Optimal. Leaf size=98 \[ \frac{3 d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}-\frac{3 d (d+e x) (a e-c d x)}{8 a^2 c \left (a+c x^2\right )}+\frac{x (d+e x)^3}{4 a \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.0380723, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {729, 723, 205} \[ \frac{3 d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}-\frac{3 d (d+e x) (a e-c d x)}{8 a^2 c \left (a+c x^2\right )}+\frac{x (d+e x)^3}{4 a \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 729
Rule 723
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a+c x^2\right )^3} \, dx &=\frac{x (d+e x)^3}{4 a \left (a+c x^2\right )^2}+\frac{(3 d) \int \frac{(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{4 a}\\ &=\frac{x (d+e x)^3}{4 a \left (a+c x^2\right )^2}-\frac{3 d (a e-c d x) (d+e x)}{8 a^2 c \left (a+c x^2\right )}+\frac{\left (3 d \left (c d^2+a e^2\right )\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 c}\\ &=\frac{x (d+e x)^3}{4 a \left (a+c x^2\right )^2}-\frac{3 d (a e-c d x) (d+e x)}{8 a^2 c \left (a+c x^2\right )}+\frac{3 d \left (c d^2+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.103615, size = 127, normalized size = 1.3 \[ \frac{\frac{\sqrt{a} \left (-a^2 c e \left (6 d^2+3 d e x+4 e^2 x^2\right )-2 a^3 e^3+a c^2 d x \left (5 d^2+3 e^2 x^2\right )+3 c^3 d^3 x^3\right )}{\left (a+c x^2\right )^2}+3 \sqrt{c} d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 133, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{3\,d \left ( a{e}^{2}+c{d}^{2} \right ){x}^{3}}{8\,{a}^{2}}}-{\frac{{e}^{3}{x}^{2}}{2\,c}}-{\frac{d \left ( 3\,a{e}^{2}-5\,c{d}^{2} \right ) x}{8\,ac}}-{\frac{e \left ( a{e}^{2}+3\,c{d}^{2} \right ) }{4\,{c}^{2}}} \right ) }+{\frac{3\,d{e}^{2}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{3}}{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 = 2.05173, size = 822, normalized size = 8.39 \begin{align*} \left [-\frac{8 \, a^{3} c e^{3} x^{2} + 12 \, a^{3} c d^{2} e + 4 \, a^{4} e^{3} - 6 \,{\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x^{3} + 3 \,{\left (a^{2} c d^{3} + a^{3} d e^{2} +{\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{4} + 2 \,{\left (a c^{2} d^{3} + a^{2} c d e^{2}\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^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x}{16 \,{\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}, -\frac{4 \, a^{3} c e^{3} x^{2} + 6 \, a^{3} c d^{2} e + 2 \, a^{4} e^{3} - 3 \,{\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x^{3} - 3 \,{\left (a^{2} c d^{3} + a^{3} d e^{2} +{\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{4} + 2 \,{\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (5 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x}{8 \,{\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.72044, size = 272, normalized size = 2.78 \begin{align*} - \frac{3 d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log{\left (- \frac{3 a^{3} c d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right )}{3 a d e^{2} + 3 c d^{3}} + x \right )}}{16} + \frac{3 d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log{\left (\frac{3 a^{3} c d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right )}{3 a d e^{2} + 3 c d^{3}} + x \right )}}{16} + \frac{- 2 a^{3} e^{3} - 6 a^{2} c d^{2} e - 4 a^{2} c e^{3} x^{2} + x^{3} \left (3 a c^{2} d e^{2} + 3 c^{3} d^{3}\right ) + x \left (- 3 a^{2} c d e^{2} + 5 a c^{2} d^{3}\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 = 1.33972, size = 167, normalized size = 1.7 \begin{align*} \frac{3 \,{\left (c d^{3} + a d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c} + \frac{3 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d x^{3} e^{2} + 5 \, a c^{2} d^{3} x - 4 \, a^{2} c x^{2} e^{3} - 3 \, a^{2} c d x e^{2} - 6 \, a^{2} c d^{2} e - 2 \, a^{3} 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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