3.1575 \(\int \frac{(b+2 c x) (d+e x)}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{3/2}}+\frac{\sqrt{a+b x+c x^2} (-b e+4 c d+2 c e x)}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.144771, antiderivative size = 84, 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 \[ \frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{3/2}}+\frac{\sqrt{a+b x+c x^2} (-b e+4 c d+2 c e x)}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.3203, size = 76, normalized size = 0.9 \[ - \frac{\sqrt{a + b x + c x^{2}} \left (b e - 4 c d - 2 c e x\right )}{2 c} + \frac{e \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{4 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.215914, size = 80, normalized size = 0.95 \[ \frac{e \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{4 c^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-b e+4 c d+2 c e x)}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 117, normalized size = 1.4 \[ -{\frac{be}{2\,c}\sqrt{c{x}^{2}+bx+a}}+2\,d\sqrt{c{x}^{2}+bx+a}+{\frac{{b}^{2}e}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+ex\sqrt{c{x}^{2}+bx+a}-{ae\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.380958, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 4 \, a c\right )} e \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) - 4 \,{\left (2 \, c e x + 4 \, c d - b e\right )} \sqrt{c x^{2} + b x + a} \sqrt{c}}{8 \, c^{\frac{3}{2}}}, \frac{{\left (b^{2} - 4 \, a c\right )} e \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) + 2 \,{\left (2 \, c e x + 4 \, c d - b e\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c}}{4 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.289002, size = 113, normalized size = 1.35 \[ \frac{1}{2} \, \sqrt{c x^{2} + b x + a}{\left (2 \, x e + \frac{4 \, c d - b e}{c}\right )} - \frac{{\left (b^{2} e - 4 \, a c e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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