Optimal. Leaf size=371 \[ \frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{40 a^2 x^4}-\frac {\sqrt {a+b x+c x^2} \left (40 a^2 b (55 c e-36 a g)+256 a^2 c (4 c d-5 a f)-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+945 b^4 d\right )}{1920 a^5 x}+\frac {\sqrt {a+b x+c x^2} \left (120 a^2 (3 c e-4 a g)-350 a b^2 e-4 a b (161 c d-100 a f)+315 b^3 d\right )}{960 a^4 x^2}-\frac {\sqrt {a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{240 a^3 x^3}+\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right ) \left (-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)+48 a^2 b c (5 c d-4 a f)-70 a b^4 e-40 a b^3 (7 c d-2 a f)+63 b^5 d\right )}{256 a^{11/2}}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5} \]
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Rubi [A] time = 0.82, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1650, 834, 806, 724, 206} \[ -\frac {\sqrt {a+b x+c x^2} \left (40 a^2 b (55 c e-36 a g)+256 a^2 c (4 c d-5 a f)-60 a b^2 (49 c d-20 a f)-1050 a b^3 e+945 b^4 d\right )}{1920 a^5 x}+\frac {\sqrt {a+b x+c x^2} \left (120 a^2 (3 c e-4 a g)-350 a b^2 e-4 a b (161 c d-100 a f)+315 b^3 d\right )}{960 a^4 x^2}+\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right ) \left (48 a^2 b^2 (5 c e-2 a g)+48 a^2 b c (5 c d-4 a f)-32 a^3 c (3 c e-4 a g)-40 a b^3 (7 c d-2 a f)-70 a b^4 e+63 b^5 d\right )}{256 a^{11/2}}-\frac {\sqrt {a+b x+c x^2} \left (80 a^2 f-70 a b e-64 a c d+63 b^2 d\right )}{240 a^3 x^3}+\frac {\sqrt {a+b x+c x^2} (9 b d-10 a e)}{40 a^2 x^4}-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 834
Rule 1650
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2+g x^3}{x^6 \sqrt {a+b x+c x^2}} \, dx &=-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}-\frac {\int \frac {\frac {1}{2} (9 b d-10 a e)+(4 c d-5 a f) x-5 a g x^2}{x^5 \sqrt {a+b x+c x^2}} \, dx}{5 a}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 b d-10 a e) \sqrt {a+b x+c x^2}}{40 a^2 x^4}+\frac {\int \frac {\frac {1}{4} \left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right )+\frac {1}{2} \left (27 b c d-30 a c e+40 a^2 g\right ) x}{x^4 \sqrt {a+b x+c x^2}} \, dx}{20 a^2}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 b d-10 a e) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}-\frac {\int \frac {\frac {1}{8} \left (315 b^3 d-644 a b c d-350 a b^2 e+360 a^2 c e+400 a^2 b f-480 a^3 g\right )+\frac {1}{2} c \left (63 b^2 d-70 a b e-16 a (4 c d-5 a f)\right ) x}{x^3 \sqrt {a+b x+c x^2}} \, dx}{60 a^3}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 b d-10 a e) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}+\frac {\left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) \sqrt {a+b x+c x^2}}{960 a^4 x^2}+\frac {\int \frac {\frac {1}{16} \left (945 b^4 d-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+256 a^2 c (4 c d-5 a f)+40 a^2 b (55 c e-36 a g)\right )+\frac {1}{8} c \left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx}{120 a^4}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 b d-10 a e) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}+\frac {\left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) \sqrt {a+b x+c x^2}}{960 a^4 x^2}-\frac {\left (945 b^4 d-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+256 a^2 c (4 c d-5 a f)+40 a^2 b (55 c e-36 a g)\right ) \sqrt {a+b x+c x^2}}{1920 a^5 x}-\frac {\left (63 b^5 d-70 a b^4 e+48 a^2 b c (5 c d-4 a f)-40 a b^3 (7 c d-2 a f)-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{256 a^5}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 b d-10 a e) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}+\frac {\left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) \sqrt {a+b x+c x^2}}{960 a^4 x^2}-\frac {\left (945 b^4 d-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+256 a^2 c (4 c d-5 a f)+40 a^2 b (55 c e-36 a g)\right ) \sqrt {a+b x+c x^2}}{1920 a^5 x}+\frac {\left (63 b^5 d-70 a b^4 e+48 a^2 b c (5 c d-4 a f)-40 a b^3 (7 c d-2 a f)-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{128 a^5}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 b d-10 a e) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 b^2 d-64 a c d-70 a b e+80 a^2 f\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}+\frac {\left (315 b^3 d-350 a b^2 e-4 a b (161 c d-100 a f)+120 a^2 (3 c e-4 a g)\right ) \sqrt {a+b x+c x^2}}{960 a^4 x^2}-\frac {\left (945 b^4 d-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+256 a^2 c (4 c d-5 a f)+40 a^2 b (55 c e-36 a g)\right ) \sqrt {a+b x+c x^2}}{1920 a^5 x}+\frac {\left (63 b^5 d-70 a b^4 e+48 a^2 b c (5 c d-4 a f)-40 a b^3 (7 c d-2 a f)-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 299, normalized size = 0.81 \[ \frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right ) \left (32 a^3 c (4 a g-3 c e)-48 a^2 b^2 (2 a g-5 c e)-48 a^2 b c (4 a f-5 c d)-70 a b^4 e+40 a b^3 (2 a f-7 c d)+63 b^5 d\right )}{256 a^{11/2}}-\frac {\sqrt {a+x (b+c x)} \left (32 a^4 \left (12 d+5 x \left (3 e+4 f x+6 g x^2\right )\right )-16 a^3 x (b (27 d+5 x (7 e+2 x (5 f+9 g x)))+c x (32 d+5 x (9 e+16 f x)))+4 a^2 x^2 \left (b^2 (126 d+25 x (7 e+12 f x))+2 b c x (161 d+275 e x)+256 c^2 d x^2\right )-210 a b^2 x^3 (3 b d+5 b e x+14 c d x)+945 b^4 d x^4\right )}{1920 a^5 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 42.47, size = 727, normalized size = 1.96 \[ \left [\frac {15 \, {\left ({\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} d - 2 \, {\left (35 \, a b^{4} - 120 \, a^{2} b^{2} c + 48 \, a^{3} c^{2}\right )} e + 16 \, {\left (5 \, a^{2} b^{3} - 12 \, a^{3} b c\right )} f - 32 \, {\left (3 \, a^{3} b^{2} - 4 \, a^{4} c\right )} g\right )} \sqrt {a} x^{5} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, {\left (384 \, a^{5} d - {\left (1440 \, a^{4} b g - {\left (945 \, a b^{4} - 2940 \, a^{2} b^{2} c + 1024 \, a^{3} c^{2}\right )} d + 50 \, {\left (21 \, a^{2} b^{3} - 44 \, a^{3} b c\right )} e - 80 \, {\left (15 \, a^{3} b^{2} - 16 \, a^{4} c\right )} f\right )} x^{4} - 2 \, {\left (400 \, a^{4} b f - 480 \, a^{5} g + 7 \, {\left (45 \, a^{2} b^{3} - 92 \, a^{3} b c\right )} d - 10 \, {\left (35 \, a^{3} b^{2} - 36 \, a^{4} c\right )} e\right )} x^{3} - 8 \, {\left (70 \, a^{4} b e - 80 \, a^{5} f - {\left (63 \, a^{3} b^{2} - 64 \, a^{4} c\right )} d\right )} x^{2} - 48 \, {\left (9 \, a^{4} b d - 10 \, a^{5} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, a^{6} x^{5}}, -\frac {15 \, {\left ({\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} d - 2 \, {\left (35 \, a b^{4} - 120 \, a^{2} b^{2} c + 48 \, a^{3} c^{2}\right )} e + 16 \, {\left (5 \, a^{2} b^{3} - 12 \, a^{3} b c\right )} f - 32 \, {\left (3 \, a^{3} b^{2} - 4 \, a^{4} c\right )} g\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + 2 \, {\left (384 \, a^{5} d - {\left (1440 \, a^{4} b g - {\left (945 \, a b^{4} - 2940 \, a^{2} b^{2} c + 1024 \, a^{3} c^{2}\right )} d + 50 \, {\left (21 \, a^{2} b^{3} - 44 \, a^{3} b c\right )} e - 80 \, {\left (15 \, a^{3} b^{2} - 16 \, a^{4} c\right )} f\right )} x^{4} - 2 \, {\left (400 \, a^{4} b f - 480 \, a^{5} g + 7 \, {\left (45 \, a^{2} b^{3} - 92 \, a^{3} b c\right )} d - 10 \, {\left (35 \, a^{3} b^{2} - 36 \, a^{4} c\right )} e\right )} x^{3} - 8 \, {\left (70 \, a^{4} b e - 80 \, a^{5} f - {\left (63 \, a^{3} b^{2} - 64 \, a^{4} c\right )} d\right )} x^{2} - 48 \, {\left (9 \, a^{4} b d - 10 \, a^{5} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, a^{6} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 2177, normalized size = 5.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 859, normalized size = 2.32 \[ \frac {c g \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {3}{2}}}-\frac {3 b^{2} g \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {5}{2}}}-\frac {3 b c f \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{4 a^{\frac {5}{2}}}-\frac {3 c^{2} e \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {5}{2}}}+\frac {5 b^{3} f \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {7}{2}}}+\frac {15 b^{2} c e \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {7}{2}}}+\frac {15 b \,c^{2} d \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {7}{2}}}-\frac {35 b^{4} e \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {9}{2}}}-\frac {35 b^{3} c d \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{32 a^{\frac {9}{2}}}+\frac {63 b^{5} d \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{256 a^{\frac {11}{2}}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, b g}{4 a^{2} x}+\frac {2 \sqrt {c \,x^{2}+b x +a}\, c f}{3 a^{2} x}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{2} f}{8 a^{3} x}-\frac {55 \sqrt {c \,x^{2}+b x +a}\, b c e}{48 a^{3} x}-\frac {8 \sqrt {c \,x^{2}+b x +a}\, c^{2} d}{15 a^{3} x}+\frac {35 \sqrt {c \,x^{2}+b x +a}\, b^{3} e}{64 a^{4} x}+\frac {49 \sqrt {c \,x^{2}+b x +a}\, b^{2} c d}{32 a^{4} x}-\frac {63 \sqrt {c \,x^{2}+b x +a}\, b^{4} d}{128 a^{5} x}-\frac {\sqrt {c \,x^{2}+b x +a}\, g}{2 a \,x^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b f}{12 a^{2} x^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, c e}{8 a^{2} x^{2}}-\frac {35 \sqrt {c \,x^{2}+b x +a}\, b^{2} e}{96 a^{3} x^{2}}-\frac {161 \sqrt {c \,x^{2}+b x +a}\, b c d}{240 a^{3} x^{2}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, b^{3} d}{64 a^{4} x^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, f}{3 a \,x^{3}}+\frac {7 \sqrt {c \,x^{2}+b x +a}\, b e}{24 a^{2} x^{3}}+\frac {4 \sqrt {c \,x^{2}+b x +a}\, c d}{15 a^{2} x^{3}}-\frac {21 \sqrt {c \,x^{2}+b x +a}\, b^{2} d}{80 a^{3} x^{3}}-\frac {\sqrt {c \,x^{2}+b x +a}\, e}{4 a \,x^{4}}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, b d}{40 a^{2} x^{4}}-\frac {\sqrt {c \,x^{2}+b x +a}\, d}{5 a \,x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {g\,x^3+f\,x^2+e\,x+d}{x^6\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d + e x + f x^{2} + g x^{3}}{x^{6} \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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