3.1235 \(\int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.151581, antiderivative size = 86, 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 \[ -\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.8107, size = 80, normalized size = 0.93 \[ - \frac{4 \sqrt{c} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{d \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{2}{d \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.426581, size = 130, normalized size = 1.51 \[ \frac{2 \left (-\frac{2 \sqrt{c} \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\left (4 a c-b^2\right )^{3/2}}-\frac{1}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}}+\frac{2 \sqrt{c} \log (b+2 c x)}{\left (4 a c-b^2\right )^{3/2}}\right )}{d} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(1/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)])) + (2*Sqrt[c]*Log[b + 2*c*x])/(-b^
2 + 4*a*c)^(3/2) - (2*Sqrt[c]*Log[-(b^2*Sqrt[c]) + 4*a*c^(3/2) + 2*c*Sqrt[-b^2 +
4*a*c]*Sqrt[a + x*(b + c*x)]])/(-b^2 + 4*a*c)^(3/2)))/d

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Maple [B]  time = 0.014, size = 158, normalized size = 1.8 \[ 2\,{\frac{1}{d \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+1/4\,{\frac{4\,ac-{b}^{2}}{c}}}}}}-4\,{\frac{1}{d \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({1 \left ( 1/2\,{\frac{4\,ac-{b}^{2}}{c}}+1/2\,\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}} \right ) \left ( x+1/2\,{\frac{b}{c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285707, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left ({\left (c x^{2} + b x + a\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + \sqrt{c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} +{\left (b^{3} - 4 \, a b c\right )} d x +{\left (a b^{2} - 4 \, a^{2} c\right )} d}, \frac{2 \,{\left (2 \,{\left (c x^{2} + b x + a\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}} \arctan \left (\frac{1}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{\frac{c}{b^{2} - 4 \, a c}}}\right ) - \sqrt{c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} +{\left (b^{3} - 4 \, a b c\right )} d x +{\left (a b^{2} - 4 \, a^{2} c\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: TypeError