Optimal. Leaf size=146 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (c d (2 a B e+a C d+A c d)+a e^2 (A c-3 a C)\right )}{2 a^{3/2} c^{5/2}}-\frac{(d+e x)^2 (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}-\frac{e^2 x (A c-3 a C)}{2 a c^2}+\frac{e \log \left (a+c x^2\right ) (B e+2 C d)}{2 c^2} \]
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Rubi [A] time = 0.245414, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1645, 774, 635, 205, 260} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (c d (2 a B e+a C d+A c d)+a e^2 (A c-3 a C)\right )}{2 a^{3/2} c^{5/2}}-\frac{(d+e x)^2 (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}-\frac{e^2 x (A c-3 a C)}{2 a c^2}+\frac{e \log \left (a+c x^2\right ) (B e+2 C d)}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 1645
Rule 774
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 \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}{2 a c \left (a+c x^2\right )}-\frac{\int \frac{(d+e x) (-A c d-a C d-2 a B e+(A c-3 a C) e x)}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{(A c-3 a C) e^2 x}{2 a c^2}-\frac{(a B-(A c-a C) x) (d+e x)^2}{2 a c \left (a+c x^2\right )}-\frac{\int \frac{-a (A c-3 a C) e^2+c d (-A c d-a C d-2 a B e)+c ((A c-3 a C) d e+e (-A c d-a C d-2 a B e)) x}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac{(A c-3 a C) e^2 x}{2 a c^2}-\frac{(a B-(A c-a C) x) (d+e x)^2}{2 a c \left (a+c x^2\right )}+\frac{(e (2 C d+B e)) \int \frac{x}{a+c x^2} \, dx}{c}+\frac{\left (a (A c-3 a C) e^2+c d (A c d+a C d+2 a B e)\right ) \int \frac{1}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac{(A c-3 a C) e^2 x}{2 a c^2}-\frac{(a B-(A c-a C) x) (d+e x)^2}{2 a c \left (a+c x^2\right )}+\frac{\left (a (A c-3 a C) e^2+c d (A c d+a C d+2 a B e)\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}+\frac{e (2 C d+B e) \log \left (a+c x^2\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.150622, size = 175, normalized size = 1.2 \[ \frac{\frac{\sqrt{c} \left (a^2 e (B e+2 C d+C e x)-a c \left (A e (2 d+e x)+B d (d+2 e x)+C d^2 x\right )+A c^2 d^2 x\right )}{a \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c \left (a e^2+c d^2\right )+a \left (c d (2 B e+C d)-3 a C e^2\right )\right )}{a^{3/2}}+\sqrt{c} e \log \left (a+c x^2\right ) (B e+2 C d)+2 \sqrt{c} C e^2 x}{2 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 323, normalized size = 2.2 \begin{align*}{\frac{C{e}^{2}x}{{c}^{2}}}-{\frac{A{e}^{2}x}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{xA{d}^{2}}{ \left ( 2\,c{x}^{2}+2\,a \right ) a}}-{\frac{Bdex}{c \left ( c{x}^{2}+a \right ) }}+{\frac{aC{e}^{2}x}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{C{d}^{2}x}{2\,c \left ( c{x}^{2}+a \right ) }}-{\frac{Ade}{c \left ( c{x}^{2}+a \right ) }}+{\frac{aB{e}^{2}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{B{d}^{2}}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{Cade}{{c}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{\ln \left ( c{x}^{2}+a \right ) B{e}^{2}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) Cde}{{c}^{2}}}+{\frac{A{e}^{2}}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{A{d}^{2}}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Bde}{c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{3\,aC{e}^{2}}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{C{d}^{2}}{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 [B] time = 2.1014, size = 1276, normalized size = 8.74 \begin{align*} \left [\frac{4 \, C a^{2} c^{2} e^{2} x^{3} - 2 \, B a^{2} c^{2} d^{2} + 2 \, B a^{3} c e^{2} + 4 \,{\left (C a^{3} c - A a^{2} c^{2}\right )} d e -{\left (2 \, B a^{2} c d e +{\left (C a^{2} c + A a c^{2}\right )} d^{2} -{\left (3 \, C a^{3} - A a^{2} c\right )} e^{2} +{\left (2 \, B a c^{2} d e +{\left (C a c^{2} + A c^{3}\right )} d^{2} -{\left (3 \, C a^{2} c - A a c^{2}\right )} 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 (2 \, B a^{2} c^{2} d e +{\left (C a^{2} c^{2} - A a c^{3}\right )} d^{2} -{\left (3 \, C a^{3} c - A a^{2} c^{2}\right )} e^{2}\right )} x + 2 \,{\left (2 \, C a^{3} c d e + B a^{3} c e^{2} +{\left (2 \, C a^{2} c^{2} d e + B a^{2} c^{2} e^{2}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )}{4 \,{\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}, \frac{2 \, C a^{2} c^{2} e^{2} x^{3} - B a^{2} c^{2} d^{2} + B a^{3} c e^{2} + 2 \,{\left (C a^{3} c - A a^{2} c^{2}\right )} d e +{\left (2 \, B a^{2} c d e +{\left (C a^{2} c + A a c^{2}\right )} d^{2} -{\left (3 \, C a^{3} - A a^{2} c\right )} e^{2} +{\left (2 \, B a c^{2} d e +{\left (C a c^{2} + A c^{3}\right )} d^{2} -{\left (3 \, C a^{2} c - A a c^{2}\right )} e^{2}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (2 \, B a^{2} c^{2} d e +{\left (C a^{2} c^{2} - A a c^{3}\right )} d^{2} -{\left (3 \, C a^{3} c - A a^{2} c^{2}\right )} e^{2}\right )} x +{\left (2 \, C a^{3} c d e + B a^{3} c e^{2} +{\left (2 \, C a^{2} c^{2} d e + B a^{2} c^{2} e^{2}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 19.2175, size = 593, normalized size = 4.06 \begin{align*} \frac{C e^{2} x}{c^{2}} + \left (\frac{e \left (B e + 2 C d\right )}{2 c^{2}} - \frac{\sqrt{- a^{3} c^{5}} \left (- A a c e^{2} - A c^{2} d^{2} - 2 B a c d e + 3 C a^{2} e^{2} - C a c d^{2}\right )}{4 a^{3} c^{5}}\right ) \log{\left (x + \frac{2 B a^{2} e^{2} + 4 C a^{2} d e - 4 a^{2} c^{2} \left (\frac{e \left (B e + 2 C d\right )}{2 c^{2}} - \frac{\sqrt{- a^{3} c^{5}} \left (- A a c e^{2} - A c^{2} d^{2} - 2 B a c d e + 3 C a^{2} e^{2} - C a c d^{2}\right )}{4 a^{3} c^{5}}\right )}{- A a c e^{2} - A c^{2} d^{2} - 2 B a c d e + 3 C a^{2} e^{2} - C a c d^{2}} \right )} + \left (\frac{e \left (B e + 2 C d\right )}{2 c^{2}} + \frac{\sqrt{- a^{3} c^{5}} \left (- A a c e^{2} - A c^{2} d^{2} - 2 B a c d e + 3 C a^{2} e^{2} - C a c d^{2}\right )}{4 a^{3} c^{5}}\right ) \log{\left (x + \frac{2 B a^{2} e^{2} + 4 C a^{2} d e - 4 a^{2} c^{2} \left (\frac{e \left (B e + 2 C d\right )}{2 c^{2}} + \frac{\sqrt{- a^{3} c^{5}} \left (- A a c e^{2} - A c^{2} d^{2} - 2 B a c d e + 3 C a^{2} e^{2} - C a c d^{2}\right )}{4 a^{3} c^{5}}\right )}{- A a c e^{2} - A c^{2} d^{2} - 2 B a c d e + 3 C a^{2} e^{2} - C a c d^{2}} \right )} + \frac{- 2 A a c d e + B a^{2} e^{2} - B a c d^{2} + 2 C a^{2} d e + x \left (- A a c e^{2} + A c^{2} d^{2} - 2 B a c d e + C a^{2} e^{2} - C a c d^{2}\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 = 1.16413, size = 248, normalized size = 1.7 \begin{align*} \frac{C x e^{2}}{c^{2}} + \frac{{\left (2 \, C d e + B e^{2}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{{\left (C a c d^{2} + A c^{2} d^{2} + 2 \, B a c d e - 3 \, C a^{2} e^{2} + A a c e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c^{2}} - \frac{B a c d^{2} - 2 \, C a^{2} d e + 2 \, A a c d e - B a^{2} e^{2} +{\left (C a c d^{2} - A c^{2} d^{2} + 2 \, B a c d e - C a^{2} e^{2} + A a c e^{2}\right )} x}{2 \,{\left (c x^{2} + a\right )} a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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