Optimal. Leaf size=48 \[ 16 c d^3 \sqrt{a+b x+c x^2}-\frac{2 d^3 (b+2 c x)^2}{\sqrt{a+b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.0700331, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ 16 c d^3 \sqrt{a+b x+c x^2}-\frac{2 d^3 (b+2 c x)^2}{\sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 18.0924, size = 46, normalized size = 0.96 \[ 16 c d^{3} \sqrt{a + b x + c x^{2}} - \frac{2 d^{3} \left (b + 2 c x\right )^{2}}{\sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**3/(c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0658147, size = 40, normalized size = 0.83 \[ \frac{d^3 \left (8 c \left (2 a+c x^2\right )-2 b^2+8 b c x\right )}{\sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 41, normalized size = 0.9 \[ 2\,{\frac{{d}^{3} \left ( 4\,{c}^{2}{x}^{2}+4\,bxc+8\,ac-{b}^{2} \right ) }{\sqrt{c{x}^{2}+bx+a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^3/(c*x^2+b*x+a)^(3/2),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)^3/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.286586, size = 63, normalized size = 1.31 \[ \frac{2 \,{\left (4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x -{\left (b^{2} - 8 \, a c\right )} d^{3}\right )}}{\sqrt{c x^{2} + b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3/(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.8826, size = 92, normalized size = 1.92 \[ \frac{16 a c d^{3}}{\sqrt{a + b x + c x^{2}}} - \frac{2 b^{2} d^{3}}{\sqrt{a + b x + c x^{2}}} + \frac{8 b c d^{3} x}{\sqrt{a + b x + c x^{2}}} + \frac{8 c^{2} d^{3} x^{2}}{\sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**3/(c*x**2+b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.229593, size = 188, normalized size = 3.92 \[ \frac{2 \,{\left (4 \,{\left (\frac{{\left (b^{2} c^{3} d^{3} - 4 \, a c^{4} d^{3}\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac{b^{3} c^{2} d^{3} - 4 \, a b c^{3} d^{3}}{b^{2} c - 4 \, a c^{2}}\right )} x - \frac{b^{4} c d^{3} - 12 \, a b^{2} c^{2} d^{3} + 32 \, a^{2} c^{3} d^{3}}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt{c x^{2} + b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")
[Out]