Optimal. Leaf size=120 \[ -\frac{3 (d+e x) \left (a e^2+c d^2\right ) (a e-c d x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{3 \left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}-\frac{(d+e x)^3 (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.0479793, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {723, 205} \[ -\frac{3 (d+e x) \left (a e^2+c d^2\right ) (a e-c d x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{3 \left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}-\frac{(d+e x)^3 (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 723
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\left (a+c x^2\right )^3} \, dx &=-\frac{(a e-c d x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}+\frac{\left (3 \left (c d^2+a e^2\right )\right ) \int \frac{(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac{(a e-c d x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac{3 \left (c d^2+a e^2\right ) (a e-c d x) (d+e x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\left (3 \left (c d^2+a e^2\right )^2\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac{(a e-c d x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac{3 \left (c d^2+a e^2\right ) (a e-c d x) (d+e x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{3 \left (c d^2+a e^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.101979, size = 148, normalized size = 1.23 \[ \frac{-a^2 c e \left (6 d^2 e x+8 d^3+16 d e^2 x^2+5 e^3 x^3\right )-a^3 e^3 (8 d+3 e x)+a c^2 d^2 x \left (5 d^2+6 e^2 x^2\right )+3 c^3 d^4 x^3}{8 a^2 c^2 \left (a+c x^2\right )^2}+\frac{3 \left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 189, normalized size = 1.6 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ( -{\frac{ \left ( 5\,{a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}-3\,{c}^{2}{d}^{4} \right ){x}^{3}}{8\,{a}^{2}c}}-2\,{\frac{d{e}^{3}{x}^{2}}{c}}-{\frac{ \left ( 3\,{a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}-5\,{c}^{2}{d}^{4} \right ) x}{8\,a{c}^{2}}}-{\frac{de \left ( a{e}^{2}+c{d}^{2} \right ) }{{c}^{2}}} \right ) }+{\frac{3\,{e}^{4}}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{2}{e}^{2}}{4\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{4}}{8\,{a}^{2}}\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 = 1.90785, size = 1104, normalized size = 9.2 \begin{align*} \left [-\frac{32 \, a^{3} c^{2} d e^{3} x^{2} + 16 \, a^{3} c^{2} d^{3} e + 16 \, a^{4} c d e^{3} - 2 \,{\left (3 \, a c^{4} d^{4} + 6 \, a^{2} c^{3} d^{2} e^{2} - 5 \, a^{3} c^{2} e^{4}\right )} x^{3} + 3 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4} + 2 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} 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 ) - 2 \,{\left (5 \, a^{2} c^{3} d^{4} - 6 \, a^{3} c^{2} d^{2} e^{2} - 3 \, a^{4} c e^{4}\right )} x}{16 \,{\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac{16 \, a^{3} c^{2} d e^{3} x^{2} + 8 \, a^{3} c^{2} d^{3} e + 8 \, a^{4} c d e^{3} -{\left (3 \, a c^{4} d^{4} + 6 \, a^{2} c^{3} d^{2} e^{2} - 5 \, a^{3} c^{2} e^{4}\right )} x^{3} - 3 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4} + 2 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (5 \, a^{2} c^{3} d^{4} - 6 \, a^{3} c^{2} d^{2} e^{2} - 3 \, a^{4} c e^{4}\right )} x}{8 \,{\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.63037, size = 328, normalized size = 2.73 \begin{align*} - \frac{3 \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2} \log{\left (- \frac{3 a^{3} c^{2} \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2}}{3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 3 c^{2} d^{4}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2} \log{\left (\frac{3 a^{3} c^{2} \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2}}{3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 3 c^{2} d^{4}} + x \right )}}{16} - \frac{8 a^{3} d e^{3} + 8 a^{2} c d^{3} e + 16 a^{2} c d e^{3} x^{2} + x^{3} \left (5 a^{2} c e^{4} - 6 a c^{2} d^{2} e^{2} - 3 c^{3} d^{4}\right ) + x \left (3 a^{3} e^{4} + 6 a^{2} c d^{2} e^{2} - 5 a c^{2} d^{4}\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33361, size = 217, normalized size = 1.81 \begin{align*} \frac{3 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c^{2}} + \frac{3 \, c^{3} d^{4} x^{3} + 6 \, a c^{2} d^{2} x^{3} e^{2} + 5 \, a c^{2} d^{4} x - 5 \, a^{2} c x^{3} e^{4} - 16 \, a^{2} c d x^{2} e^{3} - 6 \, a^{2} c d^{2} x e^{2} - 8 \, a^{2} c d^{3} e - 3 \, a^{3} x e^{4} - 8 \, a^{3} d e^{3}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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