Optimal. Leaf size=156 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (c d (2 a B e+a C d+3 A c d)+a e^2 (3 a C+A c)\right )}{8 a^{5/2} c^{5/2}}-\frac{(d+e x) (a e (3 a C+A c)-c x (2 a B e+a C d+3 A c d))}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^2 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.230675, antiderivative size = 175, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1645, 778, 205} \[ -\frac{x \left (a e^2 (3 a C+A c)-c d (2 a B e+a C d+3 A c d)\right )+2 a e (a B e+2 a C d+2 A c d)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (c d (2 a B e+a C d+3 A c d)+a e^2 (3 a C+A c)\right )}{8 a^{5/2} c^{5/2}}-\frac{(d+e x)^2 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1645
Rule 778
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 \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)^2}{4 a c \left (a+c x^2\right )^2}-\frac{\int \frac{(d+e x) (-3 A c d-a C d-2 a B e-(A c+3 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)^2}{4 a c \left (a+c x^2\right )^2}-\frac{2 a e (2 A c d+2 a C d+a B e)+\left (a (A c+3 a C) e^2-c d (3 A c d+a C d+2 a B e)\right ) x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\left (a (A c+3 a C) e^2+c d (3 A c d+a C d+2 a B e)\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^2}{4 a c \left (a+c x^2\right )^2}-\frac{2 a e (2 A c d+2 a C d+a B e)+\left (a (A c+3 a C) e^2-c d (3 A c d+a C d+2 a B e)\right ) x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\left (a (A c+3 a C) e^2+c d (3 A c d+a C d+2 a B e)\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.153778, size = 211, normalized size = 1.35 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c \left (a e^2+3 c d^2\right )+a \left (3 a C e^2+c d (2 B e+C d)\right )\right )}{8 a^{5/2} c^{5/2}}+\frac{a^2 (-e) (4 B e+8 C d+5 C e x)+a c x \left (e (A e+2 B d)+C d^2\right )+3 A c^2 d^2 x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{a^2 e (B e+2 C d+C e x)-a c \left (A e (2 d+e x)+B d (d+2 e x)+C d^2 x\right )+A c^2 d^2 x}{4 a c^2 \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 283, normalized size = 1.8 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( aA{e}^{2}c+3\,A{c}^{2}{d}^{2}+2\,Bacde-5\,{a}^{2}C{e}^{2}+Cac{d}^{2} \right ){x}^{3}}{8\,{a}^{2}c}}-{\frac{e \left ( Be+2\,Cd \right ){x}^{2}}{2\,c}}-{\frac{ \left ( aA{e}^{2}c-5\,A{c}^{2}{d}^{2}+2\,Bacde+3\,{a}^{2}C{e}^{2}+Cac{d}^{2} \right ) x}{8\,a{c}^{2}}}-{\frac{2\,Acde+aB{e}^{2}+Bc{d}^{2}+2\,Cade}{4\,{c}^{2}}} \right ) }+{\frac{A{e}^{2}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,A{d}^{2}}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Bde}{4\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,C{e}^{2}}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{C{d}^{2}}{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 [B] time = 2.48735, size = 1644, normalized size = 10.54 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 163.039, size = 389, normalized size = 2.49 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{5} c^{5}}} \left (A a c e^{2} + 3 A c^{2} d^{2} + 2 B a c d e + 3 C a^{2} e^{2} + C a c d^{2}\right ) \log{\left (- a^{3} c^{2} \sqrt{- \frac{1}{a^{5} c^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} c^{5}}} \left (A a c e^{2} + 3 A c^{2} d^{2} + 2 B a c d e + 3 C a^{2} e^{2} + C a c d^{2}\right ) \log{\left (a^{3} c^{2} \sqrt{- \frac{1}{a^{5} c^{5}}} + x \right )}}{16} - \frac{4 A a^{2} c d e + 2 B a^{3} e^{2} + 2 B a^{2} c d^{2} + 4 C a^{3} d e + x^{3} \left (- A a c^{2} e^{2} - 3 A c^{3} d^{2} - 2 B a c^{2} d e + 5 C a^{2} c e^{2} - C a c^{2} d^{2}\right ) + x^{2} \left (4 B a^{2} c e^{2} + 8 C a^{2} c d e\right ) + x \left (A a^{2} c e^{2} - 5 A a c^{2} d^{2} + 2 B a^{2} c d e + 3 C a^{3} e^{2} + C a^{2} c d^{2}\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 = 1.15984, size = 343, normalized size = 2.2 \begin{align*} \frac{{\left (C a c d^{2} + 3 \, 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 )}{8 \, \sqrt{a c} a^{2} c^{2}} + \frac{C a c^{2} d^{2} x^{3} + 3 \, A c^{3} d^{2} x^{3} + 2 \, B a c^{2} d x^{3} e - 5 \, C a^{2} c x^{3} e^{2} + A a c^{2} x^{3} e^{2} - 8 \, C a^{2} c d x^{2} e - C a^{2} c d^{2} x + 5 \, A a c^{2} d^{2} x - 4 \, B a^{2} c x^{2} e^{2} - 2 \, B a^{2} c d x e - 2 \, B a^{2} c d^{2} - 3 \, C a^{3} x e^{2} - A a^{2} c x e^{2} - 4 \, C a^{3} d e - 4 \, A a^{2} c d e - 2 \, B a^{3} e^{2}}{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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