Optimal. Leaf size=160 \[ 84 c^2 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^3+252 c^2 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)-252 c^2 d^{10} \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\frac{252}{5} c^2 d^{10} (b+2 c x)^5 \]
[Out]
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Rubi [A] time = 0.344062, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ 84 c^2 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^3+252 c^2 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)-252 c^2 d^{10} \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\frac{252}{5} c^2 d^{10} (b+2 c x)^5 \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^10/(a + b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 92.1499, size = 175, normalized size = 1.09 \[ 252 b c^{2} d^{10} \left (- 4 a c + b^{2}\right )^{2} + 504 c^{3} d^{10} x \left (- 4 a c + b^{2}\right )^{2} + \frac{252 c^{2} d^{10} \left (b + 2 c x\right )^{5}}{5} + 84 c^{2} d^{10} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right ) - 252 c^{2} d^{10} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )} - \frac{9 c d^{10} \left (b + 2 c x\right )^{7}}{a + b x + c x^{2}} - \frac{d^{10} \left (b + 2 c x\right )^{9}}{2 \left (a + b x + c x^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**10/(c*x**2+b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.212649, size = 192, normalized size = 1.2 \[ d^{10} \left (128 c^3 x \left (48 a^2 c^2-30 a b^2 c+5 b^4\right )-256 c^5 x^3 \left (4 a c-3 b^2\right )+128 b c^4 x^2 \left (5 b^2-12 a c\right )-252 c^2 \left (4 a c-b^2\right )^{5/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+\frac{17 c \left (4 a c-b^2\right )^3 (b+2 c x)}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^4 (b+2 c x)}{2 (a+x (b+c x))^2}+512 b c^6 x^4+\frac{1024 c^7 x^5}{5}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^10/(a + b*x + c*x^2)^3,x]
[Out]
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Maple [B] time = 0.021, size = 751, normalized size = 4.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^10/(c*x^2+b*x+a)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^10/(c*x^2 + b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226396, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^10/(c*x^2 + b*x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 38.7825, size = 660, normalized size = 4.12 \[ 512 b c^{6} d^{10} x^{4} + \frac{1024 c^{7} d^{10} x^{5}}{5} + 126 c^{2} d^{10} \sqrt{- \left (4 a c - b^{2}\right )^{5}} \log{\left (x + \frac{2016 a^{2} b c^{4} d^{10} - 1008 a b^{3} c^{3} d^{10} + 126 b^{5} c^{2} d^{10} - 126 c^{2} d^{10} \sqrt{- \left (4 a c - b^{2}\right )^{5}}}{4032 a^{2} c^{5} d^{10} - 2016 a b^{2} c^{4} d^{10} + 252 b^{4} c^{3} d^{10}} \right )} - 126 c^{2} d^{10} \sqrt{- \left (4 a c - b^{2}\right )^{5}} \log{\left (x + \frac{2016 a^{2} b c^{4} d^{10} - 1008 a b^{3} c^{3} d^{10} + 126 b^{5} c^{2} d^{10} + 126 c^{2} d^{10} \sqrt{- \left (4 a c - b^{2}\right )^{5}}}{4032 a^{2} c^{5} d^{10} - 2016 a b^{2} c^{4} d^{10} + 252 b^{4} c^{3} d^{10}} \right )} + x^{3} \left (- 1024 a c^{6} d^{10} + 768 b^{2} c^{5} d^{10}\right ) + x^{2} \left (- 1536 a b c^{5} d^{10} + 640 b^{3} c^{4} d^{10}\right ) + x \left (6144 a^{2} c^{5} d^{10} - 3840 a b^{2} c^{4} d^{10} + 640 b^{4} c^{3} d^{10}\right ) + \frac{1920 a^{4} b c^{4} d^{10} - 1376 a^{3} b^{3} c^{3} d^{10} + 312 a^{2} b^{5} c^{2} d^{10} - 18 a b^{7} c d^{10} - b^{9} d^{10} + x^{3} \left (4352 a^{3} c^{6} d^{10} - 3264 a^{2} b^{2} c^{5} d^{10} + 816 a b^{4} c^{4} d^{10} - 68 b^{6} c^{3} d^{10}\right ) + x^{2} \left (6528 a^{3} b c^{5} d^{10} - 4896 a^{2} b^{3} c^{4} d^{10} + 1224 a b^{5} c^{3} d^{10} - 102 b^{7} c^{2} d^{10}\right ) + x \left (3840 a^{4} c^{5} d^{10} - 576 a^{3} b^{2} c^{4} d^{10} - 1008 a^{2} b^{4} c^{3} d^{10} + 372 a b^{6} c^{2} d^{10} - 36 b^{8} c d^{10}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**10/(c*x**2+b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.219268, size = 621, normalized size = 3.88 \[ \frac{252 \,{\left (b^{6} c^{2} d^{10} - 12 \, a b^{4} c^{3} d^{10} + 48 \, a^{2} b^{2} c^{4} d^{10} - 64 \, a^{3} c^{5} d^{10}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{68 \, b^{6} c^{3} d^{10} x^{3} - 816 \, a b^{4} c^{4} d^{10} x^{3} + 3264 \, a^{2} b^{2} c^{5} d^{10} x^{3} - 4352 \, a^{3} c^{6} d^{10} x^{3} + 102 \, b^{7} c^{2} d^{10} x^{2} - 1224 \, a b^{5} c^{3} d^{10} x^{2} + 4896 \, a^{2} b^{3} c^{4} d^{10} x^{2} - 6528 \, a^{3} b c^{5} d^{10} x^{2} + 36 \, b^{8} c d^{10} x - 372 \, a b^{6} c^{2} d^{10} x + 1008 \, a^{2} b^{4} c^{3} d^{10} x + 576 \, a^{3} b^{2} c^{4} d^{10} x - 3840 \, a^{4} c^{5} d^{10} x + b^{9} d^{10} + 18 \, a b^{7} c d^{10} - 312 \, a^{2} b^{5} c^{2} d^{10} + 1376 \, a^{3} b^{3} c^{3} d^{10} - 1920 \, a^{4} b c^{4} d^{10}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} + \frac{128 \,{\left (8 \, c^{22} d^{10} x^{5} + 20 \, b c^{21} d^{10} x^{4} + 30 \, b^{2} c^{20} d^{10} x^{3} - 40 \, a c^{21} d^{10} x^{3} + 25 \, b^{3} c^{19} d^{10} x^{2} - 60 \, a b c^{20} d^{10} x^{2} + 25 \, b^{4} c^{18} d^{10} x - 150 \, a b^{2} c^{19} d^{10} x + 240 \, a^{2} c^{20} d^{10} x\right )}}{5 \, c^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^10/(c*x^2 + b*x + a)^3,x, algorithm="giac")
[Out]