Optimal. Leaf size=291 \[ \frac{\left (-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )+2 a^3 c d^2+2 a b^2 e (b d+2 c e)+b^4 \left (-e^2\right )\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{(a d-b e) \left (a b d+2 a c e+b^2 (-e)\right ) \log \left (a x^2+b x+c\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}-\frac{e^3}{d^2 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac{e^3 \log (d+e x) \left (4 a d^2-e (3 b d-2 c e)\right )}{d^3 \left (a d^2-e (b d-c e)\right )^2}-\frac{\log (x) (b d+2 c e)}{c^2 d^3}-\frac{1}{c d^2 x} \]
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Rubi [A] time = 0.563207, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1569, 893, 634, 618, 206, 628} \[ \frac{\left (-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )+2 a^3 c d^2+2 a b^2 e (b d+2 c e)+b^4 \left (-e^2\right )\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{(a d-b e) \left (a b d+2 a c e+b^2 (-e)\right ) \log \left (a x^2+b x+c\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}-\frac{e^3}{d^2 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac{e^3 \log (d+e x) \left (4 a d^2-e (3 b d-2 c e)\right )}{d^3 \left (a d^2-e (b d-c e)\right )^2}-\frac{\log (x) (b d+2 c e)}{c^2 d^3}-\frac{1}{c d^2 x} \]
Antiderivative was successfully verified.
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Rule 1569
Rule 893
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) x^4 (d+e x)^2} \, dx &=\int \frac{1}{x^2 (d+e x)^2 \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{1}{c d^2 x^2}+\frac{-b d-2 c e}{c^2 d^3 x}+\frac{e^4}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)^2}+\frac{e^4 \left (4 a d^2-e (3 b d-2 c e)\right )}{d^3 \left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac{-a^3 c d^2+b^4 e^2-a b^2 e (2 b d+3 c e)+a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )+a (a d-b e) \left (a b d-b^2 e+2 a c e\right ) x}{c^2 \left (a d^2-e (b d-c e)\right )^2 \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac{1}{c d^2 x}-\frac{e^3}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac{(b d+2 c e) \log (x)}{c^2 d^3}+\frac{e^3 \left (4 a d^2-e (3 b d-2 c e)\right ) \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )^2}+\frac{\int \frac{-a^3 c d^2+b^4 e^2-a b^2 e (2 b d+3 c e)+a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )+a (a d-b e) \left (a b d-b^2 e+2 a c e\right ) x}{c+b x+a x^2} \, dx}{c^2 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac{1}{c d^2 x}-\frac{e^3}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac{(b d+2 c e) \log (x)}{c^2 d^3}+\frac{e^3 \left (4 a d^2-e (3 b d-2 c e)\right ) \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )^2}+\frac{\left ((a d-b e) \left (a b d-b^2 e+2 a c e\right )\right ) \int \frac{b+2 a x}{c+b x+a x^2} \, dx}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}-\frac{\left (2 a^3 c d^2-b^4 e^2+2 a b^2 e (b d+2 c e)-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )\right ) \int \frac{1}{c+b x+a x^2} \, dx}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac{1}{c d^2 x}-\frac{e^3}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac{(b d+2 c e) \log (x)}{c^2 d^3}+\frac{e^3 \left (4 a d^2-e (3 b d-2 c e)\right ) \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )^2}+\frac{(a d-b e) \left (a b d-b^2 e+2 a c e\right ) \log \left (c+b x+a x^2\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (2 a^3 c d^2-b^4 e^2+2 a b^2 e (b d+2 c e)-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{c^2 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac{1}{c d^2 x}-\frac{e^3}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{\left (2 a^3 c d^2-b^4 e^2+2 a b^2 e (b d+2 c e)-a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )\right ) \tanh ^{-1}\left (\frac{b+2 a x}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac{(b d+2 c e) \log (x)}{c^2 d^3}+\frac{e^3 \left (4 a d^2-e (3 b d-2 c e)\right ) \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )^2}+\frac{(a d-b e) \left (a b d-b^2 e+2 a c e\right ) \log \left (c+b x+a x^2\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )^2}\\ \end{align*}
Mathematica [A] time = 0.380898, size = 287, normalized size = 0.99 \[ \frac{\left (a^2 \left (b^2 d^2+6 b c d e+2 c^2 e^2\right )-2 a^3 c d^2-2 a b^2 e (b d+2 c e)+b^4 e^2\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{c^2 \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )^2}+\frac{(a d-b e) \left (a b d+2 a c e+b^2 (-e)\right ) \log (x (a x+b)+c)}{2 c^2 \left (a d^2+e (c e-b d)\right )^2}-\frac{e^3}{d^2 (d+e x) \left (a d^2+e (c e-b d)\right )}+\frac{e^3 \log (d+e x) \left (4 a d^2+e (2 c e-3 b d)\right )}{d^3 \left (a d^2+e (c e-b d)\right )^2}-\frac{\log (x) (b d+2 c e)}{c^2 d^3}-\frac{1}{c d^2 x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 791, normalized size = 2.7 \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 = 1.10928, size = 657, normalized size = 2.26 \begin{align*} -\frac{{\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} c d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 6 \, a^{2} b c d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} \arctan \left (-\frac{{\left (2 \, a d - \frac{2 \, a d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{2} c^{2} d^{4} - 2 \, a b c^{2} d^{3} e + b^{2} c^{2} d^{2} e^{2} + 2 \, a c^{3} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + c^{4} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (a^{2} b d^{2} - 2 \, a b^{2} d e + 2 \, a^{2} c d e + b^{3} e^{2} - 2 \, a b c e^{2}\right )} \log \left (-a + \frac{2 \, a d}{x e + d} - \frac{a d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} - \frac{c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (a^{2} c^{2} d^{4} - 2 \, a b c^{2} d^{3} e + b^{2} c^{2} d^{2} e^{2} + 2 \, a c^{3} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + c^{4} e^{4}\right )}} - \frac{e^{7}}{{\left (a d^{4} e^{4} - b d^{3} e^{5} + c d^{2} e^{6}\right )}{\left (x e + d\right )}} - \frac{{\left (b d e + 2 \, c e^{2}\right )} e^{\left (-1\right )} \log \left ({\left | -\frac{d}{x e + d} + 1 \right |}\right )}{c^{2} d^{3}} + \frac{e}{c d^{3}{\left (\frac{d}{x e + d} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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