Optimal. Leaf size=158 \[ -\frac{6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 (d+e x) (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]
[Out]
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Rubi [A] time = 0.223801, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 (d+e x) (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/(a + b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 33.4721, size = 146, normalized size = 0.92 \[ - \frac{\left (b + 2 c x\right ) \left (d + e x\right )^{3}}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} + \frac{3 \left (d + e x\right ) \left (\frac{b e}{2} - c d\right ) \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{\left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} + \frac{6 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(c*x**2+b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.890309, size = 308, normalized size = 1.95 \[ \frac{1}{2} \left (\frac{2 c \left (a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right )+b^2 e^2 (3 c d x-a e)+b c \left (3 a e^2 (d+e x)+c d^2 (d-3 e x)\right )-b^3 e^3 x}{c^2 \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{4 c^2 \left (-4 a^2 e^3+3 a c d e^2 x+3 c^2 d^3 x\right )+b^2 c e \left (5 a e^2-9 c d^2+6 c d e x\right )+6 b c^2 \left (a e^2 (d-e x)+c d^2 (d-3 e x)\right )+b^4 \left (-e^3\right )+3 b^3 c d e^2}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{12 (2 c d-b e) \left (e (a e-b d)+c d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(a + b*x + c*x^2)^3,x]
[Out]
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Maple [B] time = 0.017, size = 695, normalized size = 4.4 \[{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ( -3\,{\frac{c \left ( ab{e}^{3}-2\,ad{e}^{2}c-{b}^{2}d{e}^{2}+3\,bc{d}^{2}e-2\,{c}^{2}{d}^{3} \right ){x}^{3}}{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}}-{\frac{ \left ( 16\,{a}^{2}{c}^{2}{e}^{3}+a{b}^{2}c{e}^{3}-18\,ab{c}^{2}d{e}^{2}+{b}^{4}{e}^{3}-9\,{b}^{3}cd{e}^{2}+27\,{b}^{2}{c}^{2}{d}^{2}e-18\,b{c}^{3}{d}^{3} \right ){x}^{2}}{2\,c \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }}-{\frac{ \left ( 5\,{a}^{2}bc{e}^{3}+6\,{a}^{2}{c}^{2}d{e}^{2}+a{b}^{3}{e}^{3}-15\,a{b}^{2}cd{e}^{2}+15\,ab{c}^{2}{d}^{2}e-10\,a{c}^{3}{d}^{3}+3\,{b}^{3}c{d}^{2}e-2\,{b}^{2}{c}^{2}{d}^{3} \right ) x}{c \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }}-{\frac{8\,{a}^{3}c{e}^{3}+{a}^{2}{b}^{2}{e}^{3}-18\,{a}^{2}bcd{e}^{2}+24\,{a}^{2}{c}^{2}{d}^{2}e+3\,a{b}^{2}c{d}^{2}e-10\,ab{c}^{2}{d}^{3}+{b}^{3}c{d}^{3}}{2\,c \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }} \right ) }-6\,{\frac{ab{e}^{3}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{ad{e}^{2}c}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+6\,{\frac{{b}^{2}d{e}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-18\,{\frac{bc{d}^{2}e}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{{c}^{2}{d}^{3}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/(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((e*x + d)^3/(c*x^2 + b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236369, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + b*x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 36.2289, size = 1180, normalized size = 7.47 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/(c*x**2+b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.208002, size = 602, normalized size = 3.81 \[ \frac{6 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{4} d^{3} x^{3} - 18 \, b c^{3} d^{2} x^{3} e + 18 \, b c^{3} d^{3} x^{2} + 6 \, b^{2} c^{2} d x^{3} e^{2} + 12 \, a c^{3} d x^{3} e^{2} - 27 \, b^{2} c^{2} d^{2} x^{2} e + 4 \, b^{2} c^{2} d^{3} x + 20 \, a c^{3} d^{3} x - 6 \, a b c^{2} x^{3} e^{3} + 9 \, b^{3} c d x^{2} e^{2} + 18 \, a b c^{2} d x^{2} e^{2} - 6 \, b^{3} c d^{2} x e - 30 \, a b c^{2} d^{2} x e - b^{3} c d^{3} + 10 \, a b c^{2} d^{3} - b^{4} x^{2} e^{3} - a b^{2} c x^{2} e^{3} - 16 \, a^{2} c^{2} x^{2} e^{3} + 30 \, a b^{2} c d x e^{2} - 12 \, a^{2} c^{2} d x e^{2} - 3 \, a b^{2} c d^{2} e - 24 \, a^{2} c^{2} d^{2} e - 2 \, a b^{3} x e^{3} - 10 \, a^{2} b c x e^{3} + 18 \, a^{2} b c d e^{2} - a^{2} b^{2} e^{3} - 8 \, a^{3} c e^{3}}{2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + b*x + a)^3,x, algorithm="giac")
[Out]