Optimal. Leaf size=414 \[ \frac{\log \left (a+b x+c x^2\right ) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac{\log (d+e x) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-b^2 e^2 (a B e+3 A c d)+b c \left (-3 a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )-2 c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )+A b^3 e^3\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}+\frac{B d-A e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{A e (2 c d-b e)-B \left (c d^2-a e^2\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]
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Rubi [A] time = 0.803191, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {800, 634, 618, 206, 628} \[ \frac{\log \left (a+b x+c x^2\right ) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac{\log (d+e x) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-b^2 e^2 (a B e+3 A c d)+b c \left (-3 a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )-2 c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )+A b^3 e^3\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}+\frac{B d-A e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{A e (2 c d-b e)-B \left (c d^2-a e^2\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac{e (-B d+A e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{e \left (-B \left (c^2 d^3-3 a c d e^2+a b e^3\right )+A e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right )}{\left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{a B e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )+A \left (c^3 d^3-b^3 e^3-3 c^2 d e (b d+a e)+b c e^2 (3 b d+2 a e)\right )+c \left (B \left (c^2 d^3-3 a c d e^2+a b e^3\right )-A e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) x}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{B d-A e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{A e (2 c d-b e)-B \left (c d^2-a e^2\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{\left (B \left (c^2 d^3-3 a c d e^2+a b e^3\right )-A e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\int \frac{a B e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )+A \left (c^3 d^3-b^3 e^3-3 c^2 d e (b d+a e)+b c e^2 (3 b d+2 a e)\right )+c \left (B \left (c^2 d^3-3 a c d e^2+a b e^3\right )-A e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) x}{a+b x+c x^2} \, dx}{\left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{B d-A e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{A e (2 c d-b e)-B \left (c d^2-a e^2\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{\left (B \left (c^2 d^3-3 a c d e^2+a b e^3\right )-A e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\left (B \left (c^2 d^3-3 a c d e^2+a b e^3\right )-A e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac{\left (A b^3 e^3-b^2 e^2 (3 A c d+a B e)+b c \left (B c d^3+3 A c d^2 e+3 a B d e^2-3 a A e^3\right )-2 c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{B d-A e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{A e (2 c d-b e)-B \left (c d^2-a e^2\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{\left (B \left (c^2 d^3-3 a c d e^2+a b e^3\right )-A e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\left (B \left (c^2 d^3-3 a c d e^2+a b e^3\right )-A e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac{\left (A b^3 e^3-b^2 e^2 (3 A c d+a B e)+b c \left (B c d^3+3 A c d^2 e+3 a B d e^2-3 a A e^3\right )-2 c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{B d-A e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{A e (2 c d-b e)-B \left (c d^2-a e^2\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\left (A b^3 e^3-b^2 e^2 (3 A c d+a B e)+b c \left (B c d^3+3 A c d^2 e+3 a B d e^2-3 a A e^3\right )-2 c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3}-\frac{\left (B \left (c^2 d^3-3 a c d e^2+a b e^3\right )-A e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\left (B \left (c^2 d^3-3 a c d e^2+a b e^3\right )-A e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.614782, size = 413, normalized size = 1. \[ -\frac{\log (d+e x) \left (A e \left (c e (a e+3 b d)-b^2 e^2-3 c^2 d^2\right )+B \left (a b e^3-3 a c d e^2+c^2 d^3\right )\right )}{\left (e (a e-b d)+c d^2\right )^3}+\frac{\log (a+x (b+c x)) \left (A e \left (c e (a e+3 b d)-b^2 e^2-3 c^2 d^2\right )+B \left (a b e^3-3 a c d e^2+c^2 d^3\right )\right )}{2 \left (e (a e-b d)+c d^2\right )^3}+\frac{\tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (-b^2 e^2 (a B e+3 A c d)+b c \left (-3 a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+2 c \left (A c d \left (3 a e^2-c d^2\right )+a B e \left (a e^2-3 c d^2\right )\right )+A b^3 e^3\right )}{\sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )^3}+\frac{B \left (c d^2-a e^2\right )+A e (b e-2 c d)}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac{B d-A e}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 1339, normalized size = 3.2 \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 = 1.12516, size = 1076, normalized size = 2.6 \begin{align*} \frac{{\left (B c^{2} d^{3} - 3 \, A c^{2} d^{2} e - 3 \, B a c d e^{2} + 3 \, A b c d e^{2} + B a b e^{3} - A b^{2} e^{3} + A a c e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )}} - \frac{{\left (B c^{2} d^{3} e - 3 \, A c^{2} d^{2} e^{2} - 3 \, B a c d e^{3} + 3 \, A b c d e^{3} + B a b e^{4} - A b^{2} e^{4} + A a c e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} + 3 \, a c^{2} d^{4} e^{3} - b^{3} d^{3} e^{4} - 6 \, a b c d^{3} e^{4} + 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} c d^{2} e^{5} - 3 \, a^{2} b d e^{6} + a^{3} e^{7}} - \frac{{\left (B b c^{2} d^{3} - 2 \, A c^{3} d^{3} - 6 \, B a c^{2} d^{2} e + 3 \, A b c^{2} d^{2} e + 3 \, B a b c d e^{2} - 3 \, A b^{2} c d e^{2} + 6 \, A a c^{2} d e^{2} - B a b^{2} e^{3} + A b^{3} e^{3} + 2 \, B a^{2} c e^{3} - 3 \, A a b c e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{3 \, B c^{2} d^{5} - 4 \, B b c d^{4} e - 5 \, A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, B a c d^{3} e^{2} + 8 \, A b c d^{3} e^{2} - 3 \, A b^{2} d^{2} e^{3} - 6 \, A a c d^{2} e^{3} - B a^{2} d e^{4} + 4 \, A a b d e^{4} - A a^{2} e^{5} + 2 \,{\left (B c^{2} d^{4} e - B b c d^{3} e^{2} - 2 \, A c^{2} d^{3} e^{2} + 3 \, A b c d^{2} e^{3} + B a b d e^{4} - A b^{2} d e^{4} - 2 \, A a c d e^{4} - B a^{2} e^{5} + A a b e^{5}\right )} x}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}^{3}{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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