Optimal. Leaf size=346 \[ -\frac{\sqrt{a+b x+c x^2} \left (-2 c x \left (-40 c^2 (9 a f+10 b e)+14 b c (46 a g+25 b f)-315 b^3 g+480 c^3 d\right )-60 b^2 c (20 c e-49 a g)+40 b c^2 (36 c d-55 a f)+256 a c^2 (5 c e-4 a g)+1050 b^3 c f-945 b^4 g\right )}{1920 c^5}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (48 b^2 c^2 (2 c d-5 a f)-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)-32 a c^3 (4 c d-3 a f)+70 b^4 c f-63 b^5 g\right )}{256 c^{11/2}}+\frac{x^2 \sqrt{a+b x+c x^2} \left (-64 a c g+63 b^2 g-70 b c f+80 c^2 e\right )}{240 c^3}+\frac{x^3 \sqrt{a+b x+c x^2} (10 c f-9 b g)}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c} \]
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Rubi [A] time = 0.811677, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {1653, 832, 779, 621, 206} \[ -\frac{\sqrt{a+b x+c x^2} \left (-2 c x \left (-40 c^2 (9 a f+10 b e)+14 b c (46 a g+25 b f)-315 b^3 g+480 c^3 d\right )-60 b^2 c (20 c e-49 a g)+40 b c^2 (36 c d-55 a f)+256 a c^2 (5 c e-4 a g)+1050 b^3 c f-945 b^4 g\right )}{1920 c^5}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (48 b^2 c^2 (2 c d-5 a f)-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)-32 a c^3 (4 c d-3 a f)+70 b^4 c f-63 b^5 g\right )}{256 c^{11/2}}+\frac{x^2 \sqrt{a+b x+c x^2} \left (-64 a c g+63 b^2 g-70 b c f+80 c^2 e\right )}{240 c^3}+\frac{x^3 \sqrt{a+b x+c x^2} (10 c f-9 b g)}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 832
Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2 \left (d+e x+f x^2+g x^3\right )}{\sqrt{a+b x+c x^2}} \, dx &=\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x^2 \left (5 c d+(5 c e-4 a g) x+\frac{1}{2} (10 c f-9 b g) x^2\right )}{\sqrt{a+b x+c x^2}} \, dx}{5 c}\\ &=\frac{(10 c f-9 b g) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x^2 \left (\frac{1}{2} \left (40 c^2 d-30 a c f+27 a b g\right )+\frac{1}{4} \left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{20 c^2}\\ &=\frac{\left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}+\frac{(10 c f-9 b g) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x \left (-\frac{1}{2} a \left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right )+\frac{1}{8} \left (480 c^3 d-40 c^2 (10 b e+9 a f)-315 b^3 g+14 b c (25 b f+46 a g)\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{60 c^3}\\ &=\frac{\left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}+\frac{(10 c f-9 b g) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}-\frac{\left (1050 b^3 c f+40 b c^2 (36 c d-55 a f)-945 b^4 g-60 b^2 c (20 c e-49 a g)+256 a c^2 (5 c e-4 a g)-2 c \left (480 c^3 d-40 c^2 (10 b e+9 a f)-315 b^3 g+14 b c (25 b f+46 a g)\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}+\frac{\left (70 b^4 c f+48 b^2 c^2 (2 c d-5 a f)-32 a c^3 (4 c d-3 a f)-63 b^5 g-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{256 c^5}\\ &=\frac{\left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}+\frac{(10 c f-9 b g) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}-\frac{\left (1050 b^3 c f+40 b c^2 (36 c d-55 a f)-945 b^4 g-60 b^2 c (20 c e-49 a g)+256 a c^2 (5 c e-4 a g)-2 c \left (480 c^3 d-40 c^2 (10 b e+9 a f)-315 b^3 g+14 b c (25 b f+46 a g)\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}+\frac{\left (70 b^4 c f+48 b^2 c^2 (2 c d-5 a f)-32 a c^3 (4 c d-3 a f)-63 b^5 g-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{128 c^5}\\ &=\frac{\left (80 c^2 e-70 b c f+63 b^2 g-64 a c g\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}+\frac{(10 c f-9 b g) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}-\frac{\left (1050 b^3 c f+40 b c^2 (36 c d-55 a f)-945 b^4 g-60 b^2 c (20 c e-49 a g)+256 a c^2 (5 c e-4 a g)-2 c \left (480 c^3 d-40 c^2 (10 b e+9 a f)-315 b^3 g+14 b c (25 b f+46 a g)\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}+\frac{\left (70 b^4 c f+48 b^2 c^2 (2 c d-5 a f)-32 a c^3 (4 c d-3 a f)-63 b^5 g-40 b^3 c (2 c e-7 a g)+48 a b c^2 (4 c e-5 a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.72651, size = 282, normalized size = 0.82 \[ \frac{\sqrt{a+x (b+c x)} \left (16 c^2 \left (64 a^2 g-a c (80 e+x (45 f+32 g x))+2 c^2 x (30 d+x (20 e+3 x (5 f+4 g x)))\right )+4 b^2 c (-735 a g+300 c e+7 c x (25 f+18 g x))-8 b c^2 \left (2 c \left (90 d+x \left (50 e+35 f x+27 g x^2\right )\right )-a (275 f+161 g x)\right )-210 b^3 c (5 f+3 g x)+945 b^4 g\right )}{1920 c^5}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (-48 b^2 c^2 (2 c d-5 a f)+40 b^3 c (2 c e-7 a g)+48 a b c^2 (5 a g-4 c e)+32 a c^3 (4 c d-3 a f)-70 b^4 c f+63 b^5 g\right )}{256 c^{11/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.058, size = 783, normalized size = 2.3 \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 = 2.15884, size = 1666, normalized size = 4.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (d + e x + f x^{2} + g x^{3}\right )}{\sqrt{a + b x + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23068, size = 446, normalized size = 1.29 \begin{align*} \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (\frac{8 \, g x}{c} + \frac{10 \, c^{4} f - 9 \, b c^{3} g}{c^{5}}\right )} x - \frac{70 \, b c^{3} f - 63 \, b^{2} c^{2} g + 64 \, a c^{3} g - 80 \, c^{4} e}{c^{5}}\right )} x + \frac{480 \, c^{4} d + 350 \, b^{2} c^{2} f - 360 \, a c^{3} f - 315 \, b^{3} c g + 644 \, a b c^{2} g - 400 \, b c^{3} e}{c^{5}}\right )} x - \frac{1440 \, b c^{3} d + 1050 \, b^{3} c f - 2200 \, a b c^{2} f - 945 \, b^{4} g + 2940 \, a b^{2} c g - 1024 \, a^{2} c^{2} g - 1200 \, b^{2} c^{2} e + 1280 \, a c^{3} e}{c^{5}}\right )} - \frac{{\left (96 \, b^{2} c^{3} d - 128 \, a c^{4} d + 70 \, b^{4} c f - 240 \, a b^{2} c^{2} f + 96 \, a^{2} c^{3} f - 63 \, b^{5} g + 280 \, a b^{3} c g - 240 \, a^{2} b c^{2} g - 80 \, b^{3} c^{2} e + 192 \, a b c^{3} e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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