3.1236 \(\int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{16 c \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)}-\frac{2}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}} \]

[Out]

-2/((b^2 - 4*a*c)*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2]) - (16*c*Sqrt[a + b*x +
c*x^2])/((b^2 - 4*a*c)^2*d^2*(b + 2*c*x))

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Rubi [A]  time = 0.110402, antiderivative size = 76, 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 \[ -\frac{16 c \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)}-\frac{2}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(3/2)),x]

[Out]

-2/((b^2 - 4*a*c)*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2]) - (16*c*Sqrt[a + b*x +
c*x^2])/((b^2 - 4*a*c)^2*d^2*(b + 2*c*x))

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Rubi in Sympy [A]  time = 27.0709, size = 71, normalized size = 0.93 \[ - \frac{16 c \sqrt{a + b x + c x^{2}}}{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2}} - \frac{2}{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a)**(3/2),x)

[Out]

-16*c*sqrt(a + b*x + c*x**2)/(d**2*(b + 2*c*x)*(-4*a*c + b**2)**2) - 2/(d**2*(b
+ 2*c*x)*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2))

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Mathematica [A]  time = 0.0917567, size = 56, normalized size = 0.74 \[ -\frac{2 \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b^2 + 8*b*c*x + 4*c*(a + 2*c*x^2)))/((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)*Sqrt[a
+ x*(b + c*x)])

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Maple [A]  time = 0.009, size = 68, normalized size = 0.9 \[ -2\,{\frac{8\,{c}^{2}{x}^{2}+8\,bxc+4\,ac+{b}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ){d}^{2} \left ( 2\,cx+b \right ) \sqrt{c{x}^{2}+bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a)^(3/2),x)

[Out]

-2*(8*c^2*x^2+8*b*c*x+4*a*c+b^2)/(2*c*x+b)/d^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(c*x^2
+b*x+a)^(1/2)

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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(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.297302, size = 209, normalized size = 2.75 \[ -\frac{2 \,{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c x^{2} + b x + a}}{2 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} x^{3} + 3 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{2} x^{2} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d^{2} x +{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^(3/2)),x, algorithm="fricas")

[Out]

-2*(8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c*x^2 + b*x + a)/(2*(b^4*c^2 - 8*a*b
^2*c^3 + 16*a^2*c^4)*d^2*x^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2*x^2 +
(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2*x + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{a b^{2} \sqrt{a + b x + c x^{2}} + 4 a b c x \sqrt{a + b x + c x^{2}} + 4 a c^{2} x^{2} \sqrt{a + b x + c x^{2}} + b^{3} x \sqrt{a + b x + c x^{2}} + 5 b^{2} c x^{2} \sqrt{a + b x + c x^{2}} + 8 b c^{2} x^{3} \sqrt{a + b x + c x^{2}} + 4 c^{3} x^{4} \sqrt{a + b x + c x^{2}}}\, dx}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(1/(a*b**2*sqrt(a + b*x + c*x**2) + 4*a*b*c*x*sqrt(a + b*x + c*x**2) + 4
*a*c**2*x**2*sqrt(a + b*x + c*x**2) + b**3*x*sqrt(a + b*x + c*x**2) + 5*b**2*c*x
**2*sqrt(a + b*x + c*x**2) + 8*b*c**2*x**3*sqrt(a + b*x + c*x**2) + 4*c**3*x**4*
sqrt(a + b*x + c*x**2)), x)/d**2

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{2}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((2*c*d*x + b*d)^2*(c*x^2 + b*x + a)^(3/2)), x)