3.1189 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^6} \, dx\)

Optimal. Leaf size=79 \[ \frac{4 \left (a+b x+c x^2\right )^{3/2}}{15 d^6 \left (b^2-4 a c\right )^2 (b+2 c x)^3}+\frac{2 \left (a+b x+c x^2\right )^{3/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]

[Out]

(2*(a + b*x + c*x^2)^(3/2))/(5*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^5) + (4*(a + b*x +
c*x^2)^(3/2))/(15*(b^2 - 4*a*c)^2*d^6*(b + 2*c*x)^3)

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Rubi [A]  time = 0.108503, 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 \[ \frac{4 \left (a+b x+c x^2\right )^{3/2}}{15 d^6 \left (b^2-4 a c\right )^2 (b+2 c x)^3}+\frac{2 \left (a+b x+c x^2\right )^{3/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^6,x]

[Out]

(2*(a + b*x + c*x^2)^(3/2))/(5*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^5) + (4*(a + b*x +
c*x^2)^(3/2))/(15*(b^2 - 4*a*c)^2*d^6*(b + 2*c*x)^3)

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Rubi in Sympy [A]  time = 27.3871, size = 75, normalized size = 0.95 \[ \frac{4 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{15 d^{6} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )^{2}} + \frac{2 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{5 d^{6} \left (b + 2 c x\right )^{5} \left (- 4 a c + b^{2}\right )} \]

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)**6,x)

[Out]

4*(a + b*x + c*x**2)**(3/2)/(15*d**6*(b + 2*c*x)**3*(-4*a*c + b**2)**2) + 2*(a +
b*x + c*x**2)**(3/2)/(5*d**6*(b + 2*c*x)**5*(-4*a*c + b**2))

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Mathematica [A]  time = 0.145267, size = 72, normalized size = 0.91 \[ \frac{\sqrt{a+x (b+c x)} \left (\frac{2 (b+2 c x)^4}{\left (b^2-4 a c\right )^2}+\frac{(b+2 c x)^2}{b^2-4 a c}-3\right )}{30 c d^6 (b+2 c x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^6,x]

[Out]

(Sqrt[a + x*(b + c*x)]*(-3 + (b + 2*c*x)^2/(b^2 - 4*a*c) + (2*(b + 2*c*x)^4)/(b^
2 - 4*a*c)^2))/(30*c*d^6*(b + 2*c*x)^5)

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Maple [A]  time = 0.01, size = 70, normalized size = 0.9 \[ -{\frac{-16\,{c}^{2}{x}^{2}-16\,bxc+24\,ac-10\,{b}^{2}}{15\, \left ( 2\,cx+b \right ) ^{5}{d}^{6} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}} \]

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)^6,x)

[Out]

-2/15*(-8*c^2*x^2-8*b*c*x+12*a*c-5*b^2)*(c*x^2+b*x+a)^(3/2)/(2*c*x+b)^5/d^6/(16*
a^2*c^2-8*a*b^2*c+b^4)

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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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.675457, size = 370, normalized size = 4.68 \[ \frac{2 \,{\left (8 \, c^{3} x^{4} + 16 \, b c^{2} x^{3} + 5 \, a b^{2} - 12 \, a^{2} c +{\left (13 \, b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (5 \, b^{3} - 4 \, a b c\right )} x\right )} \sqrt{c x^{2} + b x + a}}{15 \,{\left (32 \,{\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} d^{6} x^{5} + 80 \,{\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} d^{6} x^{4} + 80 \,{\left (b^{6} c^{3} - 8 \, a b^{4} c^{4} + 16 \, a^{2} b^{2} c^{5}\right )} d^{6} x^{3} + 40 \,{\left (b^{7} c^{2} - 8 \, a b^{5} c^{3} + 16 \, a^{2} b^{3} c^{4}\right )} d^{6} x^{2} + 10 \,{\left (b^{8} c - 8 \, a b^{6} c^{2} + 16 \, a^{2} b^{4} c^{3}\right )} d^{6} x +{\left (b^{9} - 8 \, a b^{7} c + 16 \, a^{2} b^{5} c^{2}\right )} d^{6}\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

2/15*(8*c^3*x^4 + 16*b*c^2*x^3 + 5*a*b^2 - 12*a^2*c + (13*b^2*c - 4*a*c^2)*x^2 +
(5*b^3 - 4*a*b*c)*x)*sqrt(c*x^2 + b*x + a)/(32*(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*
c^7)*d^6*x^5 + 80*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^6*x^4 + 80*(b^6*c^3 -
8*a*b^4*c^4 + 16*a^2*b^2*c^5)*d^6*x^3 + 40*(b^7*c^2 - 8*a*b^5*c^3 + 16*a^2*b^3*
c^4)*d^6*x^2 + 10*(b^8*c - 8*a*b^6*c^2 + 16*a^2*b^4*c^3)*d^6*x + (b^9 - 8*a*b^7*
c + 16*a^2*b^5*c^2)*d^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{\sqrt{a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \]

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)**6,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b*
*3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x)/d**6

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GIAC/XCAS [A]  time = 0.652983, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

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)^6,x, algorithm="giac")

[Out]

sage0*x