3.1225 \(\int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=75 \[ \frac{d^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c}}+d^2 (b+2 c x) \sqrt{a+b x+c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0924616, antiderivative size = 75, 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 \[ \frac{d^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c}}+d^2 (b+2 c x) \sqrt{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.4236, size = 70, normalized size = 0.93 \[ d^{2} \left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} + \frac{d^{2} \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 \sqrt{c}} \]

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)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.168582, size = 69, normalized size = 0.92 \[ d^2 \left (\frac{\left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{2 \sqrt{c}}+(b+2 c x) \sqrt{a+x (b+c x)}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 108, normalized size = 1.4 \[{\frac{{b}^{2}{d}^{2}}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+2\,{d}^{2}cx\sqrt{c{x}^{2}+bx+a}+{d}^{2}b\sqrt{c{x}^{2}+bx+a}-2\,{d}^{2}\sqrt{c}a\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) \]

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)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272406, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 4 \, a c\right )} d^{2} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) - 4 \,{\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{c}}{4 \, \sqrt{c}}, \frac{{\left (b^{2} - 4 \, a c\right )} d^{2} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) + 2 \,{\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c}}{2 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.239237, size = 105, normalized size = 1.4 \[{\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt{c x^{2} + b x + a} - \frac{{\left (b^{2} d^{2} - 4 \, a c d^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2 \, \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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