Optimal. Leaf size=484 \[ \frac {4 c \text {Li}_3\left (-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{d^3 \log ^3(f) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {4 c \text {Li}_3\left (-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{d^3 \log ^3(f) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}-\frac {4 c x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{d^2 \log ^2(f) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}+\frac {4 c x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{d^2 \log ^2(f) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c x^2 \log \left (\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}+1\right )}{d \log (f) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 c x^2 \log \left (\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}+1\right )}{d \log (f) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c x^3}{3 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}-\frac {2 c x^3}{3 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )} \]
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Rubi [A] time = 0.87, antiderivative size = 484, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2263, 2184, 2190, 2531, 2282, 6589} \[ -\frac {4 c x \text {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{d^2 \log ^2(f) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}+\frac {4 c x \text {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}\right )}{d^2 \log ^2(f) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}+\frac {4 c \text {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{d^3 \log ^3(f) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {4 c \text {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}\right )}{d^3 \log ^3(f) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c x^2 \log \left (\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}+1\right )}{d \log (f) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 c x^2 \log \left (\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}+1\right )}{d \log (f) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c x^3}{3 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}-\frac {2 c x^3}{3 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2263
Rule 2282
Rule 2531
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx &=\frac {(2 c) \int \frac {x^2}{b-\sqrt {b^2-4 a c}+2 c f^{c+d x}} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {x^2}{b+\sqrt {b^2-4 a c}+2 c f^{c+d x}} \, dx}{\sqrt {b^2-4 a c}}\\ &=-\frac {2 c x^3}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )}+\frac {\left (4 c^2\right ) \int \frac {f^{c+d x} x^2}{b-\sqrt {b^2-4 a c}+2 c f^{c+d x}} \, dx}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {\left (4 c^2\right ) \int \frac {f^{c+d x} x^2}{b+\sqrt {b^2-4 a c}+2 c f^{c+d x}} \, dx}{b^2-4 a c+b \sqrt {b^2-4 a c}}\\ &=-\frac {2 c x^3}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )}+\frac {2 c x^2 \log \left (1+\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {2 c x^2 \log \left (1+\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \log (f)}-\frac {(4 c) \int x \log \left (1+\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \log (f)}-\frac {(4 c) \int x \log \left (1+\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \log (f)}\\ &=-\frac {2 c x^3}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )}+\frac {2 c x^2 \log \left (1+\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {2 c x^2 \log \left (1+\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {4 c x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}+\frac {4 c x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}-\frac {(4 c) \int \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}-\frac {(4 c) \int \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}\\ &=-\frac {2 c x^3}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )}+\frac {2 c x^2 \log \left (1+\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {2 c x^2 \log \left (1+\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {4 c x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}+\frac {4 c x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}-\frac {(4 c) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,f^{c+d x}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d^3 \log ^3(f)}-\frac {(4 c) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,f^{c+d x}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d^3 \log ^3(f)}\\ &=-\frac {2 c x^3}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )}+\frac {2 c x^2 \log \left (1+\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {2 c x^2 \log \left (1+\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {4 c x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}+\frac {4 c x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}-\frac {4 c \text {Li}_3\left (-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d^3 \log ^3(f)}-\frac {4 c \text {Li}_3\left (-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d^3 \log ^3(f)}\\ \end {align*}
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Mathematica [F] time = 2.83, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.44, size = 694, normalized size = 1.43 \[ \frac {2 \, {\left (b^{2} - 4 \, a c\right )} d^{3} x^{3} \log \relax (f)^{3} - 6 \, {\left (a b d x \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \relax (f) + {\left (b^{2} - 4 \, a c\right )} d x \log \relax (f)\right )} {\rm Li}_2\left (-\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right )} f^{d x + c} + 2 \, a}{2 \, a} + 1\right ) + 6 \, {\left (a b d x \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \relax (f) - {\left (b^{2} - 4 \, a c\right )} d x \log \relax (f)\right )} {\rm Li}_2\left (\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b\right )} f^{d x + c} - 2 \, a}{2 \, a} + 1\right ) + 3 \, {\left (a b c^{2} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \relax (f)^{2} - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \log \relax (f)^{2}\right )} \log \left (2 \, c f^{d x + c} + a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right ) - 3 \, {\left (a b c^{2} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \relax (f)^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \log \relax (f)^{2}\right )} \log \left (2 \, c f^{d x + c} - a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right ) - 3 \, {\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \relax (f)^{2} + {\left ({\left (b^{2} - 4 \, a c\right )} d^{2} x^{2} - b^{2} c^{2} + 4 \, a c^{3}\right )} \log \relax (f)^{2}\right )} \log \left (\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right )} f^{d x + c} + 2 \, a}{2 \, a}\right ) + 3 \, {\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \relax (f)^{2} - {\left ({\left (b^{2} - 4 \, a c\right )} d^{2} x^{2} - b^{2} c^{2} + 4 \, a c^{3}\right )} \log \relax (f)^{2}\right )} \log \left (-\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b\right )} f^{d x + c} - 2 \, a}{2 \, a}\right ) + 6 \, {\left (a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 4 \, a c\right )} {\rm polylog}\left (3, -\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right )} f^{d x + c}}{2 \, a}\right ) - 6 \, {\left (a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b^{2} + 4 \, a c\right )} {\rm polylog}\left (3, \frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b\right )} f^{d x + c}}{2 \, a}\right )}{6 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{3} \log \relax (f)^{3}} \]
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 \[ \int \frac {x^{2}}{c f^{2 \, d x + 2 \, c} + b f^{d x + c} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{b \,f^{d x +c}+c \,f^{2 d x +2 c}+a}\, dx \]
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 \[ \int \frac {x^2}{a+b\,f^{c+d\,x}+c\,f^{2\,c+2\,d\,x}} \,d x \]
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 \[ \int \frac {x^{2}}{a + b f^{c} f^{d x} + c f^{2 c} f^{2 d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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