Optimal. Leaf size=97 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+a C d+A c d)}{2 a^{3/2} c^{3/2}}-\frac{(d+e x) (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}+\frac{C e \log \left (a+c x^2\right )}{2 c^2} \]
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Rubi [A] time = 0.0824126, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1645, 635, 205, 260} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+a C d+A c d)}{2 a^{3/2} c^{3/2}}-\frac{(d+e x) (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}+\frac{C e \log \left (a+c x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 1645
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^2} \, dx &=-\frac{(a B-(A c-a C) x) (d+e x)}{2 a c \left (a+c x^2\right )}-\frac{\int \frac{-A c d-a (C d+B e)-2 a C e x}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac{(C e) \int \frac{x}{a+c x^2} \, dx}{c}+\frac{(A c d+a C d+a B e) \int \frac{1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac{(A c d+a C d+a B e) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}+\frac{C e \log \left (a+c x^2\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.104366, size = 102, normalized size = 1.05 \[ \frac{\frac{a^2 C e-a c (A e+B (d+e x)+C d x)+A c^2 d x}{a \left (a+c x^2\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+a C d+A c d)}{a^{3/2}}+C e \log \left (a+c x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 134, normalized size = 1.4 \begin{align*}{\frac{1}{c{x}^{2}+a} \left ({\frac{ \left ( Acd-aBe-Cad \right ) x}{2\,ac}}-{\frac{Ace+Bcd-aCe}{2\,{c}^{2}}} \right ) }+{\frac{Ce\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{2}}}+{\frac{Ad}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Be}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Cd}{2\,c}\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 = 1.76906, size = 717, normalized size = 7.39 \begin{align*} \left [-\frac{2 \, B a^{2} c d +{\left (B a^{2} e +{\left (B a c e +{\left (C a c + A c^{2}\right )} d\right )} x^{2} +{\left (C a^{2} + A a c\right )} d\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) - 2 \,{\left (C a^{3} - A a^{2} c\right )} e + 2 \,{\left (B a^{2} c e +{\left (C a^{2} c - A a c^{2}\right )} d\right )} x - 2 \,{\left (C a^{2} c e x^{2} + C a^{3} e\right )} \log \left (c x^{2} + a\right )}{4 \,{\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, -\frac{B a^{2} c d -{\left (B a^{2} e +{\left (B a c e +{\left (C a c + A c^{2}\right )} d\right )} x^{2} +{\left (C a^{2} + A a c\right )} d\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (C a^{3} - A a^{2} c\right )} e +{\left (B a^{2} c e +{\left (C a^{2} c - A a c^{2}\right )} d\right )} x -{\left (C a^{2} c e x^{2} + C a^{3} e\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.4949, size = 318, normalized size = 3.28 \begin{align*} \left (\frac{C e}{2 c^{2}} - \frac{\sqrt{- a^{3} c^{5}} \left (A c d + B a e + C a d\right )}{4 a^{3} c^{4}}\right ) \log{\left (x + \frac{- 2 C a^{2} e + 4 a^{2} c^{2} \left (\frac{C e}{2 c^{2}} - \frac{\sqrt{- a^{3} c^{5}} \left (A c d + B a e + C a d\right )}{4 a^{3} c^{4}}\right )}{A c^{2} d + B a c e + C a c d} \right )} + \left (\frac{C e}{2 c^{2}} + \frac{\sqrt{- a^{3} c^{5}} \left (A c d + B a e + C a d\right )}{4 a^{3} c^{4}}\right ) \log{\left (x + \frac{- 2 C a^{2} e + 4 a^{2} c^{2} \left (\frac{C e}{2 c^{2}} + \frac{\sqrt{- a^{3} c^{5}} \left (A c d + B a e + C a d\right )}{4 a^{3} c^{4}}\right )}{A c^{2} d + B a c e + C a c d} \right )} - \frac{A a c e + B a c d - C a^{2} e + x \left (- A c^{2} d + B a c e + C a c d\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15104, size = 151, normalized size = 1.56 \begin{align*} \frac{C e \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{{\left (C a d + A c d + B a e\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c} - \frac{{\left (C a d - A c d + B a e\right )} x + \frac{B a c d - C a^{2} e + A a c e}{c}}{2 \,{\left (c x^{2} + a\right )} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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