Optimal. Leaf size=770 \[ \frac{6 g^2 (f+g x) \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^3 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{6 g^2 (f+g x) \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^3 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{3 g (f+g x)^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^2 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{3 g (f+g x)^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^2 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{6 g^3 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (4,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^4 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{6 g^3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (4,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^4 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{(f+g x)^3 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )}{i \left (b-\sqrt{b^2-4 a c}\right )}-\frac{(f+g x)^3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )}{i \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^4 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )}{4 g \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^4 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right )}{4 g \left (b-\sqrt{b^2-4 a c}\right )} \]
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Rubi [A] time = 1.37399, antiderivative size = 770, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.159, Rules used = {2265, 2184, 2190, 2531, 6609, 2282, 6589} \[ \frac{6 g^2 (f+g x) \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^3 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{6 g^2 (f+g x) \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^3 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{3 g (f+g x)^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^2 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{3 g (f+g x)^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^2 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{6 g^3 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (4,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^4 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{6 g^3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (4,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^4 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{(f+g x)^3 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )}{i \left (b-\sqrt{b^2-4 a c}\right )}-\frac{(f+g x)^3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )}{i \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^4 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )}{4 g \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^4 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right )}{4 g \left (b-\sqrt{b^2-4 a c}\right )} \]
Antiderivative was successfully verified.
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Rule 2265
Rule 2184
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (d+e e^{h+572 x}\right ) (f+g x)^3}{a+b e^{h+572 x}+c e^{2 h+1144 x}} \, dx &=-\left (\left (-e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{(f+g x)^3}{b+\sqrt{b^2-4 a c}+2 c e^{h+572 x}} \, dx\right )+\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{(f+g x)^3}{b-\sqrt{b^2-4 a c}+2 c e^{h+572 x}} \, dx\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt{b^2-4 a c}\right ) g}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt{b^2-4 a c}\right ) g}-\frac{\left (2 c \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{e^{h+572 x} (f+g x)^3}{b+\sqrt{b^2-4 a c}+2 c e^{h+572 x}} \, dx}{b+\sqrt{b^2-4 a c}}-\frac{\left (2 c \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{e^{h+572 x} (f+g x)^3}{b-\sqrt{b^2-4 a c}+2 c e^{h+572 x}} \, dx}{b-\sqrt{b^2-4 a c}}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt{b^2-4 a c}\right ) g}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt{b^2-4 a c}\right ) g}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{572 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{572 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g\right ) \int (f+g x)^2 \log \left (1+\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right ) \, dx}{572 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g\right ) \int (f+g x)^2 \log \left (1+\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right ) \, dx}{572 \left (b-\sqrt{b^2-4 a c}\right )}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt{b^2-4 a c}\right ) g}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt{b^2-4 a c}\right ) g}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{572 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{572 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x)^2 \text{Li}_2\left (-\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{327184 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x)^2 \text{Li}_2\left (-\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{327184 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2\right ) \int (f+g x) \text{Li}_2\left (-\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right ) \, dx}{163592 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2\right ) \int (f+g x) \text{Li}_2\left (-\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right ) \, dx}{163592 \left (b-\sqrt{b^2-4 a c}\right )}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt{b^2-4 a c}\right ) g}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt{b^2-4 a c}\right ) g}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{572 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{572 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x)^2 \text{Li}_2\left (-\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{327184 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x)^2 \text{Li}_2\left (-\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{327184 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2 (f+g x) \text{Li}_3\left (-\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{93574624 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2 (f+g x) \text{Li}_3\left (-\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{93574624 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{\left (3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^3\right ) \int \text{Li}_3\left (-\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right ) \, dx}{93574624 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{\left (3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^3\right ) \int \text{Li}_3\left (-\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right ) \, dx}{93574624 \left (b-\sqrt{b^2-4 a c}\right )}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt{b^2-4 a c}\right ) g}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt{b^2-4 a c}\right ) g}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{572 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{572 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x)^2 \text{Li}_2\left (-\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{327184 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x)^2 \text{Li}_2\left (-\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{327184 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2 (f+g x) \text{Li}_3\left (-\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{93574624 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2 (f+g x) \text{Li}_3\left (-\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{93574624 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{\left (3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{x} \, dx,x,e^{h+572 x}\right )}{53524684928 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{\left (3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{2 c x}{-b+\sqrt{b^2-4 a c}}\right )}{x} \, dx,x,e^{h+572 x}\right )}{53524684928 \left (b-\sqrt{b^2-4 a c}\right )}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt{b^2-4 a c}\right ) g}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt{b^2-4 a c}\right ) g}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{572 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{572 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x)^2 \text{Li}_2\left (-\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{327184 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x)^2 \text{Li}_2\left (-\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{327184 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2 (f+g x) \text{Li}_3\left (-\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{93574624 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2 (f+g x) \text{Li}_3\left (-\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{93574624 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{3 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^3 \text{Li}_4\left (-\frac{2 c e^{h+572 x}}{b-\sqrt{b^2-4 a c}}\right )}{53524684928 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^3 \text{Li}_4\left (-\frac{2 c e^{h+572 x}}{b+\sqrt{b^2-4 a c}}\right )}{53524684928 \left (b+\sqrt{b^2-4 a c}\right )}\\ \end{align*}
Mathematica [B] time = 4.47916, size = 2441, normalized size = 3.17 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d+e{{\rm e}^{ix+h}} \right ) \left ( gx+f \right ) ^{3}}{a+b{{\rm e}^{ix+h}}+c{{\rm e}^{2\,ix+2\,h}}}}\, dx \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 [C] time = 1.7657, size = 4178, normalized size = 5.43 \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 [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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}^{3}{\left (e e^{\left (i x + h\right )} + d\right )}}{c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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