Optimal. Leaf size=191 \[ \frac {(c+d x)^2}{2 a^2 d}-\frac {d x}{a^2 f g n \log (F)}+\frac {c+d x}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}-\frac {d \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)} \]
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Rubi [A]
time = 0.24, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2216, 2215,
2221, 2317, 2438, 2222, 2320, 272, 36, 29, 31} \begin {gather*} -\frac {d \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 {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}+\frac {d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {(c+d x)^2}{2 a^2 d}-\frac {d x}{a^2 f g n \log (F)}+\frac {c+d x}{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 29
Rule 31
Rule 36
Rule 272
Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rubi steps
\begin {align*} \int \frac {c+d x}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx &=\frac {\int \frac {c+d x}{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)}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {c+d x}{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)}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2}-\frac {d \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a f g n \log (F)}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {c+d x}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}-\frac {d \text {Subst}\left (\int \frac {1}{x \left (a+b x^n\right )} \, dx,x,F^{g (e+f x)}\right )}{a f^2 g^2 n \log ^2(F)}+\frac {d \int \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^2 f g n \log (F)}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {c+d x}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {d \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^2 g^2 n^2 \log ^2(F)}-\frac {d \text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {c+d x}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}-\frac {d \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 {d \text {Subst}\left (\int \frac {1}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {(b d) \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {(c+d x)^2}{2 a^2 d}-\frac {d x}{a^2 f g n \log (F)}+\frac {c+d x}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {d \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}-\frac {d \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)}\\ \end {align*}
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Mathematica [F]
time = 0.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x}{\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. \(590\) vs.
\(2(189)=378\).
time = 0.06, size = 591, normalized size = 3.09
method | result | size |
risch | \(\frac {d x +c}{a f \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right ) g n \ln \left (F \right )}+\frac {d \ln \left (F^{g \left (f x +e \right )}\right )^{2}}{2 \ln \left (F \right )^{2} f^{2} g^{2} a^{2}}-\frac {d \ln \left (1+\frac {b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right ) \ln \left (F^{g \left (f x +e \right )}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n \,a^{2}}+\frac {d \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right ) x}{\ln \left (F \right ) f g n \,a^{2}}-\frac {d \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right ) \ln \left (F^{g \left (f x +e \right )}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n \,a^{2}}-\frac {d \ln \left (a +b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right ) x}{\ln \left (F \right ) f g n \,a^{2}}+\frac {d \ln \left (a +b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right ) \ln \left (F^{g \left (f x +e \right )}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n \,a^{2}}-\frac {d \polylog \left (2, -\frac {b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n^{2} a^{2}}-\frac {d \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n^{2} a^{2}}+\frac {d \ln \left (a +b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n^{2} a^{2}}+\frac {c \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right ) f g n \,a^{2}}-\frac {c \ln \left (a +b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right ) f g n \,a^{2}}\) | \(591\) |
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 \begin {gather*} \text {Failed to integrate} \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. 428 vs.
\(2 (192) = 384\).
time = 0.36, size = 428, normalized size = 2.24 \begin {gather*} \frac {{\left (a d f^{2} g^{2} n^{2} x^{2} + 2 \, a c f^{2} g^{2} n^{2} x + 2 \, a c f g^{2} n^{2} e - a d g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} + {\left ({\left (b d f^{2} g^{2} n^{2} x^{2} + 2 \, b c f^{2} g^{2} n^{2} x + 2 \, b c f g^{2} n^{2} e - b d g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (b d f g n x + b d g n e\right )} \log \left (F\right )\right )} F^{f g n x + g n e} - 2 \, {\left (F^{f g n x + g n e} b d + a d\right )} {\rm Li}_2\left (-\frac {F^{f g n x + g n e} b + a}{a} + 1\right ) + 2 \, {\left ({\left (b d - {\left (b c f g n - b d g n e\right )} \log \left (F\right )\right )} F^{f g n x + g n e} + a d - {\left (a c f g n - a d g n e\right )} \log \left (F\right )\right )} \log \left (F^{f g n x + g n e} b + a\right ) + 2 \, {\left (a c f g n - a d g n e\right )} \log \left (F\right ) - 2 \, {\left ({\left (b d f g n x + b d g n e\right )} F^{f g n x + g n e} \log \left (F\right ) + {\left (a d f g n x + a d g n e\right )} \log \left (F\right )\right )} \log \left (\frac {F^{f g n x + g n e} b + a}{a}\right )}{2 \, {\left (F^{f g n x + g n e} a^{2} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{3} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\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 + d x}{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 {d}{a + b e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\right )\, dx + \int \frac {c 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 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 = -0.01 \begin {gather*} \int \frac {c+d\,x}{{\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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