Optimal. Leaf size=134 \[ 140 c^2 d^8 \left (b^2-4 a c\right ) (b+2 c x)-140 c^2 d^8 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}+\frac{140}{3} c^2 d^8 (b+2 c x)^3 \]
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Rubi [A] time = 0.263537, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ 140 c^2 d^8 \left (b^2-4 a c\right ) (b+2 c x)-140 c^2 d^8 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}+\frac{140}{3} c^2 d^8 (b+2 c x)^3 \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 70.4638, size = 146, normalized size = 1.09 \[ 140 b c^{2} d^{8} \left (- 4 a c + b^{2}\right ) + 280 c^{3} d^{8} x \left (- 4 a c + b^{2}\right ) + \frac{140 c^{2} d^{8} \left (b + 2 c x\right )^{3}}{3} - 140 c^{2} d^{8} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )} - \frac{7 c d^{8} \left (b + 2 c x\right )^{5}}{a + b x + c x^{2}} - \frac{d^{8} \left (b + 2 c x\right )^{7}}{2 \left (a + b x + c x^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**8/(c*x**2+b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.162455, size = 142, normalized size = 1.06 \[ d^8 \left (-256 c^3 x \left (3 a c-b^2\right )+140 c^2 \left (4 a c-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )-\frac{13 c \left (b^2-4 a c\right )^2 (b+2 c x)}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^3 (b+2 c x)}{2 (a+x (b+c x))^2}+128 b c^4 x^2+\frac{256 c^5 x^3}{3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x]
[Out]
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Maple [B] time = 0.02, size = 526, normalized size = 3.9 \[{\frac{256\,{d}^{8}{c}^{5}{x}^{3}}{3}}+128\,{d}^{8}b{c}^{4}{x}^{2}-768\,{d}^{8}xa{c}^{4}+256\,{d}^{8}x{b}^{2}{c}^{3}-416\,{\frac{{d}^{8}{x}^{3}{a}^{2}{c}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+208\,{\frac{{d}^{8}{x}^{3}a{b}^{2}{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-26\,{\frac{{d}^{8}{x}^{3}{b}^{4}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-624\,{\frac{{d}^{8}{x}^{2}{a}^{2}b{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+312\,{\frac{{d}^{8}a{b}^{3}{c}^{3}{x}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-39\,{\frac{{d}^{8}{x}^{2}{b}^{5}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-352\,{\frac{{d}^{8}{a}^{3}{c}^{4}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-48\,{\frac{{d}^{8}{a}^{2}{b}^{2}{c}^{3}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+90\,{\frac{{d}^{8}{c}^{2}a{b}^{4}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-14\,{\frac{{d}^{8}{b}^{6}cx}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-176\,{\frac{{d}^{8}{a}^{3}b{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+80\,{\frac{{d}^{8}{a}^{2}{b}^{3}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-7\,{\frac{{d}^{8}a{b}^{5}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{8}{b}^{7}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+2240\,{\frac{{c}^{4}{d}^{8}{a}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-1120\,{\frac{{d}^{8}{c}^{3}a{b}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+140\,{\frac{{d}^{8}{c}^{2}{b}^{4}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^8/(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)^8/(c*x^2 + b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236594, 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)^8/(c*x^2 + b*x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 29.9584, size = 469, normalized size = 3.5 \[ 128 b c^{4} d^{8} x^{2} + \frac{256 c^{5} d^{8} x^{3}}{3} - 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{280 a b c^{3} d^{8} - 70 b^{3} c^{2} d^{8} - 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{560 a c^{4} d^{8} - 140 b^{2} c^{3} d^{8}} \right )} + 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{280 a b c^{3} d^{8} - 70 b^{3} c^{2} d^{8} + 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{560 a c^{4} d^{8} - 140 b^{2} c^{3} d^{8}} \right )} + x \left (- 768 a c^{4} d^{8} + 256 b^{2} c^{3} d^{8}\right ) - \frac{352 a^{3} b c^{3} d^{8} - 160 a^{2} b^{3} c^{2} d^{8} + 14 a b^{5} c d^{8} + b^{7} d^{8} + x^{3} \left (832 a^{2} c^{5} d^{8} - 416 a b^{2} c^{4} d^{8} + 52 b^{4} c^{3} d^{8}\right ) + x^{2} \left (1248 a^{2} b c^{4} d^{8} - 624 a b^{3} c^{3} d^{8} + 78 b^{5} c^{2} d^{8}\right ) + x \left (704 a^{3} c^{4} d^{8} + 96 a^{2} b^{2} c^{3} d^{8} - 180 a b^{4} c^{2} d^{8} + 28 b^{6} c d^{8}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**8/(c*x**2+b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.218748, size = 425, normalized size = 3.17 \[ \frac{140 \,{\left (b^{4} c^{2} d^{8} - 8 \, a b^{2} c^{3} d^{8} + 16 \, a^{2} c^{4} d^{8}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{52 \, b^{4} c^{3} d^{8} x^{3} - 416 \, a b^{2} c^{4} d^{8} x^{3} + 832 \, a^{2} c^{5} d^{8} x^{3} + 78 \, b^{5} c^{2} d^{8} x^{2} - 624 \, a b^{3} c^{3} d^{8} x^{2} + 1248 \, a^{2} b c^{4} d^{8} x^{2} + 28 \, b^{6} c d^{8} x - 180 \, a b^{4} c^{2} d^{8} x + 96 \, a^{2} b^{2} c^{3} d^{8} x + 704 \, a^{3} c^{4} d^{8} x + b^{7} d^{8} + 14 \, a b^{5} c d^{8} - 160 \, a^{2} b^{3} c^{2} d^{8} + 352 \, a^{3} b c^{3} d^{8}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} + \frac{128 \,{\left (2 \, c^{14} d^{8} x^{3} + 3 \, b c^{13} d^{8} x^{2} + 6 \, b^{2} c^{12} d^{8} x - 18 \, a c^{13} d^{8} x\right )}}{3 \, c^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^8/(c*x^2 + b*x + a)^3,x, algorithm="giac")
[Out]