Optimal. Leaf size=270 \[ \frac {\sqrt {a+b x+c x^2} (7 b d-8 a e)}{24 a^2 x^3}-\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right ) \left (32 a^2 b (3 c e-2 a g)+16 a^2 c (3 c d-4 a f)-40 a b^3 e-24 a b^2 (5 c d-2 a f)+35 b^4 d\right )}{128 a^{9/2}}+\frac {\sqrt {a+b x+c x^2} \left (64 a^2 (2 c e-3 a g)-120 a b^2 e-4 a b (55 c d-36 a f)+105 b^3 d\right )}{192 a^4 x}-\frac {\sqrt {a+b x+c x^2} \left (48 a^2 f-40 a b e-36 a c d+35 b^2 d\right )}{96 a^3 x^2}-\frac {d \sqrt {a+b x+c x^2}}{4 a x^4} \]
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Rubi [A] time = 0.49, antiderivative size = 270, normalized size of antiderivative = 1.00, 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 = {1650, 834, 806, 724, 206} \[ \frac {\sqrt {a+b x+c x^2} \left (64 a^2 (2 c e-3 a g)-120 a b^2 e-4 a b (55 c d-36 a f)+105 b^3 d\right )}{192 a^4 x}-\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right ) \left (32 a^2 b (3 c e-2 a g)+16 a^2 c (3 c d-4 a f)-24 a b^2 (5 c d-2 a f)-40 a b^3 e+35 b^4 d\right )}{128 a^{9/2}}-\frac {\sqrt {a+b x+c x^2} \left (48 a^2 f-40 a b e-36 a c d+35 b^2 d\right )}{96 a^3 x^2}+\frac {\sqrt {a+b x+c x^2} (7 b d-8 a e)}{24 a^2 x^3}-\frac {d \sqrt {a+b x+c x^2}}{4 a x^4} \]
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^5 \sqrt {a+b x+c x^2}} \, dx &=-\frac {d \sqrt {a+b x+c x^2}}{4 a x^4}-\frac {\int \frac {\frac {1}{2} (7 b d-8 a e)+(3 c d-4 a f) x-4 a g x^2}{x^4 \sqrt {a+b x+c x^2}} \, dx}{4 a}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 b d-8 a e) \sqrt {a+b x+c x^2}}{24 a^2 x^3}+\frac {\int \frac {\frac {1}{4} \left (35 b^2 d-40 a b e-12 a (3 c d-4 a f)\right )+\left (7 b c d-8 a c e+12 a^2 g\right ) x}{x^3 \sqrt {a+b x+c x^2}} \, dx}{12 a^2}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 b d-8 a e) \sqrt {a+b x+c x^2}}{24 a^2 x^3}-\frac {\left (35 b^2 d-36 a c d-40 a b e+48 a^2 f\right ) \sqrt {a+b x+c x^2}}{96 a^3 x^2}-\frac {\int \frac {\frac {1}{8} \left (105 b^3 d-220 a b c d-120 a b^2 e+128 a^2 c e+144 a^2 b f-192 a^3 g\right )+\frac {1}{4} c \left (35 b^2 d-40 a b e-12 a (3 c d-4 a f)\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx}{24 a^3}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 b d-8 a e) \sqrt {a+b x+c x^2}}{24 a^2 x^3}-\frac {\left (35 b^2 d-36 a c d-40 a b e+48 a^2 f\right ) \sqrt {a+b x+c x^2}}{96 a^3 x^2}+\frac {\left (105 b^3 d-120 a b^2 e-4 a b (55 c d-36 a f)+64 a^2 (2 c e-3 a g)\right ) \sqrt {a+b x+c x^2}}{192 a^4 x}+\frac {\left (35 b^4 d-40 a b^3 e+16 a^2 c (3 c d-4 a f)-24 a b^2 (5 c d-2 a f)+32 a^2 b (3 c e-2 a g)\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{128 a^4}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 b d-8 a e) \sqrt {a+b x+c x^2}}{24 a^2 x^3}-\frac {\left (35 b^2 d-36 a c d-40 a b e+48 a^2 f\right ) \sqrt {a+b x+c x^2}}{96 a^3 x^2}+\frac {\left (105 b^3 d-120 a b^2 e-4 a b (55 c d-36 a f)+64 a^2 (2 c e-3 a g)\right ) \sqrt {a+b x+c x^2}}{192 a^4 x}-\frac {\left (35 b^4 d-40 a b^3 e+16 a^2 c (3 c d-4 a f)-24 a b^2 (5 c d-2 a f)+32 a^2 b (3 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 )}{64 a^4}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 b d-8 a e) \sqrt {a+b x+c x^2}}{24 a^2 x^3}-\frac {\left (35 b^2 d-36 a c d-40 a b e+48 a^2 f\right ) \sqrt {a+b x+c x^2}}{96 a^3 x^2}+\frac {\left (105 b^3 d-120 a b^2 e-4 a b (55 c d-36 a f)+64 a^2 (2 c e-3 a g)\right ) \sqrt {a+b x+c x^2}}{192 a^4 x}-\frac {\left (35 b^4 d-40 a b^3 e+16 a^2 c (3 c d-4 a f)-24 a b^2 (5 c d-2 a f)+32 a^2 b (3 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 )}{128 a^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 212, normalized size = 0.79 \[ \frac {\sqrt {a+x (b+c x)} \left (-16 a^3 \left (3 d+4 e x+6 x^2 (f+2 g x)\right )+8 a^2 x (7 b d+2 b x (5 e+9 f x)+c x (9 d+16 e x))-10 a b x^2 (7 b d+12 b e x+22 c d x)+105 b^3 d x^3\right )}{192 a^4 x^4}-\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right ) \left (32 a^2 b (3 c e-2 a g)+16 a^2 c (3 c d-4 a f)-40 a b^3 e+24 a b^2 (2 a f-5 c d)+35 b^4 d\right )}{128 a^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 10.83, size = 525, normalized size = 1.94 \[ \left [\frac {3 \, {\left (64 \, a^{3} b g - {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} d + 8 \, {\left (5 \, a b^{3} - 12 \, a^{2} b c\right )} e - 16 \, {\left (3 \, a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} \sqrt {a} x^{4} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, {\left (48 \, a^{4} d - {\left (144 \, a^{3} b f - 192 \, a^{4} g + 5 \, {\left (21 \, a b^{3} - 44 \, a^{2} b c\right )} d - 8 \, {\left (15 \, a^{2} b^{2} - 16 \, a^{3} c\right )} e\right )} x^{3} - 2 \, {\left (40 \, a^{3} b e - 48 \, a^{4} f - {\left (35 \, a^{2} b^{2} - 36 \, a^{3} c\right )} d\right )} x^{2} - 8 \, {\left (7 \, a^{3} b d - 8 \, a^{4} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, a^{5} x^{4}}, -\frac {3 \, {\left (64 \, a^{3} b g - {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} d + 8 \, {\left (5 \, a b^{3} - 12 \, a^{2} b c\right )} e - 16 \, {\left (3 \, a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + 2 \, {\left (48 \, a^{4} d - {\left (144 \, a^{3} b f - 192 \, a^{4} g + 5 \, {\left (21 \, a b^{3} - 44 \, a^{2} b c\right )} d - 8 \, {\left (15 \, a^{2} b^{2} - 16 \, a^{3} c\right )} e\right )} x^{3} - 2 \, {\left (40 \, a^{3} b e - 48 \, a^{4} f - {\left (35 \, a^{2} b^{2} - 36 \, a^{3} c\right )} d\right )} x^{2} - 8 \, {\left (7 \, a^{3} b d - 8 \, a^{4} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, a^{5} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 1448, normalized size = 5.36 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 591, normalized size = 2.19 \[ \frac {b g \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {3}{2}}}+\frac {c f \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {3}{2}}}-\frac {3 b^{2} f \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {5}{2}}}-\frac {3 b c e \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{4 a^{\frac {5}{2}}}-\frac {3 c^{2} d \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {5}{2}}}+\frac {5 b^{3} e \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {7}{2}}}+\frac {15 b^{2} c d \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {7}{2}}}-\frac {35 b^{4} d \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {9}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}\, g}{a x}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, b f}{4 a^{2} x}+\frac {2 \sqrt {c \,x^{2}+b x +a}\, c e}{3 a^{2} x}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{2} e}{8 a^{3} x}-\frac {55 \sqrt {c \,x^{2}+b x +a}\, b c d}{48 a^{3} x}+\frac {35 \sqrt {c \,x^{2}+b x +a}\, b^{3} d}{64 a^{4} x}-\frac {\sqrt {c \,x^{2}+b x +a}\, f}{2 a \,x^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b e}{12 a^{2} x^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, c d}{8 a^{2} x^{2}}-\frac {35 \sqrt {c \,x^{2}+b x +a}\, b^{2} d}{96 a^{3} x^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, e}{3 a \,x^{3}}+\frac {7 \sqrt {c \,x^{2}+b x +a}\, b d}{24 a^{2} x^{3}}-\frac {\sqrt {c \,x^{2}+b x +a}\, d}{4 a \,x^{4}} \]
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 \[ \int \frac {g\,x^3+f\,x^2+e\,x+d}{x^5\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {d + e x + f x^{2} + g x^{3}}{x^{5} \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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