3.2324 \(\int (d+e x) \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=115 \[ -\frac{\left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 c} \]

[Out]

((2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8*c^2) + (e*(a + b*x + c*x^2)
^(3/2))/(3*c) - ((b^2 - 4*a*c)*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])])/(16*c^(5/2))

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Rubi [A]  time = 0.125053, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

((2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8*c^2) + (e*(a + b*x + c*x^2)
^(3/2))/(3*c) - ((b^2 - 4*a*c)*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])])/(16*c^(5/2))

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Rubi in Sympy [A]  time = 12.8335, size = 104, normalized size = 0.9 \[ \frac{e \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{3 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{8 c^{2}} + \frac{\left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(a + b*x + c*x**2)**(3/2)/(3*c) - (b + 2*c*x)*(b*e - 2*c*d)*sqrt(a + b*x + c*x
**2)/(8*c**2) + (-4*a*c + b**2)*(b*e - 2*c*d)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(
a + b*x + c*x**2)))/(16*c**(5/2))

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Mathematica [A]  time = 0.211466, size = 112, normalized size = 0.97 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 c (2 a e+c x (3 d+2 e x))-3 b^2 e+2 b c (3 d+e x)\right )+3 \left (b^2-4 a c\right ) (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{48 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3*b^2*e + 2*b*c*(3*d + e*x) + 4*c*(2*a*e + c*
x*(3*d + 2*e*x))) + 3*(b^2 - 4*a*c)*(-2*c*d + b*e)*Log[b + 2*c*x + 2*Sqrt[c]*Sqr
t[a + x*(b + c*x)]])/(48*c^(5/2))

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Maple [B]  time = 0.009, size = 229, normalized size = 2. \[{\frac{dx}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{bd}{4\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{ad}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{b}^{2}d}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{e}{3\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{bex}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{b}^{2}e}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{bea}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{e{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*(c*x^2+b*x+a)^(1/2)*x+1/4*d/c*(c*x^2+b*x+a)^(1/2)*b+1/2*d/c^(1/2)*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/8*d/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))*b^2+1/3*e*(c*x^2+b*x+a)^(3/2)/c-1/4*e*b/c*(c*x^2+b*x+a)^(1/2)*x-
1/8*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)-1/4*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))*a+1/16*e*b^3/c^(5/2)*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(sqrt(c*x^2 + b*x + a)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239606, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, c^{2} e x^{2} + 6 \, b c d -{\left (3 \, b^{2} - 8 \, a c\right )} e + 2 \,{\left (6 \, c^{2} d + b c e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d -{\left (b^{3} - 4 \, a b c\right )} e\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{96 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (8 \, c^{2} e x^{2} + 6 \, b c d -{\left (3 \, b^{2} - 8 \, a c\right )} e + 2 \,{\left (6 \, c^{2} d + b c e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 3 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d -{\left (b^{3} - 4 \, a b c\right )} e\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{48 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*c^2*e*x^2 + 6*b*c*d - (3*b^2 - 8*a*c)*e + 2*(6*c^2*d + b*c*e)*x)*sqr
t(c*x^2 + b*x + a)*sqrt(c) + 3*(2*(b^2*c - 4*a*c^2)*d - (b^3 - 4*a*b*c)*e)*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)*sqr
t(c)))/c^(5/2), 1/48*(2*(8*c^2*e*x^2 + 6*b*c*d - (3*b^2 - 8*a*c)*e + 2*(6*c^2*d
+ b*c*e)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*(2*(b^2*c - 4*a*c^2)*d - (b^3 - 4
*a*b*c)*e)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)
*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221383, size = 174, normalized size = 1.51 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \, x e + \frac{6 \, c^{2} d + b c e}{c^{2}}\right )} x + \frac{6 \, b c d - 3 \, b^{2} e + 8 \, a c e}{c^{2}}\right )} + \frac{{\left (2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(c*x^2 + b*x + a)*(2*(4*x*e + (6*c^2*d + b*c*e)/c^2)*x + (6*b*c*d - 3*b
^2*e + 8*a*c*e)/c^2) + 1/16*(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e)*ln(abs(-
2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2)