Optimal. Leaf size=79 \[ \frac{4 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac{2 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]
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Rubi [A] time = 0.110892, antiderivative size = 79, 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 \[ \frac{4 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac{2 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]
Antiderivative was successfully verified.
[In] Int[1/((b*d + 2*c*d*x)^4*Sqrt[a + b*x + c*x^2]),x]
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Rubi in Sympy [A] time = 27.5203, size = 73, normalized size = 0.92 \[ \frac{4 \sqrt{a + b x + c x^{2}}}{3 d^{4} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2}} + \frac{2 \sqrt{a + b x + c x^{2}}}{3 d^{4} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(1/2),x)
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Mathematica [A] time = 0.0851347, size = 60, normalized size = 0.76 \[ \frac{2 \sqrt{a+x (b+c x)} \left (-4 c \left (a-2 c x^2\right )+3 b^2+8 b c x\right )}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((b*d + 2*c*d*x)^4*Sqrt[a + b*x + c*x^2]),x]
[Out]
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Maple [A] time = 0.01, size = 70, normalized size = 0.9 \[ -{\frac{-16\,{c}^{2}{x}^{2}-16\,bxc+8\,ac-6\,{b}^{2}}{3\, \left ( 2\,cx+b \right ) ^{3}{d}^{4} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }\sqrt{c{x}^{2}+bx+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*c*d*x+b*d)^4/(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(1/((2*c*d*x + b*d)^4*sqrt(c*x^2 + b*x + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.324849, size = 223, normalized size = 2.82 \[ \frac{2 \,{\left (8 \, c^{2} x^{2} + 8 \, b c x + 3 \, b^{2} - 4 \, a c\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (8 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 12 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 6 \,{\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{4} x +{\left (b^{7} - 8 \, a b^{5} c + 16 \, a^{2} b^{3} c^{2}\right )} d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^4*sqrt(c*x^2 + b*x + a)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{b^{4} \sqrt{a + b x + c x^{2}} + 8 b^{3} c x \sqrt{a + b x + c x^{2}} + 24 b^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 32 b c^{3} x^{3} \sqrt{a + b x + c x^{2}} + 16 c^{4} x^{4} \sqrt{a + b x + c x^{2}}}\, dx}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.641744, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^4*sqrt(c*x^2 + b*x + a)),x, algorithm="giac")
[Out]