Optimal. Leaf size=59 \[ \frac{4}{3} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}+\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2} \]
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Rubi [A] time = 0.0800732, antiderivative size = 59, 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 \[ \frac{4}{3} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}+\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^3/Sqrt[a + b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 17.5402, size = 56, normalized size = 0.95 \[ \frac{2 d^{3} \left (b + 2 c x\right )^{2} \sqrt{a + b x + c x^{2}}}{3} + \frac{4 d^{3} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{3} \]
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)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0689426, size = 43, normalized size = 0.73 \[ \frac{2}{3} d^3 \sqrt{a+x (b+c x)} \left (4 c \left (c x^2-2 a\right )+3 b^2+4 b c x\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^3/Sqrt[a + b*x + c*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 41, normalized size = 0.7 \[ -{\frac{2\,{d}^{3} \left ( -4\,{c}^{2}{x}^{2}-4\,bxc+8\,ac-3\,{b}^{2} \right ) }{3}\sqrt{c{x}^{2}+bx+a}} \]
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)^(1/2),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/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.266223, size = 65, normalized size = 1.1 \[ \frac{2}{3} \,{\left (4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (3 \, b^{2} - 8 \, a c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3/sqrt(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.711822, size = 97, normalized size = 1.64 \[ - \frac{16 a c d^{3} \sqrt{a + b x + c x^{2}}}{3} + 2 b^{2} d^{3} \sqrt{a + b x + c x^{2}} + \frac{8 b c d^{3} x \sqrt{a + b x + c x^{2}}}{3} + \frac{8 c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{3} \]
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)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.228044, size = 78, normalized size = 1.32 \[ \frac{2}{3} \, \sqrt{c x^{2} + b x + a}{\left (4 \,{\left (c^{2} d^{3} x + b c d^{3}\right )} x + \frac{3 \, b^{2} c^{2} d^{3} - 8 \, a c^{3} d^{3}}{c^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3/sqrt(c*x^2 + b*x + a),x, algorithm="giac")
[Out]