Optimal. Leaf size=103 \[ \frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c f^{g+h x}}{\sqrt{b^2-4 a c}}\right )}{a h \log (f) \sqrt{b^2-4 a c}}-\frac{d \log \left (a+b f^{g+h x}+c f^{2 g+2 h x}\right )}{2 a h \log (f)}+\frac{d x}{a} \]
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Rubi [A] time = 0.156565, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {2282, 800, 634, 618, 206, 628} \[ \frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c f^{g+h x}}{\sqrt{b^2-4 a c}}\right )}{a h \log (f) \sqrt{b^2-4 a c}}-\frac{d \log \left (a+b f^{g+h x}+c f^{2 g+2 h x}\right )}{2 a h \log (f)}+\frac{d x}{a} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{d+e f^{g+h x}}{a+b f^{g+h x}+c f^{2 g+2 h x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{d+e x}{x \left (a+b x+c x^2\right )} \, dx,x,f^{g+h x}\right )}{h \log (f)}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{d}{a x}+\frac{-b d+a e-c d x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,f^{g+h x}\right )}{h \log (f)}\\ &=\frac{d x}{a}+\frac{\operatorname{Subst}\left (\int \frac{-b d+a e-c d x}{a+b x+c x^2} \, dx,x,f^{g+h x}\right )}{a h \log (f)}\\ &=\frac{d x}{a}-\frac{d \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,f^{g+h x}\right )}{2 a h \log (f)}-\frac{(b d-2 a e) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,f^{g+h x}\right )}{2 a h \log (f)}\\ &=\frac{d x}{a}-\frac{d \log \left (a+b f^{g+h x}+c f^{2 g+2 h x}\right )}{2 a h \log (f)}+\frac{(b d-2 a e) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c f^{g+h x}\right )}{a h \log (f)}\\ &=\frac{d x}{a}+\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 \sqrt{b^2-4 a c} h \log (f)}-\frac{d \log \left (a+b f^{g+h x}+c f^{2 g+2 h x}\right )}{2 a h \log (f)}\\ \end{align*}
Mathematica [A] time = 0.157817, 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.
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Maple [B] time = 0.134, size = 993, normalized size = 9.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33769, size = 755, normalized size = 7.33 \begin{align*} \left [\frac{2 \,{\left (b^{2} - 4 \, a c\right )} d h x \log \left (f\right ) -{\left (b^{2} - 4 \, a c\right )} d \log \left (c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a\right ) - \sqrt{b^{2} - 4 \, a c}{\left (b d - 2 \, a e\right )} \log \left (\frac{2 \, c^{2} f^{2 \, h x + 2 \, g} + b^{2} - 2 \, a c + 2 \,{\left (b c - \sqrt{b^{2} - 4 \, a c} c\right )} f^{h x + g} - \sqrt{b^{2} - 4 \, a c} b}{c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a}\right )}{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} h \log \left (f\right )}, \frac{2 \,{\left (b^{2} - 4 \, a c\right )} d h x \log \left (f\right ) -{\left (b^{2} - 4 \, a c\right )} d \log \left (c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a\right ) + 2 \, \sqrt{-b^{2} + 4 \, a c}{\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 \,{\left (a b^{2} - 4 \, a^{2} c\right )} h \log \left (f\right )}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.878698, size = 139, normalized size = 1.35 \begin{align*} \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 (f^{g + h x} + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e f^{h x + g} + d}{c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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