Optimal. Leaf size=294 \[ \frac {(c+d x)^3}{3 a^2 d}-\frac {(c+d x)^2}{a^2 f g n \log (F)}+\frac {(c+d x)^2}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {2 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {2 d^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)} \]
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Rubi [A]
time = 0.48, 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 = {2216, 2215,
2221, 2611, 2320, 6724, 2222, 2317, 2438} \begin {gather*} -\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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 6724
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 ) \text {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 ) \text {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 = 1.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1649\) vs.
\(2(292)=584\).
time = 0.08, size = 1650, normalized size = 5.61
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1650\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.41, size = 478, normalized size = 1.63 \begin {gather*} c^{2} {\left (\frac {f g n x + g n e}{a^{2} f g n} + \frac {1}{{\left (F^{f g n x + g n e} a b + a^{2}\right )} f g n \log \left (F\right )} - \frac {\log \left (F^{f g n x + g n e} b + a\right )}{a^{2} f g n \log \left (F\right )}\right )} + \frac {d^{2} x^{2} + 2 \, c d x}{F^{f g n x} F^{g n e} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )} - \frac {2 \, c d x}{a^{2} f g n \log \left (F\right )} + \frac {2 \, c d \log \left (F^{f g n x} F^{g n e} b + a\right )}{a^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2}} - \frac {{\left (f^{2} g^{2} n^{2} x^{2} \log \left (\frac {F^{f g n x} F^{g n e} b}{a} + 1\right ) \log \left (F\right )^{2} + 2 \, f g n x {\rm Li}_2\left (-\frac {F^{f g n x} F^{g n e} b}{a}\right ) \log \left (F\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{f g n x} F^{g n e} b}{a})\right )} d^{2}}{a^{2} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} - \frac {2 \, {\left (c d f g n \log \left (F\right ) - d^{2}\right )} {\left (f g n x \log \left (\frac {F^{f g n x} F^{g n e} b}{a} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{f g n x} F^{g n e} b}{a}\right )\right )}}{a^{2} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {d^{2} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} + 3 \, {\left (c d f g n \log \left (F\right ) - d^{2}\right )} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2}}{3 \, a^{2} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 918 vs.
\(2 (296) = 592\).
time = 0.37, size = 918, normalized size = 3.12 \begin {gather*} \frac {{\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 + 3 \, a c^{2} f^{2} g^{3} n^{3} e - 3 \, a c d f g^{3} n^{3} e^{2} + a d^{2} g^{3} n^{3} e^{3}\right )} \log \left (F\right )^{3} + 3 \, {\left (a c^{2} f^{2} g^{2} n^{2} - 2 \, a c d f g^{2} n^{2} e + a d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} + {\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 + 3 \, b c^{2} f^{2} g^{3} n^{3} e - 3 \, b c d f g^{3} n^{3} e^{2} + b d^{2} g^{3} n^{3} e^{3}\right )} \log \left (F\right )^{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 + 2 \, b c d f g^{2} n^{2} e - b d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2}\right )} F^{f g n x + g n e} + 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 \left (F\right )\right )} F^{f g n x + g n e} - {\left (a d^{2} f g n x + a c d f g n\right )} \log \left (F\right )\right )} {\rm Li}_2\left (-\frac {F^{f g n x + g n e} b + a}{a} + 1\right ) - 3 \, {\left ({\left (a c^{2} f^{2} g^{2} n^{2} - 2 \, a c d f g^{2} n^{2} e + a d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} + {\left ({\left (b c^{2} f^{2} g^{2} n^{2} - 2 \, b c d f g^{2} n^{2} e + b d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (b c d f g n - b d^{2} g n e\right )} \log \left (F\right )\right )} F^{f g n x + g n e} - 2 \, {\left (a c d f g n - a d^{2} g n e\right )} \log \left (F\right )\right )} \log \left (F^{f g n x + g n e} 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 + 2 \, a c d f g^{2} n^{2} e - a d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{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 + 2 \, b c d f g^{2} n^{2} e - b d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (b d^{2} f g n x + b d^{2} g n e\right )} \log \left (F\right )\right )} F^{f g n x + g n e} - 2 \, {\left (a d^{2} f g n x + a d^{2} g n e\right )} \log \left (F\right )\right )} \log \left (\frac {F^{f g n x + g n e} b + a}{a}\right ) + 6 \, {\left (F^{f g n x + g n e} b d^{2} + a d^{2}\right )} {\rm polylog}\left (3, -\frac {F^{f g n x + g n e} b}{a}\right )}{3 \, {\left (F^{f g n x + g n e} a^{2} b f^{3} g^{3} n^{3} \log \left (F\right )^{3} + a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}\right )}} \end {gather*}
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 \begin {gather*} \frac {c^{2} + 2 c d x + d^{2} x^{2}}{a^{2} f g n \log {\left (F \right )} + a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log {\left (F \right )}} + \frac {\int \left (- \frac {2 c d}{a + b e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\right )\, dx + \int \left (- \frac {2 d^{2} x}{a + b e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\right )\, dx + \int \frac {c^{2} f g n \log {\left (F \right )}}{a + b e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx + \int \frac {d^{2} f g n x^{2} \log {\left (F \right )}}{a + b e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx + \int \frac {2 c d f g n x \log {\left (F \right )}}{a + b e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx}{a f g n \log {\left (F \right )}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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