Optimal. Leaf size=428 \[ -\frac {(f+g x) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{i \left (b-\sqrt {b^2-4 a c}\right )}-\frac {(f+g x) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{i \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{2 g \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{2 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {g \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^2 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {g \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^2 \left (\sqrt {b^2-4 a c}+b\right )} \]
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Rubi [A] time = 0.58, antiderivative size = 428, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2265, 2184, 2190, 2279, 2391} \[ -\frac {g \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \text {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^2 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {g \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}\right )}{i^2 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {(f+g x) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{i \left (b-\sqrt {b^2-4 a c}\right )}-\frac {(f+g x) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{i \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{2 g \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{2 g \left (b-\sqrt {b^2-4 a c}\right )} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2265
Rule 2279
Rule 2391
Rubi steps
\begin {align*} \int \frac {\left (d+e e^{h+574 x}\right ) (f+g x)}{a+b e^{h+574 x}+c e^{2 h+1148 x}} \, dx &=-\left (\left (-e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {f+g x}{b+\sqrt {b^2-4 a c}+2 c e^{h+574 x}} \, dx\right )+\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {f+g x}{b-\sqrt {b^2-4 a c}+2 c e^{h+574 x}} \, dx\\ &=\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^2}{2 \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}{2 \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+574 x} (f+g x)}{b+\sqrt {b^2-4 a c}+2 c e^{h+574 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+574 x} (f+g x)}{b-\sqrt {b^2-4 a c}+2 c e^{h+574 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)^2}{2 \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}{2 \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) \log \left (1+\frac {2 c e^{h+574 x}}{b-\sqrt {b^2-4 a c}}\right )}{574 \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) \log \left (1+\frac {2 c e^{h+574 x}}{b+\sqrt {b^2-4 a c}}\right )}{574 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g\right ) \int \log \left (1+\frac {2 c e^{h+574 x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{574 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g\right ) \int \log \left (1+\frac {2 c e^{h+574 x}}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{574 \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}{2 \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}{2 \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) \log \left (1+\frac {2 c e^{h+574 x}}{b-\sqrt {b^2-4 a c}}\right )}{574 \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) \log \left (1+\frac {2 c e^{h+574 x}}{b+\sqrt {b^2-4 a c}}\right )}{574 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,e^{h+574 x}\right )}{329476 \left (b+\sqrt {b^2-4 a c}\right )}+\frac {\left (\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,e^{h+574 x}\right )}{329476 \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}{2 \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}{2 \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) \log \left (1+\frac {2 c e^{h+574 x}}{b-\sqrt {b^2-4 a c}}\right )}{574 \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) \log \left (1+\frac {2 c e^{h+574 x}}{b+\sqrt {b^2-4 a c}}\right )}{574 \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 ) g \text {Li}_2\left (-\frac {2 c e^{h+574 x}}{b-\sqrt {b^2-4 a c}}\right )}{329476 \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 ) g \text {Li}_2\left (-\frac {2 c e^{h+574 x}}{b+\sqrt {b^2-4 a c}}\right )}{329476 \left (b+\sqrt {b^2-4 a c}\right )}\\ \end {align*}
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Mathematica [A] time = 1.75, size = 677, normalized size = 1.58 \[ -\frac {i \left (d f \sqrt {-\left (b^2-4 a c\right )^2} \log \left (a+e^{h+i x} \left (b+c e^{h+i x}\right )\right )+2 b d f \sqrt {b^2-4 a c} \tan ^{-1}\left (\frac {b+2 c e^{h+i x}}{\sqrt {4 a c-b^2}}\right )-2 d f i x \sqrt {-\left (b^2-4 a c\right )^2}+d g x \sqrt {-\left (b^2-4 a c\right )^2} \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )+b d g x \sqrt {4 a c-b^2} \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )+d g x \sqrt {-\left (b^2-4 a c\right )^2} \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )-b d g x \sqrt {4 a c-b^2} \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )+d g i x^2 \left (-\sqrt {-\left (b^2-4 a c\right )^2}\right )+4 a e f \sqrt {4 a c-b^2} \tanh ^{-1}\left (\frac {b+2 c e^{h+i x}}{\sqrt {b^2-4 a c}}\right )-2 a e g x \sqrt {4 a c-b^2} \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )+2 a e g x \sqrt {4 a c-b^2} \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )\right )+g \left (d \sqrt {-\left (b^2-4 a c\right )^2}+b d \sqrt {4 a c-b^2}-2 a e \sqrt {4 a c-b^2}\right ) \text {Li}_2\left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}-b}\right )+g \left (d \sqrt {-\left (b^2-4 a c\right )^2}-b d \sqrt {4 a c-b^2}+2 a e \sqrt {4 a c-b^2}\right ) \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{2 a i^2 \sqrt {-\left (b^2-4 a c\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 651, normalized size = 1.52 \[ \frac {{\left (b^{2} - 4 \, a c\right )} d g i^{2} x^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} d f i^{2} x - {\left ({\left (b^{2} - 4 \, a c\right )} d g + {\left (a b d - 2 \, a^{2} e\right )} g \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} {\rm Li}_2\left (-\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{\left (i x + h\right )} + b e^{\left (i x + h\right )} + 2 \, a}{2 \, a} + 1\right ) - {\left ({\left (b^{2} - 4 \, a c\right )} d g - {\left (a b d - 2 \, a^{2} e\right )} g \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} {\rm Li}_2\left (\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{\left (i x + h\right )} - b e^{\left (i x + h\right )} - 2 \, a}{2 \, a} + 1\right ) + {\left ({\left (b^{2} - 4 \, a c\right )} d g h - {\left (b^{2} - 4 \, a c\right )} d f i - {\left ({\left (a b d - 2 \, a^{2} e\right )} g h - {\left (a b d - 2 \, a^{2} e\right )} f i\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \log \left (2 \, c e^{\left (i x + h\right )} + a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right ) + {\left ({\left (b^{2} - 4 \, a c\right )} d g h - {\left (b^{2} - 4 \, a c\right )} d f i + {\left ({\left (a b d - 2 \, a^{2} e\right )} g h - {\left (a b d - 2 \, a^{2} e\right )} f i\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \log \left (2 \, c e^{\left (i x + h\right )} - a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right ) - {\left ({\left (b^{2} - 4 \, a c\right )} d g i x + {\left (b^{2} - 4 \, a c\right )} d g h + {\left ({\left (a b d - 2 \, a^{2} e\right )} g i x + {\left (a b d - 2 \, a^{2} e\right )} g h\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \log \left (\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{\left (i x + h\right )} + b e^{\left (i x + h\right )} + 2 \, a}{2 \, a}\right ) - {\left ({\left (b^{2} - 4 \, a c\right )} d g i x + {\left (b^{2} - 4 \, a c\right )} d g h - {\left ({\left (a b d - 2 \, a^{2} e\right )} g i x + {\left (a b d - 2 \, a^{2} e\right )} g h\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \log \left (-\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{\left (i x + h\right )} - b e^{\left (i x + h\right )} - 2 \, a}{2 \, a}\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} i^{2}} \]
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 \[ \int \frac {{\left (g x + f\right )} {\left (e e^{\left (i x + h\right )} + d\right )}}{c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1249, normalized size = 2.92 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (f+g\,x\right )\,\left (d+e\,{\mathrm {e}}^{h+i\,x}\right )}{a+b\,{\mathrm {e}}^{h+i\,x}+c\,{\mathrm {e}}^{2\,h+2\,i\,x}} \,d x \]
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 \[ \int \frac {\left (d + e e^{h} e^{i x}\right ) \left (f + g x\right )}{a + b e^{h} e^{i x} + c e^{2 h} e^{2 i x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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