3.1186 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^3} \, dx\)

Optimal. Leaf size=91 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{8 c^{3/2} d^3 \sqrt{b^2-4 a c}}-\frac{\sqrt{a+b x+c x^2}}{4 c d^3 (b+2 c x)^2} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(4*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]]/(8*c^(3/2)*Sqrt[b^2 - 4*a*c]*d^3)

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Rubi [A]  time = 0.150021, antiderivative size = 91, 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 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{8 c^{3/2} d^3 \sqrt{b^2-4 a c}}-\frac{\sqrt{a+b x+c x^2}}{4 c d^3 (b+2 c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + b*x + c*x^2]/(4*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]]/(8*c^(3/2)*Sqrt[b^2 - 4*a*c]*d^3)

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Rubi in Sympy [A]  time = 37.3943, size = 83, normalized size = 0.91 \[ - \frac{\sqrt{a + b x + c x^{2}}}{4 c d^{3} \left (b + 2 c x\right )^{2}} + \frac{\operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{8 c^{\frac{3}{2}} d^{3} \sqrt{- 4 a c + b^{2}}} \]

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)**3,x)

[Out]

-sqrt(a + b*x + c*x**2)/(4*c*d**3*(b + 2*c*x)**2) + atan(2*sqrt(c)*sqrt(a + b*x
+ c*x**2)/sqrt(-4*a*c + b**2))/(8*c**(3/2)*d**3*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.230418, size = 145, normalized size = 1.59 \[ \frac{-(b+2 c x)^2 \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )-2 \sqrt{c} \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+(b+2 c x)^2 \log (b+2 c x)}{8 c^{3/2} d^3 \sqrt{4 a c-b^2} (b+2 c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)] + (b + 2*c*x)^2*Log[b + 2*c
*x] - (b + 2*c*x)^2*Log[-(b^2*Sqrt[c]) + 4*a*c^(3/2) + 2*c*Sqrt[-b^2 + 4*a*c]*Sq
rt[a + x*(b + c*x)]])/(8*c^(3/2)*Sqrt[-b^2 + 4*a*c]*d^3*(b + 2*c*x)^2)

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Maple [B]  time = 0.016, size = 340, normalized size = 3.7 \[ -{\frac{1}{4\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}}+{\frac{1}{8\,c{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{a}{2\,c{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{{b}^{2}}{8\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^3,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)^(3/2)
+1/8/d^3/c/(4*a*c-b^2)*(4*(x+1/2*b/c)^2*c+(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))*a+1/8/d^3/c^2/(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))*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254662, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (-\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 4 \, \sqrt{-b^{2} c + 4 \, a c^{2}} \sqrt{c x^{2} + b x + a}}{16 \,{\left (4 \, c^{3} d^{3} x^{2} + 4 \, b c^{2} d^{3} x + b^{2} c d^{3}\right )} \sqrt{-b^{2} c + 4 \, a c^{2}}}, -\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \arctan \left (\frac{\sqrt{b^{2} c - 4 \, a c^{2}}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) + 2 \, \sqrt{b^{2} c - 4 \, a c^{2}} \sqrt{c x^{2} + b x + a}}{8 \,{\left (4 \, c^{3} d^{3} x^{2} + 4 \, b c^{2} d^{3} x + b^{2} c d^{3}\right )} \sqrt{b^{2} c - 4 \, a c^{2}}}\right ] \]

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)^3,x, algorithm="fricas")

[Out]

[1/16*((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*c^3*d^3*x^2
+ 4*b*c^2*d^3*x + b^2*c*d^3)*sqrt(-b^2*c + 4*a*c^2)), -1/8*((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*c^3*d^3*x^2 + 4*b*c^2*d^3*x + 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. \[ \frac{\int \frac{\sqrt{a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx}{d^{3}} \]

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)**3,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(b**3 + 6*b**2*c*x + 12*b*c**2*x**2 + 8*c**3*x**
3), 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(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError