Optimal. Leaf size=717 \[ \frac {\left (23040 c^5 d^2 e-3465 b^5 f^3+420 b^3 c f^2 (27 b e+34 a f)-504 b c^2 f \left (70 a b e f+22 a^2 f^2+25 b^2 \left (e^2+d f\right )\right )-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (e^2+d f\right )+50 b^2 \left (e^3+6 d e f\right )\right )\right ) \sqrt {a+b x+c x^2}}{7680 c^6}+\frac {\left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x \sqrt {a+b x+c x^2}}{3840 c^5}-\frac {\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt {a+b x+c x^2}}{960 c^4}+\frac {f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}+\frac {f^2 (36 c e-11 b f) x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\left (1024 c^6 d^3+231 b^6 f^3-252 b^4 c f^2 (3 b e+5 a f)-1536 c^5 d \left (b d e+a \left (e^2+d f\right )\right )+840 b^2 c^2 f \left (4 a b e f+2 a^2 f^2+b^2 \left (e^2+d f\right )\right )+384 c^4 \left (3 b^2 d \left (e^2+d f\right )+3 a^2 f \left (e^2+d f\right )+2 a b e \left (e^2+6 d f\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{13/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.71, antiderivative size = 717, normalized size of antiderivative = 1.00, 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 = {1675, 654, 635,
212} \begin {gather*} \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 {\sqrt {a+b x+c x^2} \left (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 )-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 )+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 (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 )+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 (a^3 f^3+9 a^2 b e f^2+9 a b^2 f \left (d f+e^2\right )+b^3 \left (6 d e f+e^3\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^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 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 {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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 635
Rule 654
Rule 1675
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 ) \text {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 = 2.79, size = 615, normalized size = 0.86 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-3465 b^5 f^3+210 b^3 c f^2 (54 b e+68 a f+11 b f x)-168 b c^2 f \left (66 a^2 f^2+42 a b f (5 e+f x)+b^2 \left (75 e^2+75 d f+45 e f x+11 f^2 x^2\right )\right )+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 (20 e^3+45 e^2 f x+36 e f^2 x^2+10 f^3 x^3\right )\right )+48 c^3 \left (2 a^2 f^2 (128 e+25 f x)+b^2 \left (100 e^3+175 e^2 f x+6 e f \left (100 d+21 f x^2\right )+f^2 x \left (175 d+33 f x^2\right )\right )+2 a b f \left (275 e^2+161 e f x+f \left (275 d+39 f x^2\right )\right )\right )-64 c^4 \left (a \left (80 e^3+135 e^2 f x+96 e f \left (5 d+f x^2\right )+5 f^2 x \left (27 d+5 f x^2\right )\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 (50 e^3+105 e^2 f x+81 e f^2 x^2+22 f^3 x^3\right )\right )\right )\right )-15 \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 ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{15360 c^{13/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2206\) vs.
\(2(683)=1366\).
time = 0.15, size = 2207, normalized size = 3.08
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1387\) |
default | \(\text {Expression too large to display}\) | \(2207\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.70, size = 1569, normalized size = 2.19 \begin {gather*} \left [\frac {15 \, {\left (1024 \, c^{6} d^{3} + 384 \, {\left (3 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{2} f + 24 \, {\left (35 \, b^{4} c^{2} - 120 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} d f^{2} + {\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\right )} f^{3} - 64 \, {\left (5 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} e^{3} + 24 \, {\left (16 \, {\left (3 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d + {\left (35 \, b^{4} c^{2} - 120 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} f\right )} e^{2} - 12 \, {\left (128 \, b c^{5} d^{2} + 32 \, {\left (5 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d f + {\left (63 \, b^{5} c - 280 \, a b^{3} c^{2} + 240 \, a^{2} b c^{3}\right )} f^{2}\right )} e\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (1280 \, c^{6} f^{3} x^{5} - 1408 \, b c^{5} f^{3} x^{4} - 17280 \, b c^{5} d^{2} f - 600 \, {\left (21 \, b^{3} c^{3} - 44 \, a b c^{4}\right )} d f^{2} - 21 \, {\left (165 \, b^{5} c - 680 \, a b^{3} c^{2} + 528 \, a^{2} b c^{3}\right )} f^{3} + 16 \, {\left (360 \, c^{6} d f^{2} + {\left (99 \, b^{2} c^{4} - 100 \, a c^{5}\right )} f^{3}\right )} x^{3} - 24 \, {\left (280 \, b c^{5} d f^{2} + {\left (77 \, b^{3} c^{3} - 156 \, a b c^{4}\right )} f^{3}\right )} x^{2} + 6 \, {\left (1920 \, c^{6} d^{2} f + 40 \, {\left (35 \, b^{2} c^{4} - 36 \, a c^{5}\right )} d f^{2} + {\left (385 \, b^{4} c^{2} - 1176 \, a b^{2} c^{3} + 400 \, a^{2} c^{4}\right )} f^{3}\right )} x + 320 \, {\left (8 \, c^{6} x^{2} - 10 \, b c^{5} x + 15 \, b^{2} c^{4} - 16 \, a c^{5}\right )} e^{3} + 120 \, {\left (48 \, c^{6} f x^{3} - 56 \, b c^{5} f x^{2} - 144 \, b c^{5} d - 5 \, {\left (21 \, b^{3} c^{3} - 44 \, a b c^{4}\right )} f + 2 \, {\left (48 \, c^{6} d + {\left (35 \, b^{2} c^{4} - 36 \, a c^{5}\right )} f\right )} x\right )} e^{2} + 12 \, {\left (384 \, c^{6} f^{2} x^{4} - 432 \, b c^{5} f^{2} x^{3} + 1920 \, c^{6} d^{2} + 160 \, {\left (15 \, b^{2} c^{4} - 16 \, a c^{5}\right )} d f + {\left (945 \, b^{4} c^{2} - 2940 \, a b^{2} c^{3} + 1024 \, a^{2} c^{4}\right )} f^{2} + 8 \, {\left (160 \, c^{6} d f + {\left (63 \, b^{2} c^{4} - 64 \, a c^{5}\right )} f^{2}\right )} x^{2} - 2 \, {\left (800 \, b c^{5} d f + 7 \, {\left (45 \, b^{3} c^{3} - 92 \, a b c^{4}\right )} f^{2}\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{7}}, -\frac {15 \, {\left (1024 \, c^{6} d^{3} + 384 \, {\left (3 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{2} f + 24 \, {\left (35 \, b^{4} c^{2} - 120 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} d f^{2} + {\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\right )} f^{3} - 64 \, {\left (5 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} e^{3} + 24 \, {\left (16 \, {\left (3 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d + {\left (35 \, b^{4} c^{2} - 120 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} f\right )} e^{2} - 12 \, {\left (128 \, b c^{5} d^{2} + 32 \, {\left (5 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d f + {\left (63 \, b^{5} c - 280 \, a b^{3} c^{2} + 240 \, a^{2} b c^{3}\right )} f^{2}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (1280 \, c^{6} f^{3} x^{5} - 1408 \, b c^{5} f^{3} x^{4} - 17280 \, b c^{5} d^{2} f - 600 \, {\left (21 \, b^{3} c^{3} - 44 \, a b c^{4}\right )} d f^{2} - 21 \, {\left (165 \, b^{5} c - 680 \, a b^{3} c^{2} + 528 \, a^{2} b c^{3}\right )} f^{3} + 16 \, {\left (360 \, c^{6} d f^{2} + {\left (99 \, b^{2} c^{4} - 100 \, a c^{5}\right )} f^{3}\right )} x^{3} - 24 \, {\left (280 \, b c^{5} d f^{2} + {\left (77 \, b^{3} c^{3} - 156 \, a b c^{4}\right )} f^{3}\right )} x^{2} + 6 \, {\left (1920 \, c^{6} d^{2} f + 40 \, {\left (35 \, b^{2} c^{4} - 36 \, a c^{5}\right )} d f^{2} + {\left (385 \, b^{4} c^{2} - 1176 \, a b^{2} c^{3} + 400 \, a^{2} c^{4}\right )} f^{3}\right )} x + 320 \, {\left (8 \, c^{6} x^{2} - 10 \, b c^{5} x + 15 \, b^{2} c^{4} - 16 \, a c^{5}\right )} e^{3} + 120 \, {\left (48 \, c^{6} f x^{3} - 56 \, b c^{5} f x^{2} - 144 \, b c^{5} d - 5 \, {\left (21 \, b^{3} c^{3} - 44 \, a b c^{4}\right )} f + 2 \, {\left (48 \, c^{6} d + {\left (35 \, b^{2} c^{4} - 36 \, a c^{5}\right )} f\right )} x\right )} e^{2} + 12 \, {\left (384 \, c^{6} f^{2} x^{4} - 432 \, b c^{5} f^{2} x^{3} + 1920 \, c^{6} d^{2} + 160 \, {\left (15 \, b^{2} c^{4} - 16 \, a c^{5}\right )} d f + {\left (945 \, b^{4} c^{2} - 2940 \, a b^{2} c^{3} + 1024 \, a^{2} c^{4}\right )} f^{2} + 8 \, {\left (160 \, c^{6} d f + {\left (63 \, b^{2} c^{4} - 64 \, a c^{5}\right )} f^{2}\right )} x^{2} - 2 \, {\left (800 \, b c^{5} d f + 7 \, {\left (45 \, b^{3} c^{3} - 92 \, a b c^{4}\right )} f^{2}\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x + f x^{2}\right )^{3}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.67, size = 824, normalized size = 1.15 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (\frac {10 \, f^{3} x}{c} - \frac {11 \, b c^{4} f^{3} - 36 \, c^{5} f^{2} e}{c^{6}}\right )} x + \frac {360 \, c^{5} d f^{2} + 99 \, b^{2} c^{3} f^{3} - 100 \, a c^{4} f^{3} - 324 \, b c^{4} f^{2} e + 360 \, c^{5} f e^{2}}{c^{6}}\right )} x - \frac {840 \, b c^{4} d f^{2} + 231 \, b^{3} c^{2} f^{3} - 468 \, a b c^{3} f^{3} - 1920 \, c^{5} d f e - 756 \, b^{2} c^{3} f^{2} e + 768 \, a c^{4} f^{2} e + 840 \, b c^{4} f e^{2} - 320 \, c^{5} e^{3}}{c^{6}}\right )} x + \frac {5760 \, c^{5} d^{2} f + 4200 \, b^{2} c^{3} d f^{2} - 4320 \, a c^{4} d f^{2} + 1155 \, b^{4} c f^{3} - 3528 \, a b^{2} c^{2} f^{3} + 1200 \, a^{2} c^{3} f^{3} - 9600 \, b c^{4} d f e - 3780 \, b^{3} c^{2} f^{2} e + 7728 \, a b c^{3} f^{2} e + 5760 \, c^{5} d e^{2} + 4200 \, b^{2} c^{3} f e^{2} - 4320 \, a c^{4} f e^{2} - 1600 \, b c^{4} e^{3}}{c^{6}}\right )} x - \frac {17280 \, b c^{4} d^{2} f + 12600 \, b^{3} c^{2} d f^{2} - 26400 \, a b c^{3} d f^{2} + 3465 \, b^{5} f^{3} - 14280 \, a b^{3} c f^{3} + 11088 \, a^{2} b c^{2} f^{3} - 23040 \, c^{5} d^{2} e - 28800 \, b^{2} c^{3} d f e + 30720 \, a c^{4} d f e - 11340 \, b^{4} c f^{2} e + 35280 \, a b^{2} c^{2} f^{2} e - 12288 \, a^{2} c^{3} f^{2} e + 17280 \, b c^{4} d e^{2} + 12600 \, b^{3} c^{2} f e^{2} - 26400 \, a b c^{3} f e^{2} - 4800 \, b^{2} c^{3} e^{3} + 5120 \, a c^{4} e^{3}}{c^{6}}\right )} - \frac {{\left (1024 \, c^{6} d^{3} + 1152 \, b^{2} c^{4} d^{2} f - 1536 \, a c^{5} d^{2} f + 840 \, b^{4} c^{2} d f^{2} - 2880 \, a b^{2} c^{3} d f^{2} + 1152 \, a^{2} c^{4} d f^{2} + 231 \, b^{6} f^{3} - 1260 \, a b^{4} c f^{3} + 1680 \, a^{2} b^{2} c^{2} f^{3} - 320 \, a^{3} c^{3} f^{3} - 1536 \, b c^{5} d^{2} e - 1920 \, b^{3} c^{3} d f e + 4608 \, a b c^{4} d f e - 756 \, b^{5} c f^{2} e + 3360 \, a b^{3} c^{2} f^{2} e - 2880 \, a^{2} b c^{3} f^{2} e + 1152 \, b^{2} c^{4} d e^{2} - 1536 \, a c^{5} d e^{2} + 840 \, b^{4} c^{2} f e^{2} - 2880 \, a b^{2} c^{3} f e^{2} + 1152 \, a^{2} c^{4} f e^{2} - 320 \, b^{3} c^{3} e^{3} + 768 \, a b c^{4} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {13}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f\,x^2+e\,x+d\right )}^3}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________