Optimal. Leaf size=103 \[ \frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c f^{g+h x}}{\sqrt{b^2-4 a c}}\right )}{a h \log (f) \sqrt{b^2-4 a c}}-\frac{d \log \left (a+b f^{g+h x}+c f^{2 g+2 h x}\right )}{2 a h \log (f)}+\frac{d x}{a} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.26173, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c f^{g+h x}}{\sqrt{b^2-4 a c}}\right )}{a h \log (f) \sqrt{b^2-4 a c}}-\frac{d \log \left (a+b f^{g+h x}+c f^{2 g+2 h x}\right )}{2 a h \log (f)}+\frac{d x}{a} \]
Antiderivative was successfully verified.
[In] Int[(d + e*f^(g + h*x))/(a + b*f^(g + h*x) + c*f^(2*(g + h*x))),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 57.2282, size = 148, normalized size = 1.44 \[ \frac{d f^{- g - h x} f^{g + h x} \log{\left (f^{g + h x} \right )}}{a h \log{\left (f \right )}} - \frac{d f^{- g - h x} f^{g + h x} \log{\left (a + b f^{g + h x} + c f^{2 g + 2 h x} \right )}}{2 a h \log{\left (f \right )}} - \frac{f^{- g - h x} f^{g + h x} \left (2 a e - b d\right ) \operatorname{atanh}{\left (\frac{b + 2 c f^{g + h x}}{\sqrt{- 4 a c + b^{2}}} \right )}}{a h \sqrt{- 4 a c + b^{2}} \log{\left (f \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d+e*f**(h*x+g))/(a+b*f**(h*x+g)+c*f**(2*h*x+2*g)),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.034115, size = 102, normalized size = 0.99 \[ -\frac{\frac{2 (b d-2 a e) \tan ^{-1}\left (\frac{b+2 c f^{g+h x}}{\sqrt{4 a c-b^2}}\right )}{h \log (f) \sqrt{4 a c-b^2}}+\frac{d \log \left (a+f^{g+h x} \left (b+c f^{g+h x}\right )\right )}{h \log (f)}-2 d x}{2 a} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*f^(g + h*x))/(a + b*f^(g + h*x) + c*f^(2*(g + h*x))),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0., size = 993, normalized size = 9.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d+e*f^(h*x+g))/(a+b*f^(h*x+g)+c*f^(2*h*x+2*g)),x)
[Out]
_______________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f^(h*x + g) + d)/(c*f^(2*h*x + 2*g) + b*f^(h*x + g) + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.393964, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b^{2} - 4 \, a c} d h x \log \left (f\right ) - \sqrt{b^{2} - 4 \, a c} d \log \left (c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a\right ) -{\left (b d - 2 \, a e\right )} \log \left (\frac{2 \, \sqrt{b^{2} - 4 \, a c} c^{2} f^{2 \, h x + 2 \, g} - b^{3} + 4 \, a b c - 2 \,{\left (b^{2} c - 4 \, a c^{2} - \sqrt{b^{2} - 4 \, a c} b c\right )} f^{h x + g} +{\left (b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a}\right )}{2 \, \sqrt{b^{2} - 4 \, a c} a h \log \left (f\right )}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c} d h x \log \left (f\right ) - \sqrt{-b^{2} + 4 \, a c} d \log \left (c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a\right ) - 2 \,{\left (b d - 2 \, a e\right )} \arctan \left (-\frac{2 \, \sqrt{-b^{2} + 4 \, a c} c f^{h x + g} + \sqrt{-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a h \log \left (f\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f^(h*x + g) + d)/(c*f^(2*h*x + 2*g) + b*f^(h*x + g) + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.33486, size = 148, normalized size = 1.44 \[ \operatorname{RootSum}{\left (z^{2} \left (4 a^{2} c h^{2} \log{\left (f \right )}^{2} - a b^{2} h^{2} \log{\left (f \right )}^{2}\right ) + z \left (4 a c d h \log{\left (f \right )} - b^{2} d h \log{\left (f \right )}\right ) + a e^{2} - b d e + c d^{2}, \left ( i \mapsto i \log{\left (e^{\frac{\left (2 g + 2 h x\right ) \log{\left (f \right )}}{2}} + \frac{4 i a^{2} c h \log{\left (f \right )} - i a b^{2} h \log{\left (f \right )} + a b e + 2 a c d - b^{2} d}{2 a c e - b c d} \right )} \right )\right )} + \frac{d x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d+e*f**(h*x+g))/(a+b*f**(h*x+g)+c*f**(2*h*x+2*g)),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e f^{h x + g} + d}{c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f^(h*x + g) + d)/(c*f^(2*h*x + 2*g) + b*f^(h*x + g) + a),x, algorithm="giac")
[Out]