3.1233 \(\int \frac{(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ 4 \sqrt{c} d^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{2 d^2 (b+2 c x)}{\sqrt{a+b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0824887, antiderivative size = 66, 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 \[ 4 \sqrt{c} d^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{2 d^2 (b+2 c x)}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.9258, size = 63, normalized size = 0.95 \[ 4 \sqrt{c} d^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )} - \frac{2 d^{2} \left (b + 2 c x\right )}{\sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.111289, size = 71, normalized size = 1.08 \[ -\frac{2 d^2 \left (-2 \sqrt{c} \sqrt{a+x (b+c x)} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+b+2 c x\right )}{\sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 72, normalized size = 1.1 \[ -4\,{\frac{c{d}^{2}x}{\sqrt{c{x}^{2}+bx+a}}}-2\,{\frac{{d}^{2}b}{\sqrt{c{x}^{2}+bx+a}}}+4\,{d}^{2}\sqrt{c}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*d^2*c*x/(c*x^2+b*x+a)^(1/2)-2*d^2*b/(c*x^2+b*x+a)^(1/2)+4*d^2*c^(1/2)*ln((1/2
*b+c*x)/c^(1/2)+(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((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 = 0.288253, size = 1, normalized size = 0.02 \[ \left [\frac{2 \,{\left ({\left (c d^{2} x^{2} + b d^{2} x + a d^{2}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) -{\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt{c x^{2} + b x + a}\right )}}{c x^{2} + b x + a}, \frac{2 \,{\left (2 \,{\left (c d^{2} x^{2} + b d^{2} x + a d^{2}\right )} \sqrt{-c} \arctan \left (\frac{2 \, c x + b}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{-c}}\right ) -{\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt{c x^{2} + b x + a}\right )}}{c x^{2} + b x + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{2} \left (\int \frac{b^{2}}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx + \int \frac{4 c^{2} x^{2}}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx + \int \frac{4 b c x}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*(Integral(b**2/(a*sqrt(a + b*x + c*x**2) + b*x*sqrt(a + b*x + c*x**2) + c*x
**2*sqrt(a + b*x + c*x**2)), x) + Integral(4*c**2*x**2/(a*sqrt(a + b*x + c*x**2)
+ b*x*sqrt(a + b*x + c*x**2) + c*x**2*sqrt(a + b*x + c*x**2)), x) + Integral(4*
b*c*x/(a*sqrt(a + b*x + c*x**2) + b*x*sqrt(a + b*x + c*x**2) + c*x**2*sqrt(a + b
*x + c*x**2)), x))

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GIAC/XCAS [A]  time = 0.236541, size = 153, normalized size = 2.32 \[ -4 \, \sqrt{c} d^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right ) - \frac{2 \,{\left (\frac{2 \,{\left (b^{2} c d^{2} - 4 \, a c^{2} d^{2}\right )} x}{b^{2} - 4 \, a c} + \frac{b^{3} d^{2} - 4 \, a b c d^{2}}{b^{2} - 4 \, a c}\right )}}{\sqrt{c x^{2} + b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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