Optimal. Leaf size=294 \[ -\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {2 d (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}+\frac {2 d^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac {2 d^2 \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {(c+d x)^2}{a^2 f g n \log (F)}+\frac {(c+d x)^3}{3 a^2 d}+\frac {(c+d x)^2}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]
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Rubi [A] time = 0.67, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2185, 2184, 2190, 2531, 2282, 6589, 2191, 2279, 2391} \[ -\frac {2 d (c+d x) \text {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \text {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac {2 d^2 \text {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac {2 d (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}-\frac {(c+d x)^2}{a^2 f g n \log (F)}+\frac {(c+d x)^3}{3 a^2 d}+\frac {(c+d x)^2}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 6589
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx &=\frac {\int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a}\\ &=\frac {(c+d x)^3}{3 a^2 d}+\frac {(c+d x)^2}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2}-\frac {(2 d) \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a f g n \log (F)}\\ &=\frac {(c+d x)^3}{3 a^2 d}-\frac {(c+d x)^2}{a^2 f g n \log (F)}+\frac {(c+d x)^2}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {(2 d) \int (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^2 f g n \log (F)}+\frac {(2 b d) \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2 f g n \log (F)}\\ &=\frac {(c+d x)^3}{3 a^2 d}-\frac {(c+d x)^2}{a^2 f g n \log (F)}+\frac {(c+d x)^2}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {2 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {\left (2 d^2\right ) \int \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {\left (2 d^2\right ) \int \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^2 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {(c+d x)^3}{3 a^2 d}-\frac {(c+d x)^2}{a^2 f g n \log (F)}+\frac {(c+d x)^2}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {2 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x^n}{a}\right )}{x} \, dx,x,F^{g (e+f x)}\right )}{a^2 f^3 g^3 n^2 \log ^3(F)}\\ &=\frac {(c+d x)^3}{3 a^2 d}-\frac {(c+d x)^2}{a^2 f g n \log (F)}+\frac {(c+d x)^2}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {2 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {2 d^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}\\ \end {align*}
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Mathematica [F] time = 1.18, size = 0, normalized size = 0.00 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.44, size = 834, normalized size = 2.84 \[ \frac {3 \, {\left (a d^{2} e^{2} - 2 \, a c d e f + a c^{2} f^{2}\right )} g^{2} n^{2} \log \relax (F)^{2} + {\left (a d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, a c d f^{3} g^{3} n^{3} x^{2} + 3 \, a c^{2} f^{3} g^{3} n^{3} x + {\left (a d^{2} e^{3} - 3 \, a c d e^{2} f + 3 \, a c^{2} e f^{2}\right )} g^{3} n^{3}\right )} \log \relax (F)^{3} + {\left ({\left (b d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, b c d f^{3} g^{3} n^{3} x^{2} + 3 \, b c^{2} f^{3} g^{3} n^{3} x + {\left (b d^{2} e^{3} - 3 \, b c d e^{2} f + 3 \, b c^{2} e f^{2}\right )} g^{3} n^{3}\right )} \log \relax (F)^{3} - 3 \, {\left (b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b c d f^{2} g^{2} n^{2} x - {\left (b d^{2} e^{2} - 2 \, b c d e f\right )} g^{2} n^{2}\right )} \log \relax (F)^{2}\right )} F^{f g n x + e g n} + 6 \, {\left (a d^{2} + {\left (b d^{2} - {\left (b d^{2} f g n x + b c d f g n\right )} \log \relax (F)\right )} F^{f g n x + e g n} - {\left (a d^{2} f g n x + a c d f g n\right )} \log \relax (F)\right )} {\rm Li}_2\left (-\frac {F^{f g n x + e g n} b + a}{a} + 1\right ) - 3 \, {\left ({\left (a d^{2} e^{2} - 2 \, a c d e f + a c^{2} f^{2}\right )} g^{2} n^{2} \log \relax (F)^{2} + 2 \, {\left (a d^{2} e - a c d f\right )} g n \log \relax (F) + {\left ({\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} g^{2} n^{2} \log \relax (F)^{2} + 2 \, {\left (b d^{2} e - b c d f\right )} g n \log \relax (F)\right )} F^{f g n x + e g n}\right )} \log \left (F^{f g n x + e g n} b + a\right ) - 3 \, {\left ({\left (a d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a c d f^{2} g^{2} n^{2} x - {\left (a d^{2} e^{2} - 2 \, a c d e f\right )} g^{2} n^{2}\right )} \log \relax (F)^{2} + {\left ({\left (b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b c d f^{2} g^{2} n^{2} x - {\left (b d^{2} e^{2} - 2 \, b c d e f\right )} g^{2} n^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (b d^{2} f g n x + b d^{2} e g n\right )} \log \relax (F)\right )} F^{f g n x + e g n} - 2 \, {\left (a d^{2} f g n x + a d^{2} e g n\right )} \log \relax (F)\right )} \log \left (\frac {F^{f g n x + e g n} b + a}{a}\right ) + 6 \, {\left (F^{f g n x + e g n} b d^{2} + a d^{2}\right )} {\rm polylog}\left (3, -\frac {F^{f g n x + e g n} b}{a}\right )}{3 \, {\left (F^{f g n x + e g n} a^{2} b f^{3} g^{3} n^{3} \log \relax (F)^{3} + a^{3} f^{3} g^{3} n^{3} \log \relax (F)^{3}\right )}} \]
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 \[ \int \frac {{\left (d x + c\right )}^{2}}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1650, normalized size = 5.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} {\left (\frac {1}{{\left ({\left (F^{f g x + e g}\right )}^{n} a b n + a^{2} n\right )} f g \log \relax (F)} + \frac {\log \left (F^{f g x + e g}\right )}{a^{2} f g \log \relax (F)} - \frac {\log \left (\frac {{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{2} f g n \log \relax (F)}\right )} + \frac {d^{2} x^{2} + 2 \, c d x}{{\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} a b f g n \log \relax (F) + a^{2} f g n \log \relax (F)} + \int \frac {d^{2} f g n x^{2} \log \relax (F) - 2 \, c d + 2 \, {\left (c d f g n \log \relax (F) - d^{2}\right )} x}{{\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} a b f g n \log \relax (F) + a^{2} f g n \log \relax (F)}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^2} \,d x \]
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 \[ \frac {c^{2} + 2 c d x + d^{2} x^{2}}{a^{2} f g n \log {\relax (F )} + a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log {\relax (F )}} + \frac {\int \left (- \frac {2 c d}{a + b e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\right )\, dx + \int \left (- \frac {2 d^{2} x}{a + b e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\right )\, dx + \int \frac {c^{2} f g n \log {\relax (F )}}{a + b e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx + \int \frac {d^{2} f g n x^{2} \log {\relax (F )}}{a + b e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx + \int \frac {2 c d f g n x \log {\relax (F )}}{a + b e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx}{a f g n \log {\relax (F )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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