Optimal. Leaf size=599 \[ -\frac {2 g (f+g x) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^2 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 g (f+g x) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^2 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {(f+g x)^2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{i \left (b-\sqrt {b^2-4 a c}\right )}-\frac {(f+g x)^2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{i \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{3 g \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^3 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{3 g \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 g^2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^3 \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 g^2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^3 \left (\sqrt {b^2-4 a c}+b\right )} \]
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Rubi [A] time = 1.00, antiderivative size = 599, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2265, 2184, 2190, 2531, 2282, 6589} \[ -\frac {2 g (f+g x) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \text {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^2 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 g (f+g x) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}\right )}{i^2 \left (\sqrt {b^2-4 a c}+b\right )}+\frac {2 g^2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \text {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^3 \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 g^2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}\right )}{i^3 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {(f+g x)^2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{i \left (b-\sqrt {b^2-4 a c}\right )}-\frac {(f+g x)^2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{i \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{3 g \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^3 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{3 g \left (b-\sqrt {b^2-4 a c}\right )} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2265
Rule 2282
Rule 2531
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (d+e e^{h+573 x}\right ) (f+g x)^2}{a+b e^{h+573 x}+c e^{2 h+1146 x}} \, dx &=-\left (\left (-e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {(f+g x)^2}{b+\sqrt {b^2-4 a c}+2 c e^{h+573 x}} \, dx\right )+\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {(f+g x)^2}{b-\sqrt {b^2-4 a c}+2 c e^{h+573 x}} \, dx\\ &=\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3}{3 \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}{3 \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+573 x} (f+g x)^2}{b+\sqrt {b^2-4 a c}+2 c e^{h+573 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+573 x} (f+g x)^2}{b-\sqrt {b^2-4 a c}+2 c e^{h+573 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)^3}{3 \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}{3 \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)^2 \log \left (1+\frac {2 c e^{h+573 x}}{b-\sqrt {b^2-4 a c}}\right )}{573 \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)^2 \log \left (1+\frac {2 c e^{h+573 x}}{b+\sqrt {b^2-4 a c}}\right )}{573 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g\right ) \int (f+g x) \log \left (1+\frac {2 c e^{h+573 x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{573 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g\right ) \int (f+g x) \log \left (1+\frac {2 c e^{h+573 x}}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{573 \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}{3 \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}{3 \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)^2 \log \left (1+\frac {2 c e^{h+573 x}}{b-\sqrt {b^2-4 a c}}\right )}{573 \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)^2 \log \left (1+\frac {2 c e^{h+573 x}}{b+\sqrt {b^2-4 a c}}\right )}{573 \left (b+\sqrt {b^2-4 a c}\right )}-\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+573 x}}{b-\sqrt {b^2-4 a c}}\right )}{328329 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+573 x}}{b+\sqrt {b^2-4 a c}}\right )}{328329 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2\right ) \int \text {Li}_2\left (-\frac {2 c e^{h+573 x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{328329 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2\right ) \int \text {Li}_2\left (-\frac {2 c e^{h+573 x}}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{328329 \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}{3 \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}{3 \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)^2 \log \left (1+\frac {2 c e^{h+573 x}}{b-\sqrt {b^2-4 a c}}\right )}{573 \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)^2 \log \left (1+\frac {2 c e^{h+573 x}}{b+\sqrt {b^2-4 a c}}\right )}{573 \left (b+\sqrt {b^2-4 a c}\right )}-\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+573 x}}{b-\sqrt {b^2-4 a c}}\right )}{328329 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+573 x}}{b+\sqrt {b^2-4 a c}}\right )}{328329 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,e^{h+573 x}\right )}{188132517 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,e^{h+573 x}\right )}{188132517 \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}{3 \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}{3 \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)^2 \log \left (1+\frac {2 c e^{h+573 x}}{b-\sqrt {b^2-4 a c}}\right )}{573 \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)^2 \log \left (1+\frac {2 c e^{h+573 x}}{b+\sqrt {b^2-4 a c}}\right )}{573 \left (b+\sqrt {b^2-4 a c}\right )}-\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+573 x}}{b-\sqrt {b^2-4 a c}}\right )}{328329 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+573 x}}{b+\sqrt {b^2-4 a c}}\right )}{328329 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 \text {Li}_3\left (-\frac {2 c e^{h+573 x}}{b-\sqrt {b^2-4 a c}}\right )}{188132517 \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 \text {Li}_3\left (-\frac {2 c e^{h+573 x}}{b+\sqrt {b^2-4 a c}}\right )}{188132517 \left (b+\sqrt {b^2-4 a c}\right )}\\ \end {align*}
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Mathematica [B] time = 2.68, size = 1412, normalized size = 2.36 \[ -\frac {-2 \sqrt {-\left (b^2-4 a c\right )^2} d g^2 x^3 i^3-6 \sqrt {-\left (b^2-4 a c\right )^2} d f g x^2 i^3-6 \sqrt {-\left (b^2-4 a c\right )^2} d f^2 x i^3+6 b \sqrt {b^2-4 a c} d f^2 \tan ^{-1}\left (\frac {b+2 c e^{h+i x}}{\sqrt {4 a c-b^2}}\right ) i^2+12 a \sqrt {4 a c-b^2} e f^2 \tanh ^{-1}\left (\frac {b+2 c e^{h+i x}}{\sqrt {b^2-4 a c}}\right ) i^2+3 \sqrt {-\left (b^2-4 a c\right )^2} d g^2 x^2 \log \left (\frac {2 e^{h+i x} c}{b-\sqrt {b^2-4 a c}}+1\right ) i^2+3 b \sqrt {4 a c-b^2} d g^2 x^2 \log \left (\frac {2 e^{h+i x} c}{b-\sqrt {b^2-4 a c}}+1\right ) i^2-6 a \sqrt {4 a c-b^2} e g^2 x^2 \log \left (\frac {2 e^{h+i x} c}{b-\sqrt {b^2-4 a c}}+1\right ) i^2+6 \sqrt {-\left (b^2-4 a c\right )^2} d f g x \log \left (\frac {2 e^{h+i x} c}{b-\sqrt {b^2-4 a c}}+1\right ) i^2+6 b \sqrt {4 a c-b^2} d f g x \log \left (\frac {2 e^{h+i x} c}{b-\sqrt {b^2-4 a c}}+1\right ) i^2-12 a \sqrt {4 a c-b^2} e f g x \log \left (\frac {2 e^{h+i x} c}{b-\sqrt {b^2-4 a c}}+1\right ) i^2+3 \sqrt {-\left (b^2-4 a c\right )^2} d g^2 x^2 \log \left (\frac {2 e^{h+i x} c}{b+\sqrt {b^2-4 a c}}+1\right ) i^2-3 b \sqrt {4 a c-b^2} d g^2 x^2 \log \left (\frac {2 e^{h+i x} c}{b+\sqrt {b^2-4 a c}}+1\right ) i^2+6 a \sqrt {4 a c-b^2} e g^2 x^2 \log \left (\frac {2 e^{h+i x} c}{b+\sqrt {b^2-4 a c}}+1\right ) i^2+6 \sqrt {-\left (b^2-4 a c\right )^2} d f g x \log \left (\frac {2 e^{h+i x} c}{b+\sqrt {b^2-4 a c}}+1\right ) i^2-6 b \sqrt {4 a c-b^2} d f g x \log \left (\frac {2 e^{h+i x} c}{b+\sqrt {b^2-4 a c}}+1\right ) i^2+12 a \sqrt {4 a c-b^2} e f g x \log \left (\frac {2 e^{h+i x} c}{b+\sqrt {b^2-4 a c}}+1\right ) i^2+3 \sqrt {-\left (b^2-4 a c\right )^2} d f^2 \log \left (a+e^{h+i x} \left (b+c e^{h+i x}\right )\right ) i^2+6 \left (\sqrt {-\left (b^2-4 a c\right )^2} d+b \sqrt {4 a c-b^2} d-2 a \sqrt {4 a c-b^2} e\right ) g (f+g x) \text {Li}_2\left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}-b}\right ) i+6 \left (\sqrt {-\left (b^2-4 a c\right )^2} d-b \sqrt {4 a c-b^2} d+2 a \sqrt {4 a c-b^2} e\right ) g (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right ) i-6 \sqrt {-\left (b^2-4 a c\right )^2} d g^2 \text {Li}_3\left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}-b}\right )-6 b \sqrt {4 a c-b^2} d g^2 \text {Li}_3\left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}-b}\right )+12 a \sqrt {4 a c-b^2} e g^2 \text {Li}_3\left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}-b}\right )-6 \sqrt {-\left (b^2-4 a c\right )^2} d g^2 \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )+6 b \sqrt {4 a c-b^2} d g^2 \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )-12 a \sqrt {4 a c-b^2} e g^2 \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{6 a \sqrt {-\left (b^2-4 a c\right )^2} i^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.47, size = 1193, normalized size = 1.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{2} {\left (e e^{\left (i x + h\right )} + d\right )}}{c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,{\mathrm e}^{i x +h}+d \right ) \left (g x +f \right )^{2}}{b \,{\mathrm e}^{i x +h}+c \,{\mathrm e}^{2 i x +2 h}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^2\,\left (d+e\,{\mathrm {e}}^{h+i\,x}\right )}{a+b\,{\mathrm {e}}^{h+i\,x}+c\,{\mathrm {e}}^{2\,h+2\,i\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e e^{h} e^{i x}\right ) \left (f + g x\right )^{2}}{a + b e^{h} e^{i x} + c e^{2 h} e^{2 i x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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