Optimal. Leaf size=321 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{-c (2 a h+b f)+b^2 h+2 c^2 d}{\sqrt{b^2-4 a c}}-b h+c f\right )}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{-2 a c h+b^2 h-b c f+2 c^2 d}{\sqrt{b^2-4 a c}}-b h+c f\right )}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{2 c^2 \sqrt{b^2-4 a c}}+\frac{(c g-b i) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{h x}{c}+\frac{i x^2}{2 c} \]
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Rubi [A] time = 0.533943, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1673, 1676, 1166, 205, 1663, 1657, 634, 618, 206, 628} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{-c (2 a h+b f)+b^2 h+2 c^2 d}{\sqrt{b^2-4 a c}}-b h+c f\right )}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{-2 a c h+b^2 h-b c f+2 c^2 d}{\sqrt{b^2-4 a c}}-b h+c f\right )}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{2 c^2 \sqrt{b^2-4 a c}}+\frac{(c g-b i) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{h x}{c}+\frac{i x^2}{2 c} \]
Antiderivative was successfully verified.
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Rule 1673
Rule 1676
Rule 1166
Rule 205
Rule 1663
Rule 1657
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2+g x^3+h x^4+24 x^5}{a+b x^2+c x^4} \, dx &=\int \frac{x \left (e+g x^2+24 x^4\right )}{a+b x^2+c x^4} \, dx+\int \frac{d+f x^2+h x^4}{a+b x^2+c x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{e+g x+24 x^2}{a+b x+c x^2} \, dx,x,x^2\right )+\int \left (\frac{h}{c}+\frac{c d-a h+(c f-b h) x^2}{c \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=\frac{h x}{c}+\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{24}{c}-\frac{24 a-c e+(24 b-c g) x}{c \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )+\frac{\int \frac{c d-a h+(c f-b h) x^2}{a+b x^2+c x^4} \, dx}{c}\\ &=\frac{h x}{c}+\frac{12 x^2}{c}-\frac{\operatorname{Subst}\left (\int \frac{24 a-c e+(24 b-c g) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c}+\frac{\left (c f-b h-\frac{2 c^2 d-b c f+b^2 h-2 a c h}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{2 c}+\frac{\left (c f-b h+\frac{2 c^2 d+b^2 h-c (b f+2 a h)}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{2 c}\\ &=\frac{h x}{c}+\frac{12 x^2}{c}+\frac{\left (c f-b h+\frac{2 c^2 d+b^2 h-c (b f+2 a h)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (c f-b h-\frac{2 c^2 d-b c f+b^2 h-2 a c h}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{(24 b-c g) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}-\frac{(2 c (24 a-c e)-b (24 b-c g)) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac{h x}{c}+\frac{12 x^2}{c}+\frac{\left (c f-b h+\frac{2 c^2 d+b^2 h-c (b f+2 a h)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (c f-b h-\frac{2 c^2 d-b c f+b^2 h-2 a c h}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{(24 b-c g) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{(2 c (24 a-c e)-b (24 b-c g)) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^2}\\ &=\frac{h x}{c}+\frac{12 x^2}{c}+\frac{\left (c f-b h+\frac{2 c^2 d+b^2 h-c (b f+2 a h)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (c f-b h-\frac{2 c^2 d-b c f+b^2 h-2 a c h}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{\left (24 b^2-2 c (24 a-c e)-b c g\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c}}-\frac{(24 b-c g) \log \left (a+b x^2+c x^4\right )}{4 c^2}\\ \end{align*}
Mathematica [A] time = 0.797854, size = 441, normalized size = 1.37 \[ \frac{\frac{2 \sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (c \left (f \sqrt{b^2-4 a c}-2 a h-b f\right )+b h \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{2 \sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-c \left (f \sqrt{b^2-4 a c}+2 a h+b f\right )+b h \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right ) \left (c \left (g \sqrt{b^2-4 a c}-2 a i-b g\right )+b i \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 e\right )}{\sqrt{b^2-4 a c}}-\frac{\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (-c \left (g \sqrt{b^2-4 a c}+2 a i+b g\right )+b i \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 e\right )}{\sqrt{b^2-4 a c}}+4 c h x+2 c i x^2}{4 c^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 1435, normalized size = 4.5 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{i x^{2} + 2 \, h x}{2 \, c} - \frac{-\int \frac{{\left (c g - b i\right )} x^{3} +{\left (c f - b h\right )} x^{2} + c d - a h +{\left (c e - a i\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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