Optimal. Leaf size=145 \[ \frac {(c+d x)^3}{3 a d}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)} \]
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Rubi [A]
time = 0.20, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2215, 2221,
2611, 2320, 6724} \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 f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \text {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}-\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}+\frac {(c+d x)^3}{3 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2320
Rule 2611
Rule 6724
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx &=\frac {(c+d x)^3}{3 a d}-\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}\\ &=\frac {(c+d x)^3}{3 a d}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a 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 f g n \log (F)}\\ &=\frac {(c+d x)^3}{3 a d}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a 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 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 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {(c+d x)^3}{3 a d}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a 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 f^2 g^2 n^2 \log ^2(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 f^3 g^3 n^2 \log ^3(F)}\\ &=\frac {(c+d x)^3}{3 a d}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a 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 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 f^3 g^3 n^3 \log ^3(F)}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 121, normalized size = 0.83 \begin {gather*} \frac {-f^2 g^2 n^2 (c+d x)^2 \log ^2(F) \log \left (1+\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+2 d f g n (c+d x) \log (F) \text {Li}_2\left (-\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+2 d^2 \text {Li}_3\left (-\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )}{a f^3 g^3 n^3 \log ^3(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1676\) vs.
\(2(143)=286\).
time = 0.07, size = 1677, normalized size = 11.57
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1677\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 310 vs.
\(2 (145) = 290\).
time = 0.41, size = 310, normalized size = 2.14 \begin {gather*} c^{2} {\left (\frac {f g n x + g n e}{a f g n} - \frac {\log \left (F^{f g n x + g n e} b + a\right )}{a f g n \log \left (F\right )}\right )} - \frac {2 \, {\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 )} c d}{a 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 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 \, c d f^{3} g^{3} n^{3} x^{2} \log \left (F\right )^{3}}{3 \, a 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. 291 vs.
\(2 (145) = 290\).
time = 0.40, size = 291, normalized size = 2.01 \begin {gather*} -\frac {3 \, {\left (c^{2} f^{2} g^{2} n^{2} - 2 \, c d f g^{2} n^{2} e + d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F^{f g n x + g n e} b + a\right ) \log \left (F\right )^{2} - {\left (d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, c d f^{3} g^{3} n^{3} x^{2} + 3 \, c^{2} f^{3} g^{3} n^{3} x\right )} \log \left (F\right )^{3} + 3 \, {\left (d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, c d f^{2} g^{2} n^{2} x + 2 \, c d f g^{2} n^{2} e - d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} \log \left (\frac {F^{f g n x + g n e} b + a}{a}\right ) + 6 \, {\left (d^{2} f g n x + c d f g n\right )} {\rm Li}_2\left (-\frac {F^{f g n x + g n e} b + a}{a} + 1\right ) \log \left (F\right ) - 6 \, d^{2} {\rm polylog}\left (3, -\frac {F^{f g n x + g n e} b}{a}\right )}{3 \, a 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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{2}}{a + b \left (F^{e g} F^{f g x}\right )^{n}}\, dx \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 {{\left (c+d\,x\right )}^2}{a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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