Optimal. Leaf size=63 \[ -\frac{\text{Int}\left (\frac{1}{x^3 \left (a+b F^{c+d x}\right )^2},x\right )}{b d \log (F)}-\frac{1}{2 b d x^2 \log (F) \left (a+b F^{c+d x}\right )^2} \]
[Out]
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Rubi [A] time = 0.178992, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2},x\right ) \]
Verification is Not applicable to the result.
[In] Int[F^(c + d*x)/((a + b*F^(c + d*x))^3*x^2),x]
[Out]
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Rubi in Sympy [A] time = 0., size = 0, normalized size = 0. \[ - \frac{\int \frac{1}{x^{3} \left (F^{c + d x} b + a\right )^{2}}\, dx}{b d \log{\left (F \right )}} - \frac{1}{2 b d x^{2} \left (F^{c + d x} b + a\right )^{2} \log{\left (F \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(d*x+c)/(a+b*F**(d*x+c))**3/x**2,x)
[Out]
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Mathematica [A] time = 0.907917, size = 0, normalized size = 0. \[ \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[F^(c + d*x)/((a + b*F^(c + d*x))^3*x^2),x]
[Out]
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Maple [A] time = 0.119, size = 0, normalized size = 0. \[ \int{\frac{{F}^{dx+c}}{ \left ( a+b{F}^{dx+c} \right ) ^{3}{x}^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(d*x+c)/(a+b*F^(d*x+c))^3/x^2,x)
[Out]
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Maxima [A] time = 0., size = 0, normalized size = 0. \[ -\frac{a d x \log \left (F\right ) + 2 \, F^{d x} F^{c} b + 2 \, a}{2 \,{\left (2 \, F^{d x} F^{c} a^{2} b^{2} d^{2} x^{3} \log \left (F\right )^{2} + F^{2 \, d x} F^{2 \, c} a b^{3} d^{2} x^{3} \log \left (F\right )^{2} + a^{3} b d^{2} x^{3} \log \left (F\right )^{2}\right )}} - \int \frac{d x \log \left (F\right ) + 3}{F^{d x} F^{c} a b^{2} d^{2} x^{4} \log \left (F\right )^{2} + a^{2} b d^{2} x^{4} \log \left (F\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(d*x + c)/((F^(d*x + c)*b + a)^3*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{F^{d x + c}}{3 \, F^{d x + c} a^{2} b x^{2} + 3 \, F^{2 \, d x + 2 \, c} a b^{2} x^{2} + F^{3 \, d x + 3 \, c} b^{3} x^{2} + a^{3} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(d*x + c)/((F^(d*x + c)*b + a)^3*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0., size = 0, normalized size = 0. \[ \frac{- 2 F^{c + d x} b - a d x \log{\left (F \right )} - 2 a}{4 F^{c + d x} a^{2} b^{2} d^{2} x^{3} \log{\left (F \right )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} x^{3} \log{\left (F \right )}^{2} + 2 a^{3} b d^{2} x^{3} \log{\left (F \right )}^{2}} - \frac{\int \frac{d x \log{\left (F \right )}}{a x^{4} + b x^{4} e^{c \log{\left (F \right )}} e^{d x \log{\left (F \right )}}}\, dx + \int \frac{3}{a x^{4} + b x^{4} e^{c \log{\left (F \right )}} e^{d x \log{\left (F \right )}}}\, dx}{a b d^{2} \log{\left (F \right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(d*x+c)/(a+b*F**(d*x+c))**3/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(d*x + c)/((F^(d*x + c)*b + a)^3*x^2),x, algorithm="giac")
[Out]