3.2364 \(\int \frac{(d+e x)^2}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=127 \[ \frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}+\frac{3 e \sqrt{a+b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e (d+e x) \sqrt{a+b x+c x^2}}{2 c} \]

[Out]

(3*e*(2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(4*c^2) + (e*(d + e*x)*Sqrt[a + b*x +
c*x^2])/(2*c) + ((8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(b + 2*c*
x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2))

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Rubi [A]  time = 0.271489, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}+\frac{3 e \sqrt{a+b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e (d+e x) \sqrt{a+b x+c x^2}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

(3*e*(2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(4*c^2) + (e*(d + e*x)*Sqrt[a + b*x +
c*x^2])/(2*c) + ((8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(b + 2*c*
x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2))

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Rubi in Sympy [A]  time = 33.1775, size = 122, normalized size = 0.96 \[ \frac{e \left (d + e x\right ) \sqrt{a + b x + c x^{2}}}{2 c} - \frac{3 e \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{4 c^{2}} + \frac{\left (- 4 a c e^{2} + 3 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(d + e*x)*sqrt(a + b*x + c*x**2)/(2*c) - 3*e*(b*e - 2*c*d)*sqrt(a + b*x + c*x*
*2)/(4*c**2) + (-4*a*c*e**2 + 3*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)*atanh((b +
2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(8*c**(5/2))

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Mathematica [A]  time = 0.155086, size = 102, normalized size = 0.8 \[ \frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+2 \sqrt{c} e \sqrt{a+x (b+c x)} (-3 b e+8 c d+2 c e x)}{8 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*e*(8*c*d - 3*b*e + 2*c*e*x)*Sqrt[a + x*(b + c*x)] + (8*c^2*d^2 + 3*b^
2*e^2 - 4*c*e*(2*b*d + a*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(
8*c^(5/2))

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Maple [A]  time = 0.012, size = 198, normalized size = 1.6 \[{{d}^{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,b{e}^{2}}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}{e}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{a{e}^{2}}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+2\,{\frac{de\sqrt{c{x}^{2}+bx+a}}{c}}-{bde\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+1/2*e^2*x/c*(c*x^2+b*x+a
)^(1/2)-3/4*e^2*b/c^2*(c*x^2+b*x+a)^(1/2)+3/8*e^2*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))-1/2*e^2*a/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))+2*d*e/c*(c*x^2+b*x+a)^(1/2)-d*e*b/c^(3/2)*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((e*x + d)^2/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.315389, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, c e^{2} x + 8 \, c d e - 3 \, b e^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} -{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (3 \, b^{2} - 4 \, a c\right )} e^{2}\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{16 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (2 \, c e^{2} x + 8 \, c d e - 3 \, b e^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} +{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (3 \, b^{2} - 4 \, a c\right )} e^{2}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{8 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(2*c*e^2*x + 8*c*d*e - 3*b*e^2)*sqrt(c*x^2 + b*x + a)*sqrt(c) - (8*c^2*
d^2 - 8*b*c*d*e + (3*b^2 - 4*a*c)*e^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)))/c^(5/2), 1/8*(2*(2*c*e^2*x +
8*c*d*e - 3*b*e^2)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + (8*c^2*d^2 - 8*b*c*d*e + (3*
b^2 - 4*a*c)*e^2)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(s
qrt(-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226013, size = 142, normalized size = 1.12 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x + a}{\left (\frac{2 \, x e^{2}}{c} + \frac{8 \, c d e - 3 \, b e^{2}}{c^{2}}\right )} - \frac{{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2} - 4 \, a c e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(c*x^2 + b*x + a)*(2*x*e^2/c + (8*c*d*e - 3*b*e^2)/c^2) - 1/8*(8*c^2*d^2
- 8*b*c*d*e + 3*b^2*e^2 - 4*a*c*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*sqrt(c) - b))/c^(5/2)