3.1584 \(\int \frac{(b+2 c x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=60 \[ \frac{2 e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c}}-\frac{2 (d+e x)}{\sqrt{a+b x+c x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0894999, antiderivative size = 60, 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 \[ \frac{2 e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c}}-\frac{2 (d+e x)}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.2835, size = 58, normalized size = 0.97 \[ - \frac{2 \left (d + e x\right )}{\sqrt{a + b x + c x^{2}}} + \frac{2 e \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{c}} \]

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)**(3/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.153115, size = 56, normalized size = 0.93 \[ \frac{2 e \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}-\frac{2 (d+e x)}{\sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.01, size = 158, normalized size = 2.6 \[ 2\,{\frac{bd \left ( 2\,cx+b \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-2\,{\frac{d}{\sqrt{c{x}^{2}+bx+a}}}-4\,{\frac{bcdx}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-2\,{\frac{{b}^{2}d}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-2\,{\frac{ex}{\sqrt{c{x}^{2}+bx+a}}}+2\,{\frac{e}{\sqrt{c}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.380417, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, \sqrt{c x^{2} + b x + a}{\left (e x + d\right )} \sqrt{c} -{\left (c e x^{2} + b e x + a e\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{{\left (c x^{2} + b x + a\right )} \sqrt{c}}, -\frac{2 \,{\left (\sqrt{c x^{2} + b x + a}{\left (e x + d\right )} \sqrt{-c} -{\left (c e x^{2} + b e x + a e\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )\right )}}{{\left (c x^{2} + b x + a\right )} \sqrt{-c}}\right ] \]

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)^(3/2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b + 2 c x\right ) \left (d + e x\right )}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral((b + 2*c*x)*(d + e*x)/(a + b*x + c*x**2)**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.288673, size = 136, normalized size = 2.27 \[ -\frac{2 \, e{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{\sqrt{c}} - \frac{2 \,{\left (\frac{{\left (b^{2} e - 4 \, a c e\right )} x}{b^{2} - 4 \, a c} + \frac{b^{2} d - 4 \, a c d}{b^{2} - 4 \, a c}\right )}}{\sqrt{c x^{2} + b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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