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

Optimal. Leaf size=309 \[ \frac{2 \left (-x \left (c^2 \left (2 a^2 f^2+6 a b e f+b^2 \left (2 d f+e^2\right )\right )-2 b^2 c f (2 a f+b e)-2 c^3 \left (a \left (2 d f+e^2\right )+b d e\right )+b^4 f^2+2 c^4 d^2\right )-b c \left (-3 a^2 f^2+a c \left (2 d f+e^2\right )+c^2 d^2\right )-a b^3 f^2+2 a b^2 c e f+4 a c^2 e (c d-a f)\right )}{c^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-12 c f (a f+2 b e)+15 b^2 f^2+8 c^2 \left (2 d f+e^2\right )\right )}{8 c^{7/2}}+\frac{f \sqrt{a+b x+c x^2} (8 c e-7 b f)}{4 c^3}+\frac{f^2 x \sqrt{a+b x+c x^2}}{2 c^2} \]

[Out]

(2*(2*a*b^2*c*e*f - a*b^3*f^2 + 4*a*c^2*e*(c*d - a*f) - b*c*(c^2*d^2 - 3*a^2*f^2
 + a*c*(e^2 + 2*d*f)) - (2*c^4*d^2 + b^4*f^2 - 2*b^2*c*f*(b*e + 2*a*f) - 2*c^3*(
b*d*e + a*(e^2 + 2*d*f)) + c^2*(6*a*b*e*f + 2*a^2*f^2 + b^2*(e^2 + 2*d*f)))*x))/
(c^3*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (f*(8*c*e - 7*b*f)*Sqrt[a + b*x + c*
x^2])/(4*c^3) + (f^2*x*Sqrt[a + b*x + c*x^2])/(2*c^2) + ((15*b^2*f^2 - 12*c*f*(2
*b*e + a*f) + 8*c^2*(e^2 + 2*d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x +
 c*x^2])])/(8*c^(7/2))

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Rubi [A]  time = 0.870405, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{2 \left (-x \left (c^2 \left (2 a^2 f^2+6 a b e f+b^2 \left (2 d f+e^2\right )\right )-2 b^2 c f (2 a f+b e)-2 c^3 \left (a \left (2 d f+e^2\right )+b d e\right )+b^4 f^2+2 c^4 d^2\right )-b c \left (-3 a^2 f^2+a c \left (2 d f+e^2\right )+c^2 d^2\right )-a b^3 f^2+2 a b^2 c e f+4 a c^2 e (c d-a f)\right )}{c^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-12 c f (a f+2 b e)+15 b^2 f^2+8 c^2 \left (2 d f+e^2\right )\right )}{8 c^{7/2}}+\frac{f \sqrt{a+b x+c x^2} (8 c e-7 b f)}{4 c^3}+\frac{f^2 x \sqrt{a+b x+c x^2}}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(2*a*b^2*c*e*f - a*b^3*f^2 + 4*a*c^2*e*(c*d - a*f) - b*c*(c^2*d^2 - 3*a^2*f^2
 + a*c*(e^2 + 2*d*f)) - (2*c^4*d^2 + b^4*f^2 - 2*b^2*c*f*(b*e + 2*a*f) - 2*c^3*(
b*d*e + a*(e^2 + 2*d*f)) + c^2*(6*a*b*e*f + 2*a^2*f^2 + b^2*(e^2 + 2*d*f)))*x))/
(c^3*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (f*(8*c*e - 7*b*f)*Sqrt[a + b*x + c*
x^2])/(4*c^3) + (f^2*x*Sqrt[a + b*x + c*x^2])/(2*c^2) + ((15*b^2*f^2 - 12*c*f*(2
*b*e + a*f) + 8*c^2*(e^2 + 2*d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x +
 c*x^2])])/(8*c^(7/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.644548, size = 288, normalized size = 0.93 \[ \frac{4 b c \left (-13 a^2 f^2+a c \left (4 d f+2 e^2+20 e f x-5 f^2 x^2\right )+2 c^2 d (d-2 e x)\right )+8 c^2 \left (a^2 f (8 e+3 f x)+a c \left (x \left (-2 e^2+4 e f x+f^2 x^2\right )-4 d (e+f x)\right )+2 c^2 d^2 x\right )+b^3 f (15 a f+c x (5 f x-24 e))-2 b^2 c \left (a f (12 e+31 f x)+c x \left (-8 d f-4 e^2+4 e f x+f^2 x^2\right )\right )+15 b^4 f^2 x}{4 c^3 \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)}}+\frac{\log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (-12 c f (a f+2 b e)+15 b^2 f^2+8 c^2 \left (2 d f+e^2\right )\right )}{8 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*b^4*f^2*x + b^3*f*(15*a*f + c*x*(-24*e + 5*f*x)) + 4*b*c*(-13*a^2*f^2 + 2*c^
2*d*(d - 2*e*x) + a*c*(2*e^2 + 4*d*f + 20*e*f*x - 5*f^2*x^2)) - 2*b^2*c*(a*f*(12
*e + 31*f*x) + c*x*(-4*e^2 - 8*d*f + 4*e*f*x + f^2*x^2)) + 8*c^2*(2*c^2*d^2*x +
a^2*f*(8*e + 3*f*x) + a*c*(-4*d*(e + f*x) + x*(-2*e^2 + 4*e*f*x + f^2*x^2))))/(4
*c^3*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)]) + ((15*b^2*f^2 - 12*c*f*(2*b*e + a*f)
 + 8*c^2*(e^2 + 2*d*f))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(8*c^(
7/2))

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Maple [B]  time = 0.019, size = 1011, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*f^2/c^2*a*x/(c*x^2+b*x+a)^(1/2)-13/4*f^2*b/c^3*a/(c*x^2+b*x+a)^(1/2)+15/16*f
^2*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+1/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)
^(1/2)*e^2-5/4*f^2*b/c^2*x^2/(c*x^2+b*x+a)^(1/2)-15/8*f^2*b^2/c^3*x/(c*x^2+b*x+a
)^(1/2)-3*e*f*b/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+4*e*f/c^2*a/
(c*x^2+b*x+a)^(1/2)-2*x/c/(c*x^2+b*x+a)^(1/2)*d*f+b/c^2/(c*x^2+b*x+a)^(1/2)*d*f+
8*e*f/c*a*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-3/2*e*f*b^2/c^3/(c*x^2+b*x+a)^(1/2
)+2*e*f*x^2/c/(c*x^2+b*x+a)^(1/2)-13/2*f^2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(
1/2)*x+2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*f-3*e*f*b^3/c^2/(4*a*c-b^2)/(
c*x^2+b*x+a)^(1/2)*x+4*e*f/c^2*a*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-3/2*e*f*b^4
/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+3*e*f*b/c^2*x/(c*x^2+b*x+a)^(1/2)+b^2/c/(4*
a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*e^2+b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d*f-2
*d*e*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-4*d*e*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/
2)*x-13/4*f^2*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+2*d^2*(2*c*x+b)/(4*a*c-b
^2)/(c*x^2+b*x+a)^(1/2)+2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*
f+15/8*f^2*b^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/2*f^2/c^(5/
2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/2*f^2*x^3/c/(c*x^2+b*x+a)^(1/
2)+15/16*f^2*b^3/c^4/(c*x^2+b*x+a)^(1/2)-2*d*e/c/(c*x^2+b*x+a)^(1/2)-x/c/(c*x^2+
b*x+a)^(1/2)*e^2+1/2*b/c^2/(c*x^2+b*x+a)^(1/2)*e^2+15/8*f^2*b^4/c^3/(4*a*c-b^2)/
(c*x^2+b*x+a)^(1/2)*x+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^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)^2/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.953681, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(4*(8*b*c^3*d^2 - 32*a*c^3*d*e + 8*a*b*c^2*e^2 - 2*(b^2*c^2 - 4*a*c^3)*f^
2*x^3 + (15*a*b^3 - 52*a^2*b*c)*f^2 - (8*(b^2*c^2 - 4*a*c^3)*e*f - 5*(b^3*c - 4*
a*b*c^2)*f^2)*x^2 + 8*(2*a*b*c^2*d - (3*a*b^2*c - 8*a^2*c^2)*e)*f + (16*c^4*d^2
- 16*b*c^3*d*e + 8*(b^2*c^2 - 2*a*c^3)*e^2 + (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)*
f^2 + 8*(2*(b^2*c^2 - 2*a*c^3)*d - (3*b^3*c - 10*a*b*c^2)*e)*f)*x)*sqrt(c*x^2 +
b*x + a)*sqrt(c) + (8*(a*b^2*c^2 - 4*a^2*c^3)*e^2 + 3*(5*a*b^4 - 24*a^2*b^2*c +
16*a^3*c^2)*f^2 + (8*(b^2*c^3 - 4*a*c^4)*e^2 + 3*(5*b^4*c - 24*a*b^2*c^2 + 16*a^
2*c^3)*f^2 + 8*(2*(b^2*c^3 - 4*a*c^4)*d - 3*(b^3*c^2 - 4*a*b*c^3)*e)*f)*x^2 + 8*
(2*(a*b^2*c^2 - 4*a^2*c^3)*d - 3*(a*b^3*c - 4*a^2*b*c^2)*e)*f + (8*(b^3*c^2 - 4*
a*b*c^3)*e^2 + 3*(5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*f^2 + 8*(2*(b^3*c^2 - 4*a*b
*c^3)*d - 3*(b^4*c - 4*a*b^2*c^2)*e)*f)*x)*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^3 - 4*a^2*c^4 +
 (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)*sqrt(c)), -1/8*(2*(8*b*c^3*d
^2 - 32*a*c^3*d*e + 8*a*b*c^2*e^2 - 2*(b^2*c^2 - 4*a*c^3)*f^2*x^3 + (15*a*b^3 -
52*a^2*b*c)*f^2 - (8*(b^2*c^2 - 4*a*c^3)*e*f - 5*(b^3*c - 4*a*b*c^2)*f^2)*x^2 +
8*(2*a*b*c^2*d - (3*a*b^2*c - 8*a^2*c^2)*e)*f + (16*c^4*d^2 - 16*b*c^3*d*e + 8*(
b^2*c^2 - 2*a*c^3)*e^2 + (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)*f^2 + 8*(2*(b^2*c^2
- 2*a*c^3)*d - (3*b^3*c - 10*a*b*c^2)*e)*f)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) -
(8*(a*b^2*c^2 - 4*a^2*c^3)*e^2 + 3*(5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*f^2 + (
8*(b^2*c^3 - 4*a*c^4)*e^2 + 3*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*f^2 + 8*(2*(
b^2*c^3 - 4*a*c^4)*d - 3*(b^3*c^2 - 4*a*b*c^3)*e)*f)*x^2 + 8*(2*(a*b^2*c^2 - 4*a
^2*c^3)*d - 3*(a*b^3*c - 4*a^2*b*c^2)*e)*f + (8*(b^3*c^2 - 4*a*b*c^3)*e^2 + 3*(5
*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*f^2 + 8*(2*(b^3*c^2 - 4*a*b*c^3)*d - 3*(b^4*c
- 4*a*b^2*c^2)*e)*f)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c
)))/((a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)
*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.287578, size = 549, normalized size = 1.78 \[ \frac{{\left ({\left (\frac{2 \,{\left (b^{2} c^{2} f^{2} - 4 \, a c^{3} f^{2}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} - \frac{5 \, b^{3} c f^{2} - 20 \, a b c^{2} f^{2} - 8 \, b^{2} c^{2} f e + 32 \, a c^{3} f e}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac{16 \, c^{4} d^{2} + 16 \, b^{2} c^{2} d f - 32 \, a c^{3} d f + 15 \, b^{4} f^{2} - 62 \, a b^{2} c f^{2} + 24 \, a^{2} c^{2} f^{2} - 16 \, b c^{3} d e - 24 \, b^{3} c f e + 80 \, a b c^{2} f e + 8 \, b^{2} c^{2} e^{2} - 16 \, a c^{3} e^{2}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac{8 \, b c^{3} d^{2} + 16 \, a b c^{2} d f + 15 \, a b^{3} f^{2} - 52 \, a^{2} b c f^{2} - 32 \, a c^{3} d e - 24 \, a b^{2} c f e + 64 \, a^{2} c^{2} f e + 8 \, a b c^{2} e^{2}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt{c x^{2} + b x + a}} - \frac{{\left (16 \, c^{2} d f + 15 \, b^{2} f^{2} - 12 \, a c f^{2} - 24 \, b c f e + 8 \, c^{2} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(((2*(b^2*c^2*f^2 - 4*a*c^3*f^2)*x/(b^2*c^3 - 4*a*c^4) - (5*b^3*c*f^2 - 20*a
*b*c^2*f^2 - 8*b^2*c^2*f*e + 32*a*c^3*f*e)/(b^2*c^3 - 4*a*c^4))*x - (16*c^4*d^2
+ 16*b^2*c^2*d*f - 32*a*c^3*d*f + 15*b^4*f^2 - 62*a*b^2*c*f^2 + 24*a^2*c^2*f^2 -
 16*b*c^3*d*e - 24*b^3*c*f*e + 80*a*b*c^2*f*e + 8*b^2*c^2*e^2 - 16*a*c^3*e^2)/(b
^2*c^3 - 4*a*c^4))*x - (8*b*c^3*d^2 + 16*a*b*c^2*d*f + 15*a*b^3*f^2 - 52*a^2*b*c
*f^2 - 32*a*c^3*d*e - 24*a*b^2*c*f*e + 64*a^2*c^2*f*e + 8*a*b*c^2*e^2)/(b^2*c^3
- 4*a*c^4))/sqrt(c*x^2 + b*x + a) - 1/8*(16*c^2*d*f + 15*b^2*f^2 - 12*a*c*f^2 -
24*b*c*f*e + 8*c^2*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) -
b))/c^(7/2)

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