Optimal. Leaf size=130 \[ -\frac{2 a e (a C+A c)-c x (a B e+a C d+3 A c d)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+a C d+3 A c d)}{8 a^{5/2} c^{3/2}}-\frac{(d+e x) (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.107909, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1645, 639, 205} \[ -\frac{2 a e (a C+A c)-c x (a B e+a C d+3 A c d)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+a C d+3 A c d)}{8 a^{5/2} c^{3/2}}-\frac{(d+e x) (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1645
Rule 639
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx &=-\frac{(a B-(A c-a C) x) (d+e x)}{4 a c \left (a+c x^2\right )^2}-\frac{\int \frac{-3 A c d-a (C d+B e)-2 (A c+a C) e x}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)}{4 a c \left (a+c x^2\right )^2}-\frac{2 a (A c+a C) e-c (3 A c d+a C d+a B e) x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{(3 A c d+a C d+a B e) \int \frac{1}{a+c x^2} \, dx}{8 a^2 c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)}{4 a c \left (a+c x^2\right )^2}-\frac{2 a (A c+a C) e-c (3 A c d+a C d+a B e) x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{(3 A c d+a C d+a B e) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.112773, size = 137, normalized size = 1.05 \[ \frac{\frac{2 a^{3/2} \left (a^2 C e-a c (A e+B (d+e x)+C d x)+A c^2 d x\right )}{\left (a+c x^2\right )^2}+\frac{\sqrt{a} \left (-4 a^2 C e+a c x (B e+C d)+3 A c^2 d x\right )}{a+c x^2}+\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+a C d+3 A c d)}{8 a^{5/2} c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 157, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( 3\,Acd+aBe+Cad \right ){x}^{3}}{8\,{a}^{2}}}-{\frac{Ce{x}^{2}}{2\,c}}+{\frac{ \left ( 5\,Acd-aBe-Cad \right ) x}{8\,ac}}-{\frac{Ace+Bcd+aCe}{4\,{c}^{2}}} \right ) }+{\frac{3\,Ad}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Be}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Cd}{8\,ac}\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 = 2.00029, size = 991, normalized size = 7.62 \begin{align*} \left [-\frac{8 \, C a^{3} c e x^{2} + 4 \, B a^{3} c d - 2 \,{\left (B a^{2} c^{2} e +{\left (C a^{2} c^{2} + 3 \, A a c^{3}\right )} d\right )} x^{3} +{\left (B a^{3} e +{\left (B a c^{2} e +{\left (C a c^{2} + 3 \, A c^{3}\right )} d\right )} x^{4} + 2 \,{\left (B a^{2} c e +{\left (C a^{2} c + 3 \, A a c^{2}\right )} d\right )} x^{2} +{\left (C a^{3} + 3 \, A a^{2} c\right )} d\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 4 \,{\left (C a^{4} + A a^{3} c\right )} e + 2 \,{\left (B a^{3} c e +{\left (C a^{3} c - 5 \, A a^{2} c^{2}\right )} d\right )} x}{16 \,{\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}, -\frac{4 \, C a^{3} c e x^{2} + 2 \, B a^{3} c d -{\left (B a^{2} c^{2} e +{\left (C a^{2} c^{2} + 3 \, A a c^{3}\right )} d\right )} x^{3} -{\left (B a^{3} e +{\left (B a c^{2} e +{\left (C a c^{2} + 3 \, A c^{3}\right )} d\right )} x^{4} + 2 \,{\left (B a^{2} c e +{\left (C a^{2} c + 3 \, A a c^{2}\right )} d\right )} x^{2} +{\left (C a^{3} + 3 \, A a^{2} c\right )} d\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + 2 \,{\left (C a^{4} + A a^{3} c\right )} e +{\left (B a^{3} c e +{\left (C a^{3} c - 5 \, A a^{2} c^{2}\right )} d\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 [A] time = 34.2871, size = 240, normalized size = 1.85 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{5} c^{3}}} \left (3 A c d + B a e + C a d\right ) \log{\left (- a^{3} c \sqrt{- \frac{1}{a^{5} c^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} c^{3}}} \left (3 A c d + B a e + C a d\right ) \log{\left (a^{3} c \sqrt{- \frac{1}{a^{5} c^{3}}} + x \right )}}{16} + \frac{- 2 A a^{2} c e - 2 B a^{2} c d - 2 C a^{3} e - 4 C a^{2} c e x^{2} + x^{3} \left (3 A c^{3} d + B a c^{2} e + C a c^{2} d\right ) + x \left (5 A a c^{2} d - B a^{2} c e - C a^{2} c d\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 = 1.14763, size = 205, normalized size = 1.58 \begin{align*} \frac{{\left (C a d + 3 \, A c d + B a e\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c} + \frac{C a c^{2} d x^{3} + 3 \, A c^{3} d x^{3} + B a c^{2} x^{3} e - 4 \, C a^{2} c x^{2} e - C a^{2} c d x + 5 \, A a c^{2} d x - B a^{2} c x e - 2 \, B a^{2} c d - 2 \, C a^{3} e - 2 \, A a^{2} c e}{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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