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

Optimal. Leaf size=111 \[ \frac{2 \left (c \left (2 a e-b \left (\frac{a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.165008, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b (a f+c d)+2 a c e\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.4816, size = 105, normalized size = 0.95 \[ - \frac{2 \left (a b f - 2 a c e + b c d + x \left (- 2 a c f + b^{2} f - b c e + 2 c^{2} d\right )\right )}{c \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} + \frac{f \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.323699, size = 113, normalized size = 1.02 \[ \frac{\frac{2 \sqrt{c} \left (a b f-2 a c (e+f x)+b^2 f x+b c (d-e x)+2 c^2 d x\right )}{\sqrt{a+x (b+c x)}}-f \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{3/2} \left (4 a c-b^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.008, size = 249, normalized size = 2.2 \[ 2\,{\frac{d \left ( 2\,cx+b \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{e}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-2\,{\frac{bex}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{{b}^{2}e}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{fx}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{bf}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{b}^{2}fx}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{b}^{3}f}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{f\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.575819, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (b c d - 2 \, a c e + a b f +{\left (2 \, c^{2} d - b c e +{\left (b^{2} - 2 \, a c\right )} f\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} f x^{2} +{\left (b^{3} - 4 \, a b c\right )} f x +{\left (a b^{2} - 4 \, a^{2} c\right )} f\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{c}}, -\frac{2 \,{\left (b c d - 2 \, a c e + a b f +{\left (2 \, c^{2} d - b c e +{\left (b^{2} - 2 \, a c\right )} f\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} f x^{2} +{\left (b^{3} - 4 \, a b c\right )} f x +{\left (a b^{2} - 4 \, a^{2} c\right )} f\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.284782, size = 165, normalized size = 1.49 \[ -\frac{2 \,{\left (\frac{{\left (2 \, c^{2} d + b^{2} f - 2 \, a c f - b c e\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac{b c d + a b f - 2 \, a c e}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt{c x^{2} + b x + a}} - \frac{f{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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