}+2\,c{d}^{2} \right ) }{3\,{c}^{2}}} \right ) }+{\frac{{e}^{4}}{16\,a{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{2}{e}^{2}}{8\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,{d}^{4}}{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 [B] time = 2.30775, size = 1480, normalized size = 9.55 \begin{align*} \left [-\frac{96 \, a^{4} c^{2} d e^{3} x^{2} + 64 \, a^{4} c^{2} d^{3} e + 32 \, a^{5} c d e^{3} - 6 \,{\left (5 \, a c^{5} d^{4} + 6 \, a^{2} c^{4} d^{2} e^{2} + a^{3} c^{3} e^{4}\right )} x^{5} - 16 \,{\left (5 \, a^{2} c^{4} d^{4} + 6 \, a^{3} c^{3} d^{2} e^{2} - a^{4} c^{2} e^{4}\right )} x^{3} + 3 \,{\left (5 \, a^{3} c^{2} d^{4} + 6 \, a^{4} c d^{2} e^{2} + a^{5} e^{4} +{\left (5 \, c^{5} d^{4} + 6 \, a c^{4} d^{2} e^{2} + a^{2} c^{3} e^{4}\right )} x^{6} + 3 \,{\left (5 \, a c^{4} d^{4} + 6 \, a^{2} c^{3} d^{2} e^{2} + a^{3} c^{2} e^{4}\right )} x^{4} + 3 \,{\left (5 \, a^{2} c^{3} d^{4} + 6 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )} x^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) - 6 \,{\left (11 \, a^{3} c^{3} d^{4} - 6 \, a^{4} c^{2} d^{2} e^{2} - a^{5} c e^{4}\right )} x}{96 \,{\left (a^{4} c^{6} x^{6} + 3 \, a^{5} c^{5} x^{4} + 3 \, a^{6} c^{4} x^{2} + a^{7} c^{3}\right )}}, -\frac{48 \, a^{4} c^{2} d e^{3} x^{2} + 32 \, a^{4} c^{2} d^{3} e + 16 \, a^{5} c d e^{3} - 3 \,{\left (5 \, a c^{5} d^{4} + 6 \, a^{2} c^{4} d^{2} e^{2} + a^{3} c^{3} e^{4}\right )} x^{5} - 8 \,{\left (5 \, a^{2} c^{4} d^{4} + 6 \, a^{3} c^{3} d^{2} e^{2} - a^{4} c^{2} e^{4}\right )} x^{3} - 3 \,{\left (5 \, a^{3} c^{2} d^{4} + 6 \, a^{4} c d^{2} e^{2} + a^{5} e^{4} +{\left (5 \, c^{5} d^{4} + 6 \, a c^{4} d^{2} e^{2} + a^{2} c^{3} e^{4}\right )} x^{6} + 3 \,{\left (5 \, a c^{4} d^{4} + 6 \, a^{2} c^{3} d^{2} e^{2} + a^{3} c^{2} e^{4}\right )} x^{4} + 3 \,{\left (5 \, a^{2} c^{3} d^{4} + 6 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) - 3 \,{\left (11 \, a^{3} c^{3} d^{4} - 6 \, a^{4} c^{2} d^{2} e^{2} - a^{5} c e^{4}\right )} x}{48 \,{\left (a^{4} c^{6} x^{6} + 3 \, a^{5} c^{5} x^{4} + 3 \, a^{6} c^{4} x^{2} + a^{7} c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.74925, size = 413, normalized size = 2.66 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log{\left (- \frac{a^{4} c^{2} \sqrt{- \frac{1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{a^{2} e^{4} + 6 a c d^{2} e^{2} + 5 c^{2} d^{4}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log{\left (\frac{a^{4} c^{2} \sqrt{- \frac{1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{a^{2} e^{4} + 6 a c d^{2} e^{2} + 5 c^{2} d^{4}} + x \right )}}{32} + \frac{- 16 a^{4} d e^{3} - 32 a^{3} c d^{3} e - 48 a^{3} c d e^{3} x^{2} + x^{5} \left (3 a^{2} c^{2} e^{4} + 18 a c^{3} d^{2} e^{2} + 15 c^{4} d^{4}\right ) + x^{3} \left (- 8 a^{3} c e^{4} + 48 a^{2} c^{2} d^{2} e^{2} + 40 a c^{3} d^{4}\right ) + x \left (- 3 a^{4} e^{4} - 18 a^{3} c d^{2} e^{2} + 33 a^{2} c^{2} d^{4}\right )}{48 a^{6} c^{2} + 144 a^{5} c^{3} x^{2} + 144 a^{4} c^{4} x^{4} + 48 a^{3} c^{5} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28245, size = 277, normalized size = 1.79 \begin{align*} \frac{{\left (5 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3} c^{2}} + \frac{15 \, c^{4} d^{4} x^{5} + 18 \, a c^{3} d^{2} x^{5} e^{2} + 40 \, a c^{3} d^{4} x^{3} + 3 \, a^{2} c^{2} x^{5} e^{4} + 48 \, a^{2} c^{2} d^{2} x^{3} e^{2} + 33 \, a^{2} c^{2} d^{4} x - 8 \, a^{3} c x^{3} e^{4} - 48 \, a^{3} c d x^{2} e^{3} - 18 \, a^{3} c d^{2} x e^{2} - 32 \, a^{3} c d^{3} e - 3 \, a^{4} x e^{4} - 16 \, a^{4} d e^{3}}{48 \,{\left (c x^{2} + a\right )}^{3} a^{3} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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