3.1229 \(\int \frac{1}{(b d+2 c d x)^3 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{\sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}+\frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{c} d^3 \left (b^2-4 a c\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.157367, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}+\frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{c} d^3 \left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.7414, size = 88, normalized size = 0.93 \[ \frac{\sqrt{a + b x + c x^{2}}}{d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )} + \frac{\operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 \sqrt{c} d^{3} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a + b*x + c*x**2)/(d**3*(b + 2*c*x)**2*(-4*a*c + b**2)) + atan(2*sqrt(c)*sq
rt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2))/(2*sqrt(c)*d**3*(-4*a*c + b**2)**(3/2)
)

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Mathematica [A]  time = 0.252259, size = 139, normalized size = 1.46 \[ \frac{\frac{\log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\sqrt{c} \left (4 a c-b^2\right )^{3/2}}+\frac{2 \sqrt{a+x (b+c x)}}{\left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{\log (b+2 c x)}{\sqrt{c} \left (4 a c-b^2\right )^{3/2}}}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.016, size = 174, normalized size = 1.8 \[ -{\frac{1}{4\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}}+{\frac{1}{2\,c{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/d^3/c^2/(4*a*c-b^2)/(x+1/2*b/c)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)
+1/2/d^3/c/(4*a*c-b^2)/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b
^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29157, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (-\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} - 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 4 \, \sqrt{-b^{2} c + 4 \, a c^{2}} \sqrt{c x^{2} + b x + a}}{4 \,{\left (4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{3} x^{2} + 4 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{3} x +{\left (b^{4} - 4 \, a b^{2} c\right )} d^{3}\right )} \sqrt{-b^{2} c + 4 \, a c^{2}}}, -\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \arctan \left (\frac{\sqrt{b^{2} c - 4 \, a c^{2}}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) - 2 \, \sqrt{b^{2} c - 4 \, a c^{2}} \sqrt{c x^{2} + b x + a}}{2 \,{\left (4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{3} x^{2} + 4 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{3} x +{\left (b^{4} - 4 \, a b^{2} c\right )} d^{3}\right )} \sqrt{b^{2} c - 4 \, a c^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*((4*c^2*x^2 + 4*b*c*x + b^2)*log(-((4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c)*sqr
t(-b^2*c + 4*a*c^2) - 4*(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a))/(4*c^2*x^2 + 4*
b*c*x + b^2)) - 4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a))/((4*(b^2*c^2 - 4
*a*c^3)*d^3*x^2 + 4*(b^3*c - 4*a*b*c^2)*d^3*x + (b^4 - 4*a*b^2*c)*d^3)*sqrt(-b^2
*c + 4*a*c^2)), -1/2*((4*c^2*x^2 + 4*b*c*x + b^2)*arctan(1/2*sqrt(b^2*c - 4*a*c^
2)/(sqrt(c*x^2 + b*x + a)*c)) - 2*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a))/(
(4*(b^2*c^2 - 4*a*c^3)*d^3*x^2 + 4*(b^3*c - 4*a*b*c^2)*d^3*x + (b^4 - 4*a*b^2*c)
*d^3)*sqrt(b^2*c - 4*a*c^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{b^{3} \sqrt{a + b x + c x^{2}} + 6 b^{2} c x \sqrt{a + b x + c x^{2}} + 12 b c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 8 c^{3} x^{3} \sqrt{a + b x + c x^{2}}}\, dx}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(b**3*sqrt(a + b*x + c*x**2) + 6*b**2*c*x*sqrt(a + b*x + c*x**2) + 12
*b*c**2*x**2*sqrt(a + b*x + c*x**2) + 8*c**3*x**3*sqrt(a + b*x + c*x**2)), x)/d*
*3

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError