Optimal. Leaf size=717 \[ \frac{\sqrt{a+b x+c x^2} \left (-504 b c^2 f \left (22 a^2 f^2+70 a b e f+25 b^2 \left (d f+e^2\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (d f+e^2\right )+50 b^2 \left (6 d e f+e^3\right )\right )+420 b^3 c f^2 (34 a f+27 b e)-640 c^4 \left (8 a e \left (6 d f+e^2\right )+27 b d \left (d f+e^2\right )\right )-3465 b^5 f^3+23040 c^5 d^2 e\right )}{7680 c^6}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (840 b^2 c^2 f \left (2 a^2 f^2+4 a b e f+b^2 \left (d f+e^2\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (d f+e^2\right )+b^3 \left (6 d e f+e^3\right )\right )+384 c^4 \left (3 a^2 f \left (d f+e^2\right )+2 a b e \left (6 d f+e^2\right )+3 b^2 d \left (d f+e^2\right )\right )-252 b^4 c f^2 (5 a f+3 b e)-1536 c^5 d \left (a \left (d f+e^2\right )+b d e\right )+231 b^6 f^3+1024 c^6 d^3\right )}{1024 c^{13/2}}+\frac{x \sqrt{a+b x+c x^2} \left (24 c^2 f \left (50 a^2 f^2+322 a b e f+175 b^2 \left (d f+e^2\right )\right )-252 b^2 c f^2 (14 a f+15 b e)-160 c^3 \left (27 a f \left (d f+e^2\right )+10 b \left (6 d e f+e^3\right )\right )+1155 b^4 f^3+5760 c^4 d \left (d f+e^2\right )\right )}{3840 c^5}+\frac{f x^3 \sqrt{a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{480 c^3}-\frac{x^2 \sqrt{a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{960 c^4}+\frac{f^2 x^4 \sqrt{a+b x+c x^2} (36 c e-11 b f)}{60 c^2}+\frac{f^3 x^5 \sqrt{a+b x+c x^2}}{6 c} \]
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Rubi [A] time = 2.70964, antiderivative size = 717, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1661, 640, 621, 206} \[ \frac{\sqrt{a+b x+c x^2} \left (-504 b c^2 f \left (22 a^2 f^2+70 a b e f+25 b^2 \left (d f+e^2\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (d f+e^2\right )+50 b^2 \left (6 d e f+e^3\right )\right )+420 b^3 c f^2 (34 a f+27 b e)-640 c^4 \left (8 a e \left (6 d f+e^2\right )+27 b d \left (d f+e^2\right )\right )-3465 b^5 f^3+23040 c^5 d^2 e\right )}{7680 c^6}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (840 b^2 c^2 f \left (2 a^2 f^2+4 a b e f+b^2 \left (d f+e^2\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (d f+e^2\right )+b^3 \left (6 d e f+e^3\right )\right )+384 c^4 \left (3 a^2 f \left (d f+e^2\right )+2 a b e \left (6 d f+e^2\right )+3 b^2 d \left (d f+e^2\right )\right )-252 b^4 c f^2 (5 a f+3 b e)-1536 c^5 d \left (a \left (d f+e^2\right )+b d e\right )+231 b^6 f^3+1024 c^6 d^3\right )}{1024 c^{13/2}}+\frac{x \sqrt{a+b x+c x^2} \left (24 c^2 f \left (50 a^2 f^2+322 a b e f+175 b^2 \left (d f+e^2\right )\right )-252 b^2 c f^2 (14 a f+15 b e)-160 c^3 \left (27 a f \left (d f+e^2\right )+10 b \left (6 d e f+e^3\right )\right )+1155 b^4 f^3+5760 c^4 d \left (d f+e^2\right )\right )}{3840 c^5}+\frac{f x^3 \sqrt{a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{480 c^3}-\frac{x^2 \sqrt{a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{960 c^4}+\frac{f^2 x^4 \sqrt{a+b x+c x^2} (36 c e-11 b f)}{60 c^2}+\frac{f^3 x^5 \sqrt{a+b x+c x^2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (d+e x+f x^2\right )^3}{\sqrt{a+b x+c x^2}} \, dx &=\frac{f^3 x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\int \frac{6 c d^3+18 c d^2 e x+18 c d \left (e^2+d f\right ) x^2+6 c e \left (e^2+6 d f\right ) x^3-f \left (5 a f^2-18 c \left (e^2+d f\right )\right ) x^4+\frac{1}{2} f^2 (36 c e-11 b f) x^5}{\sqrt{a+b x+c x^2}} \, dx}{6 c}\\ &=\frac{f^2 (36 c e-11 b f) x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{f^3 x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\int \frac{30 c^2 d^3+90 c^2 d^2 e x+90 c^2 d \left (e^2+d f\right ) x^2-2 \left (36 a c e f^2-11 a b f^3-15 c^2 \left (e^3+6 d e f\right )\right ) x^3+\frac{1}{4} f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^4}{\sqrt{a+b x+c x^2}} \, dx}{30 c^2}\\ &=\frac{f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}+\frac{f^2 (36 c e-11 b f) x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{f^3 x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\int \frac{120 c^3 d^3+360 c^3 d^2 e x-\frac{3}{4} \left (99 a b^2 f^3-4 a c f^2 (81 b e+25 a f)-480 c^3 d \left (e^2+d f\right )+360 a c^2 f \left (e^2+d f\right )\right ) x^2-\frac{3}{8} \left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^3}{\sqrt{a+b x+c x^2}} \, dx}{120 c^3}\\ &=-\frac{\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt{a+b x+c x^2}}{960 c^4}+\frac{f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}+\frac{f^2 (36 c e-11 b f) x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{f^3 x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\int \frac{360 c^4 d^3+\frac{3}{4} \left (1440 c^4 d^2 e+231 a b^3 f^3-36 a b c f^2 (21 b e+13 a f)-320 a c^3 e \left (e^2+6 d f\right )+24 a c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x+\frac{3}{16} \left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x^2}{\sqrt{a+b x+c x^2}} \, dx}{360 c^4}\\ &=\frac{\left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x \sqrt{a+b x+c x^2}}{3840 c^5}-\frac{\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt{a+b x+c x^2}}{960 c^4}+\frac{f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}+\frac{f^2 (36 c e-11 b f) x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{f^3 x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\int \frac{\frac{3}{16} \left (3840 c^5 d^3-1155 a b^4 f^3+252 a b^2 c f^2 (15 b e+14 a f)-5760 a c^4 d \left (e^2+d f\right )-24 a c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )+160 a c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right )+\frac{3}{32} \left (23040 c^5 d^2 e-3465 b^5 f^3+420 b^3 c f^2 (27 b e+34 a f)-504 b c^2 f \left (70 a b e f+22 a^2 f^2+25 b^2 \left (e^2+d f\right )\right )-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (e^2+d f\right )+50 b^2 \left (e^3+6 d e f\right )\right )\right ) x}{\sqrt{a+b x+c x^2}} \, dx}{720 c^5}\\ &=\frac{\left (23040 c^5 d^2 e-3465 b^5 f^3+420 b^3 c f^2 (27 b e+34 a f)-504 b c^2 f \left (70 a b e f+22 a^2 f^2+25 b^2 \left (e^2+d f\right )\right )-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (e^2+d f\right )+50 b^2 \left (e^3+6 d e f\right )\right )\right ) \sqrt{a+b x+c x^2}}{7680 c^6}+\frac{\left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x \sqrt{a+b x+c x^2}}{3840 c^5}-\frac{\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt{a+b x+c x^2}}{960 c^4}+\frac{f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}+\frac{f^2 (36 c e-11 b f) x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{f^3 x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\left (1024 c^6 d^3+231 b^6 f^3-252 b^4 c f^2 (3 b e+5 a f)-1536 c^5 d \left (b d e+a \left (e^2+d f\right )\right )+840 b^2 c^2 f \left (4 a b e f+2 a^2 f^2+b^2 \left (e^2+d f\right )\right )+384 c^4 \left (3 b^2 d \left (e^2+d f\right )+3 a^2 f \left (e^2+d f\right )+2 a b e \left (e^2+6 d f\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{1024 c^6}\\ &=\frac{\left (23040 c^5 d^2 e-3465 b^5 f^3+420 b^3 c f^2 (27 b e+34 a f)-504 b c^2 f \left (70 a b e f+22 a^2 f^2+25 b^2 \left (e^2+d f\right )\right )-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (e^2+d f\right )+50 b^2 \left (e^3+6 d e f\right )\right )\right ) \sqrt{a+b x+c x^2}}{7680 c^6}+\frac{\left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x \sqrt{a+b x+c x^2}}{3840 c^5}-\frac{\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt{a+b x+c x^2}}{960 c^4}+\frac{f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}+\frac{f^2 (36 c e-11 b f) x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{f^3 x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\left (1024 c^6 d^3+231 b^6 f^3-252 b^4 c f^2 (3 b e+5 a f)-1536 c^5 d \left (b d e+a \left (e^2+d f\right )\right )+840 b^2 c^2 f \left (4 a b e f+2 a^2 f^2+b^2 \left (e^2+d f\right )\right )+384 c^4 \left (3 b^2 d \left (e^2+d f\right )+3 a^2 f \left (e^2+d f\right )+2 a b e \left (e^2+6 d f\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\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 )}{512 c^6}\\ &=\frac{\left (23040 c^5 d^2 e-3465 b^5 f^3+420 b^3 c f^2 (27 b e+34 a f)-504 b c^2 f \left (70 a b e f+22 a^2 f^2+25 b^2 \left (e^2+d f\right )\right )-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (e^2+d f\right )+50 b^2 \left (e^3+6 d e f\right )\right )\right ) \sqrt{a+b x+c x^2}}{7680 c^6}+\frac{\left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x \sqrt{a+b x+c x^2}}{3840 c^5}-\frac{\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt{a+b x+c x^2}}{960 c^4}+\frac{f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}+\frac{f^2 (36 c e-11 b f) x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{f^3 x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\left (1024 c^6 d^3+231 b^6 f^3-252 b^4 c f^2 (3 b e+5 a f)-1536 c^5 d \left (b d e+a \left (e^2+d f\right )\right )+840 b^2 c^2 f \left (4 a b e f+2 a^2 f^2+b^2 \left (e^2+d f\right )\right )+384 c^4 \left (3 b^2 d \left (e^2+d f\right )+3 a^2 f \left (e^2+d f\right )+2 a b e \left (e^2+6 d f\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{13/2}}\\ \end{align*}
Mathematica [A] time = 1.40472, size = 615, normalized size = 0.86 \[ \frac{\sqrt{a+x (b+c x)} \left (-168 b c^2 f \left (66 a^2 f^2+42 a b f (5 e+f x)+b^2 \left (75 d f+75 e^2+45 e f x+11 f^2 x^2\right )\right )+48 c^3 \left (2 a^2 f^2 (128 e+25 f x)+2 a b f \left (f \left (275 d+39 f x^2\right )+275 e^2+161 e f x\right )+b^2 \left (6 e f \left (100 d+21 f x^2\right )+f^2 x \left (175 d+33 f x^2\right )+175 e^2 f x+100 e^3\right )\right )+210 b^3 c f^2 (68 a f+54 b e+11 b f x)-64 c^4 \left (a \left (96 e f \left (5 d+f x^2\right )+5 f^2 x \left (27 d+5 f x^2\right )+135 e^2 f x+80 e^3\right )+b \left (270 d^2 f+15 d \left (18 e^2+20 e f x+7 f^2 x^2\right )+x \left (105 e^2 f x+50 e^3+81 e f^2 x^2+22 f^3 x^3\right )\right )\right )-3465 b^5 f^3+128 c^5 \left (90 d^2 (2 e+f x)+15 d x \left (6 e^2+8 e f x+3 f^2 x^2\right )+x^2 \left (45 e^2 f x+20 e^3+36 e f^2 x^2+10 f^3 x^3\right )\right )\right )}{7680 c^6}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (840 b^2 c^2 f \left (2 a^2 f^2+4 a b e f+b^2 \left (d f+e^2\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (d f+e^2\right )+b^3 \left (6 d e f+e^3\right )\right )+384 c^4 \left (3 a^2 f \left (d f+e^2\right )+2 a b e \left (6 d f+e^2\right )+3 b^2 d \left (d f+e^2\right )\right )-252 b^4 c f^2 (5 a f+3 b e)-1536 c^5 d \left (a \left (d f+e^2\right )+b d e\right )+231 b^6 f^3+1024 c^6 d^3\right )}{1024 c^{13/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.073, size = 1930, normalized size = 2.7 \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 = 7.39034, size = 3661, normalized size = 5.11 \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{\left (d + e x + f x^{2}\right )^{3}}{\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 = 1.36933, size = 1112, normalized size = 1.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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