3.1187 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^4} \, dx\)

Optimal. Leaf size=39 \[ \frac{2 \left (a+b x+c x^2\right )^{3/2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]

[Out]

(2*(a + b*x + c*x^2)^(3/2))/(3*(b^2 - 4*a*c)*d^4*(b + 2*c*x)^3)

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Rubi [A]  time = 0.0547347, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{2 \left (a+b x+c x^2\right )^{3/2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^4,x]

[Out]

(2*(a + b*x + c*x^2)^(3/2))/(3*(b^2 - 4*a*c)*d^4*(b + 2*c*x)^3)

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Rubi in Sympy [A]  time = 13.9145, size = 36, normalized size = 0.92 \[ \frac{2 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{3 d^{4} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**4,x)

[Out]

2*(a + b*x + c*x**2)**(3/2)/(3*d**4*(b + 2*c*x)**3*(-4*a*c + b**2))

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Mathematica [A]  time = 0.0579444, size = 38, normalized size = 0.97 \[ \frac{2 (a+x (b+c x))^{3/2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^4,x]

[Out]

(2*(a + x*(b + c*x))^(3/2))/(3*(b^2 - 4*a*c)*d^4*(b + 2*c*x)^3)

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Maple [A]  time = 0.007, size = 38, normalized size = 1. \[ -{\frac{2}{3\, \left ( 2\,cx+b \right ) ^{3}{d}^{4} \left ( 4\,ac-{b}^{2} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^4,x)

[Out]

-2/3*(c*x^2+b*x+a)^(3/2)/(2*c*x+b)^3/d^4/(4*a*c-b^2)

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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(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290701, size = 132, normalized size = 3.38 \[ \frac{2 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{3 \,{\left (8 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{3} + 12 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x^{2} + 6 \,{\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d^{4} x +{\left (b^{5} - 4 \, a b^{3} c\right )} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^4,x, algorithm="fricas")

[Out]

2/3*(c*x^2 + b*x + a)^(3/2)/(8*(b^2*c^3 - 4*a*c^4)*d^4*x^3 + 12*(b^3*c^2 - 4*a*b
*c^3)*d^4*x^2 + 6*(b^4*c - 4*a*b^2*c^2)*d^4*x + (b^5 - 4*a*b^3*c)*d^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{\sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**4,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c*
*3*x**3 + 16*c**4*x**4), x)/d**4

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GIAC/XCAS [A]  time = 0.67345, size = 4, normalized size = 0.1 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^4,x, algorithm="giac")

[Out]

sage0*x