Optimal. Leaf size=305 \[ -\frac{a^3}{d (c+d x)}+\frac{3 a^2 b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{d (c+d x)}+\frac{6 a b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}+\frac{3 b^3 f g n \log (F) \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac{c f}{d}\right )-3 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{3 f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{d (c+d x)} \]
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Rubi [A] time = 0.849095, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{a^3}{d (c+d x)}+\frac{3 a^2 b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{d (c+d x)}+\frac{6 a b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}+\frac{3 b^3 f g n \log (F) \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac{c f}{d}\right )-3 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{3 f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{d (c+d x)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(F^(g*(e + f*x)))^n)^3/(c + d*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 73.3235, size = 313, normalized size = 1.03 \[ \frac{3 F^{g n \left (- 3 e - 3 f x\right )} F^{- \frac{3 g n \left (c f - d e\right )}{d}} b^{3} f g n \left (F^{g \left (e + f x\right )}\right )^{3 n} \log{\left (F \right )} \operatorname{Ei}{\left (\frac{f g n \left (3 c + 3 d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{2}} + \frac{6 F^{g n \left (- 2 e - 2 f x\right )} F^{- \frac{2 g n \left (c f - d e\right )}{d}} a b^{2} f g n \left (F^{g \left (e + f x\right )}\right )^{2 n} \log{\left (F \right )} \operatorname{Ei}{\left (\frac{f g n \left (2 c + 2 d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{2}} + \frac{3 F^{g n \left (- e - f x\right )} F^{- \frac{g n \left (c f - d e\right )}{d}} a^{2} b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )} \operatorname{Ei}{\left (\frac{f g n \left (c + d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{2}} - \frac{a^{3}}{d \left (c + d x\right )} - \frac{3 a^{2} b \left (F^{g \left (e + f x\right )}\right )^{n}}{d \left (c + d x\right )} - \frac{3 a b^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n}}{d \left (c + d x\right )} - \frac{b^{3} \left (F^{g \left (e + f x\right )}\right )^{3 n}}{d \left (c + d x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(F**(g*(f*x+e)))**n)**3/(d*x+c)**2,x)
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Mathematica [A] time = 1.88003, size = 250, normalized size = 0.82 \[ -\frac{a^3 d-3 a^2 b f g n \log (F) (c+d x) \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )+3 a^2 b d \left (F^{g (e+f x)}\right )^n-6 a b^2 f g n \log (F) (c+d x) \left (F^{g (e+f x)}\right )^{2 n} F^{-\frac{2 f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )+3 a b^2 d \left (F^{g (e+f x)}\right )^{2 n}-3 b^3 f g n \log (F) (c+d x) \left (F^{g (e+f x)}\right )^{3 n} F^{-\frac{3 f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{3 f g n \log (F) (c+d x)}{d}\right )+b^3 d \left (F^{g (e+f x)}\right )^{3 n}}{d^2 (c+d x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*(F^(g*(e + f*x)))^n)^3/(c + d*x)^2,x]
[Out]
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Maple [F] time = 0.031, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{3}}{ \left ( dx+c \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[{\left (F^{e g}\right )}^{3 \, n} b^{3} \int \frac{{\left (F^{f g x}\right )}^{3 \, n}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} + 3 \,{\left (F^{e g}\right )}^{2 \, n} a b^{2} \int \frac{{\left (F^{f g x}\right )}^{2 \, n}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} + 3 \,{\left (F^{e g}\right )}^{n} a^{2} b \int \frac{{\left (F^{f g x}\right )}^{n}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} - \frac{a^{3}}{d^{2} x + c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)^3/(d*x + c)^2,x, algorithm="maxima")
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Fricas [A] time = 0.28413, size = 350, normalized size = 1.15 \[ -\frac{3 \, F^{f g n x + e g n} a^{2} b d + 3 \, F^{2 \, f g n x + 2 \, e g n} a b^{2} d + F^{3 \, f g n x + 3 \, e g n} b^{3} d + a^{3} d - 3 \,{\left (b^{3} d f g n x + b^{3} c f g n\right )} F^{\frac{3 \,{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{3 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right ) - 6 \,{\left (a b^{2} d f g n x + a b^{2} c f g n\right )} F^{\frac{2 \,{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{2 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right ) - 3 \,{\left (a^{2} b d f g n x + a^{2} b c f g n\right )} F^{\frac{{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right )}{d^{3} x + c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)^3/(d*x + c)^2,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(F**(g*(f*x+e)))**n)**3/(d*x+c)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{3}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)^3/(d*x + c)^2,x, algorithm="giac")
[Out]