Optimal. Leaf size=208 \[ \frac{(b+2 c x) \left (-3 b^2 f h^2+6 b c e h^2+4 c^2 \left (f g^2-h (2 d h+e g)\right )\right )}{3 c h^2 (2 c g-b h)^3 \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}+\frac{2 \left (d h^2-e g h+f g^2\right )}{3 h^3 (g+h x) (2 c g-b h) \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}-\frac{f}{c h^3 \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}} \]
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Rubi [A] time = 0.424205, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.064, Rules used = {1638, 792, 613} \[ \frac{(b+2 c x) \left (-3 b^2 f h^2+6 b c e h^2+4 c^2 \left (f g^2-h (2 d h+e g)\right )\right )}{3 c h^2 (2 c g-b h)^3 \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}+\frac{2 \left (d h^2-e g h+f g^2\right )}{3 h^3 (g+h x) (2 c g-b h) \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}-\frac{f}{c h^3 \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 1638
Rule 792
Rule 613
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2}{(g+h x) \left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx &=-\frac{f}{c h^3 \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}-\frac{\int \frac{\frac{1}{2} h^3 (b f g-2 c d h)+\frac{1}{2} h^3 (2 c f g-2 c e h+b f h) x}{(g+h x) \left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx}{c h^4}\\ &=-\frac{f}{c h^3 \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}+\frac{2 \left (f g^2-e g h+d h^2\right )}{3 h^3 (2 c g-b h) (g+h x) \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}-\frac{\left (6 b c e h^2-3 b^2 f h^2+4 c^2 \left (f g^2-h (e g+2 d h)\right )\right ) \int \frac{1}{\left (-c g^2+b g h+b h^2 x+c h^2 x^2\right )^{3/2}} \, dx}{6 c h^2 (2 c g-b h)}\\ &=-\frac{f}{c h^3 \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}+\frac{\left (6 b c e h^2-3 b^2 f h^2+4 c^2 \left (f g^2-h (e g+2 d h)\right )\right ) (b+2 c x)}{3 c h^2 (2 c g-b h)^3 \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}+\frac{2 \left (f g^2-e g h+d h^2\right )}{3 h^3 (2 c g-b h) (g+h x) \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.519542, size = 219, normalized size = 1.05 \[ \frac{2 b^2 h^2 \left (f \left (8 g^2+12 g h x+3 h^2 x^2\right )-h (d h+2 e g+3 e h x)\right )-4 b c h \left (h \left (e \left (g^2+2 g h x+3 h^2 x^2\right )-2 d h (2 g+h x)\right )+2 f g^2 (4 g+5 h x)\right )+8 c^2 \left (h \left (d h \left (-g^2+2 g h x+2 h^2 x^2\right )+e g \left (g^2+g h x+h^2 x^2\right )\right )+f g^2 \left (2 g^2+2 g h x-h^2 x^2\right )\right )}{3 h^3 (g+h x) (b h-2 c g)^3 \sqrt{(g+h x) (b h-c g+c h x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 324, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 2\,chx+2\,bh-2\,cg \right ) \left ( -3\,{b}^{2}f{h}^{4}{x}^{2}+6\,bce{h}^{4}{x}^{2}-8\,{c}^{2}d{h}^{4}{x}^{2}-4\,{c}^{2}eg{h}^{3}{x}^{2}+4\,{c}^{2}f{g}^{2}{h}^{2}{x}^{2}+3\,{b}^{2}e{h}^{4}x-12\,{b}^{2}fg{h}^{3}x-4\,bcd{h}^{4}x+4\,bceg{h}^{3}x+20\,bcf{g}^{2}{h}^{2}x-8\,{c}^{2}dg{h}^{3}x-4\,{c}^{2}e{g}^{2}{h}^{2}x-8\,{c}^{2}f{g}^{3}hx+{b}^{2}d{h}^{4}+2\,{b}^{2}eg{h}^{3}-8\,{b}^{2}f{g}^{2}{h}^{2}-8\,bcdg{h}^{3}+2\,bce{g}^{2}{h}^{2}+16\,bcf{g}^{3}h+4\,{c}^{2}d{g}^{2}{h}^{2}-4\,{c}^{2}e{g}^{3}h-8\,{c}^{2}f{g}^{4} \right ) }{ \left ( 3\,{b}^{3}{h}^{3}-18\,{b}^{2}cg{h}^{2}+36\,b{c}^{2}{g}^{2}h-24\,{c}^{3}{g}^{3} \right ){h}^{3}} \left ( c{h}^{2}{x}^{2}+b{h}^{2}x+bgh-c{g}^{2} \right ) ^{-{\frac{3}{2}}}} \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 [B] time = 134.36, size = 941, normalized size = 4.52 \begin{align*} \frac{2 \,{\left (8 \, c^{2} f g^{4} - b^{2} d h^{4} + 4 \,{\left (c^{2} e - 4 \, b c f\right )} g^{3} h - 2 \,{\left (2 \, c^{2} d + b c e - 4 \, b^{2} f\right )} g^{2} h^{2} + 2 \,{\left (4 \, b c d - b^{2} e\right )} g h^{3} -{\left (4 \, c^{2} f g^{2} h^{2} - 4 \, c^{2} e g h^{3} -{\left (8 \, c^{2} d - 6 \, b c e + 3 \, b^{2} f\right )} h^{4}\right )} x^{2} +{\left (8 \, c^{2} f g^{3} h + 4 \,{\left (c^{2} e - 5 \, b c f\right )} g^{2} h^{2} + 4 \,{\left (2 \, c^{2} d - b c e + 3 \, b^{2} f\right )} g h^{3} +{\left (4 \, b c d - 3 \, b^{2} e\right )} h^{4}\right )} x\right )} \sqrt{c h^{2} x^{2} + b h^{2} x - c g^{2} + b g h}}{3 \,{\left (8 \, c^{4} g^{6} h^{3} - 20 \, b c^{3} g^{5} h^{4} + 18 \, b^{2} c^{2} g^{4} h^{5} - 7 \, b^{3} c g^{3} h^{6} + b^{4} g^{2} h^{7} -{\left (8 \, c^{4} g^{3} h^{6} - 12 \, b c^{3} g^{2} h^{7} + 6 \, b^{2} c^{2} g h^{8} - b^{3} c h^{9}\right )} x^{3} -{\left (8 \, c^{4} g^{4} h^{5} - 4 \, b c^{3} g^{3} h^{6} - 6 \, b^{2} c^{2} g^{2} h^{7} + 5 \, b^{3} c g h^{8} - b^{4} h^{9}\right )} x^{2} +{\left (8 \, c^{4} g^{5} h^{4} - 28 \, b c^{3} g^{4} h^{5} + 30 \, b^{2} c^{2} g^{3} h^{6} - 13 \, b^{3} c g^{2} h^{7} + 2 \, b^{4} g h^{8}\right )} x\right )}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x^{2} + e x + d}{{\left (c h^{2} x^{2} + b h^{2} x - c g^{2} + b g h\right )}^{\frac{3}{2}}{\left (h x + g\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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