Optimal. Leaf size=145 \[ \frac{\left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{3/2}}+\frac{x \left (a e^2+5 c d^2\right )}{16 a^3 c \left (a+c x^2\right )}-\frac{4 a d e-x \left (a e^2+5 c d^2\right )}{24 a^2 c \left (a+c x^2\right )^2}-\frac{(d+e x) (a e-c d x)}{6 a c \left (a+c x^2\right )^3} \]
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Rubi [A] time = 0.0640524, antiderivative size = 145, 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 = {739, 639, 199, 205} \[ \frac{\left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{3/2}}+\frac{x \left (a e^2+5 c d^2\right )}{16 a^3 c \left (a+c x^2\right )}-\frac{4 a d e-x \left (a e^2+5 c d^2\right )}{24 a^2 c \left (a+c x^2\right )^2}-\frac{(d+e x) (a e-c d x)}{6 a c \left (a+c x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 739
Rule 639
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a+c x^2\right )^4} \, dx &=-\frac{(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}+\frac{\int \frac{5 c d^2+a e^2+4 c d e x}{\left (a+c x^2\right )^3} \, dx}{6 a c}\\ &=-\frac{(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac{4 a d e-\left (5 c d^2+a e^2\right ) x}{24 a^2 c \left (a+c x^2\right )^2}+\frac{\left (5 c d^2+a e^2\right ) \int \frac{1}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c}\\ &=-\frac{(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac{4 a d e-\left (5 c d^2+a e^2\right ) x}{24 a^2 c \left (a+c x^2\right )^2}+\frac{\left (5 c d^2+a e^2\right ) x}{16 a^3 c \left (a+c x^2\right )}+\frac{\left (5 c d^2+a e^2\right ) \int \frac{1}{a+c x^2} \, dx}{16 a^3 c}\\ &=-\frac{(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac{4 a d e-\left (5 c d^2+a e^2\right ) x}{24 a^2 c \left (a+c x^2\right )^2}+\frac{\left (5 c d^2+a e^2\right ) x}{16 a^3 c \left (a+c x^2\right )}+\frac{\left (5 c d^2+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0910285, size = 127, normalized size = 0.88 \[ \frac{a^2 c x \left (33 d^2+8 e^2 x^2\right )-a^3 e (16 d+3 e x)+a c^2 x^3 \left (40 d^2+3 e^2 x^2\right )+15 c^3 d^2 x^5}{48 a^3 c \left (a+c x^2\right )^3}+\frac{\left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 129, normalized size = 0.9 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}} \left ({\frac{ \left ( a{e}^{2}+5\,c{d}^{2} \right ) c{x}^{5}}{16\,{a}^{3}}}+{\frac{ \left ( a{e}^{2}+5\,c{d}^{2} \right ){x}^{3}}{6\,{a}^{2}}}-{\frac{ \left ( a{e}^{2}-11\,c{d}^{2} \right ) x}{16\,ac}}-{\frac{de}{3\,c}} \right ) }+{\frac{{e}^{2}}{16\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,{d}^{2}}{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 = 2.1203, size = 983, normalized size = 6.78 \begin{align*} \left [-\frac{32 \, a^{4} c d e - 6 \,{\left (5 \, a c^{4} d^{2} + a^{2} c^{3} e^{2}\right )} x^{5} - 16 \,{\left (5 \, a^{2} c^{3} d^{2} + a^{3} c^{2} e^{2}\right )} x^{3} + 3 \,{\left ({\left (5 \, c^{4} d^{2} + a c^{3} e^{2}\right )} x^{6} + 5 \, a^{3} c d^{2} + a^{4} e^{2} + 3 \,{\left (5 \, a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{4} + 3 \,{\left (5 \, a^{2} c^{2} d^{2} + a^{3} c 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 ) - 6 \,{\left (11 \, a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} x}{96 \,{\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}, -\frac{16 \, a^{4} c d e - 3 \,{\left (5 \, a c^{4} d^{2} + a^{2} c^{3} e^{2}\right )} x^{5} - 8 \,{\left (5 \, a^{2} c^{3} d^{2} + a^{3} c^{2} e^{2}\right )} x^{3} - 3 \,{\left ({\left (5 \, c^{4} d^{2} + a c^{3} e^{2}\right )} x^{6} + 5 \, a^{3} c d^{2} + a^{4} e^{2} + 3 \,{\left (5 \, a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{4} + 3 \,{\left (5 \, a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) - 3 \,{\left (11 \, a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} x}{48 \,{\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.4061, size = 214, normalized size = 1.48 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{7} c^{3}}} \left (a e^{2} + 5 c d^{2}\right ) \log{\left (- a^{4} c \sqrt{- \frac{1}{a^{7} c^{3}}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{a^{7} c^{3}}} \left (a e^{2} + 5 c d^{2}\right ) \log{\left (a^{4} c \sqrt{- \frac{1}{a^{7} c^{3}}} + x \right )}}{32} + \frac{- 16 a^{3} d e + x^{5} \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right ) + x^{3} \left (8 a^{2} c e^{2} + 40 a c^{2} d^{2}\right ) + x \left (- 3 a^{3} e^{2} + 33 a^{2} c d^{2}\right )}{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 = 1.31693, size = 166, normalized size = 1.14 \begin{align*} \frac{{\left (5 \, c d^{2} + a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3} c} + \frac{15 \, c^{3} d^{2} x^{5} + 3 \, a c^{2} x^{5} e^{2} + 40 \, a c^{2} d^{2} x^{3} + 8 \, a^{2} c x^{3} e^{2} + 33 \, a^{2} c d^{2} x - 3 \, a^{3} x e^{2} - 16 \, a^{3} d 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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