Optimal. Leaf size=226 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a \left (c d^2-a e^2\right ) (C d-B e)\right )}{2 a^{3/2} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{a (a C e-A c e+B c d)-c x (a B e-a C d+A c d)}{2 a c \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac{e \log \left (a+c x^2\right ) \left (A e^2-B d e+C d^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{e \log (d+e x) \left (A e^2-B d e+C d^2\right )}{\left (a e^2+c d^2\right )^2} \]
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Rubi [A] time = 0.434996, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1647, 801, 635, 205, 260} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a \left (c d^2-a e^2\right ) (C d-B e)\right )}{2 a^{3/2} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{a (a C e-A c e+B c d)-c x (a B e-a C d+A c d)}{2 a c \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac{e \log \left (a+c x^2\right ) \left (A e^2-B d e+C d^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{e \log (d+e x) \left (A e^2-B d e+C d^2\right )}{\left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1647
Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{(d+e x) \left (a+c x^2\right )^2} \, dx &=-\frac{a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\int \frac{-\frac{c \left (a d (C d-B e)+A \left (c d^2+2 a e^2\right )\right )}{c d^2+a e^2}-\frac{c e (A c d-a C d+a B e) x}{c d^2+a e^2}}{(d+e x) \left (a+c x^2\right )} \, dx}{2 a c}\\ &=-\frac{a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\int \left (-\frac{2 a c e^2 \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac{c \left (-a (C d-B e) \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )+2 a c e \left (C d^2-B d e+A e^2\right ) x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{2 a c}\\ &=-\frac{a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{e \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac{\int \frac{-a (C d-B e) \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )+2 a c e \left (C d^2-B d e+A e^2\right ) x}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=-\frac{a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{e \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac{\left (c e \left (C d^2-B d e+A e^2\right )\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac{\left (a (C d-B e) \left (c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right ) \int \frac{1}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=-\frac{a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{\left (a (C d-B e) \left (c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c} \left (c d^2+a e^2\right )^2}+\frac{e \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac{e \left (C d^2-B d e+A e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.261031, size = 195, normalized size = 0.86 \[ \frac{\frac{\left (a e^2+c d^2\right ) \left (a^2 (-C) e+a c (A e-B d+B e x-C d x)+A c^2 d x\right )}{a c \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a \left (c d^2-a e^2\right ) (C d-B e)\right )}{a^{3/2} \sqrt{c}}-e \log \left (a+c x^2\right ) \left (e (A e-B d)+C d^2\right )+2 e \log (d+e x) \left (e (A e-B d)+C d^2\right )}{2 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 742, normalized size = 3.3 \begin{align*} \text{result too large to display} \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20387, size = 473, normalized size = 2.09 \begin{align*} -\frac{{\left (C d^{2} e - B d e^{2} + A e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac{{\left (C d^{2} e^{2} - B d e^{3} + A e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}} + \frac{{\left (C a c d^{3} + A c^{2} d^{3} - B a c d^{2} e - C a^{2} d e^{2} + 3 \, A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{a c}} - \frac{B a c^{2} d^{3} + C a^{2} c d^{2} e - A a c^{2} d^{2} e + B a^{2} c d e^{2} + C a^{3} e^{3} - A a^{2} c e^{3} +{\left (C a c^{2} d^{3} - A c^{3} d^{3} - B a c^{2} d^{2} e + C a^{2} c d e^{2} - A a c^{2} d e^{2} - B a^{2} c e^{3}\right )} x}{2 \,{\left (c d^{2} + a e^{2}\right )}^{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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