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

Optimal. Leaf size=761 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c f (b e-a f)+b^2 f^2-8 c^2 \left (e^2-d f\right )\right )}{8 c^{3/2} f^3}-\frac{\left (f \left (a f \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b \left (-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-4 d e^2 f+2 d e f \sqrt{e^2-4 d f}-e^3 \sqrt{e^2-4 d f}+e^4\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (e-\sqrt{e^2-4 d f}\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (f \left (a f \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b \left (e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-4 d e^2 f-2 d e f \sqrt{e^2-4 d f}+e^3 \sqrt{e^2-4 d f}+e^4\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{a+b x+c x^2} (-b f+4 c e-2 c f x)}{4 c f^2} \]

[Out]

-((4*c*e - b*f - 2*c*f*x)*Sqrt[a + b*x + c*x^2])/(4*c*f^2) - ((b^2*f^2 + 4*c*f*(
b*e - a*f) - 8*c^2*(e^2 - d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*
x^2])])/(8*c^(3/2)*f^3) - ((c*(e^4 - 4*d*e^2*f + 2*d^2*f^2 - e^3*Sqrt[e^2 - 4*d*
f] + 2*d*e*f*Sqrt[e^2 - 4*d*f]) + f*(a*f*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - b
*(e^3 - 3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4*d*f])))*ArcTanh[(4*a*
f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2
]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a
+ b*x + c*x^2])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e
^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))]) + ((c*(e^4 - 4*d*e^2*f +
2*d^2*f^2 + e^3*Sqrt[e^2 - 4*d*f] - 2*d*e*f*Sqrt[e^2 - 4*d*f]) + f*(a*f*(e^2 - 2
*d*f + e*Sqrt[e^2 - 4*d*f]) - b*(e^3 - 3*d*e*f + e^2*Sqrt[e^2 - 4*d*f] - d*f*Sqr
t[e^2 - 4*d*f])))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + S
qrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e -
b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]
*Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f
]))])

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Rubi [A]  time = 8.37725, antiderivative size = 761, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c f (b e-a f)+b^2 f^2-8 c^2 \left (e^2-d f\right )\right )}{8 c^{3/2} f^3}-\frac{\left (f \left (a f \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b \left (-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-4 d e^2 f+2 d e f \sqrt{e^2-4 d f}-e^3 \sqrt{e^2-4 d f}+e^4\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (e-\sqrt{e^2-4 d f}\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (f \left (a f \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b \left (e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-4 d e^2 f-2 d e f \sqrt{e^2-4 d f}+e^3 \sqrt{e^2-4 d f}+e^4\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{a+b x+c x^2} (-b f+4 c e-2 c f x)}{4 c f^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((4*c*e - b*f - 2*c*f*x)*Sqrt[a + b*x + c*x^2])/(4*c*f^2) - ((b^2*f^2 + 4*c*f*(
b*e - a*f) - 8*c^2*(e^2 - d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*
x^2])])/(8*c^(3/2)*f^3) - ((c*(e^4 - 4*d*e^2*f + 2*d^2*f^2 - e^3*Sqrt[e^2 - 4*d*
f] + 2*d*e*f*Sqrt[e^2 - 4*d*f]) + f*(a*f*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - b
*(e^3 - 3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4*d*f])))*ArcTanh[(4*a*
f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2
]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a
+ b*x + c*x^2])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e
^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))]) + ((c*(e^4 - 4*d*e^2*f +
2*d^2*f^2 + e^3*Sqrt[e^2 - 4*d*f] - 2*d*e*f*Sqrt[e^2 - 4*d*f]) + f*(a*f*(e^2 - 2
*d*f + e*Sqrt[e^2 - 4*d*f]) - b*(e^3 - 3*d*e*f + e^2*Sqrt[e^2 - 4*d*f] - d*f*Sqr
t[e^2 - 4*d*f])))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + S
qrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e -
b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]
*Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f
]))])

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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(x**2*(c*x**2+b*x+a)**(1/2)/(f*x**2+e*x+d),x)

[Out]

Timed out

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Mathematica [B]  time = 6.3054, size = 1546, normalized size = 2.03 \[ \sqrt{a+x (b+c x)} \left (\frac{x}{2 f}-\frac{4 c e-b f}{4 c f^2}\right )-\frac{\left (-c e^4+b f e^3+c \sqrt{e^2-4 d f} e^3-a f^2 e^2+4 c d f e^2-b f \sqrt{e^2-4 d f} e^2-3 b d f^2 e+a f^2 \sqrt{e^2-4 d f} e-2 c d f \sqrt{e^2-4 d f} e+2 a d f^3-2 c d^2 f^2+b d f^2 \sqrt{e^2-4 d f}\right ) \sqrt{a+x (b+c x)} \log \left (-e-2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}-\frac{\left (c e^4-b f e^3+c \sqrt{e^2-4 d f} e^3+a f^2 e^2-4 c d f e^2-b f \sqrt{e^2-4 d f} e^2+3 b d f^2 e+a f^2 \sqrt{e^2-4 d f} e-2 c d f \sqrt{e^2-4 d f} e-2 a d f^3+2 c d^2 f^2+b d f^2 \sqrt{e^2-4 d f}\right ) \sqrt{a+x (b+c x)} \log \left (e+2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}+\frac{\left (8 e^2 c^2-8 d f c^2+4 a f^2 c-4 b e f c-b^2 f^2\right ) \sqrt{a+x (b+c x)} \log \left (b+2 c x+2 \sqrt{c} \sqrt{c x^2+b x+a}\right )}{8 c^{3/2} f^3 \sqrt{c x^2+b x+a}}+\frac{\left (c e^4-b f e^3+c \sqrt{e^2-4 d f} e^3+a f^2 e^2-4 c d f e^2-b f \sqrt{e^2-4 d f} e^2+3 b d f^2 e+a f^2 \sqrt{e^2-4 d f} e-2 c d f \sqrt{e^2-4 d f} e-2 a d f^3+2 c d^2 f^2+b d f^2 \sqrt{e^2-4 d f}\right ) \sqrt{a+x (b+c x)} \log \left (-b e^2-2 c x e^2-2 c \sqrt{e^2-4 d f} x e-b \sqrt{e^2-4 d f} e+4 b d f+8 c d f x+2 b f \sqrt{e^2-4 d f} x+4 a f \sqrt{e^2-4 d f}+2 \sqrt{2} \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}+\frac{\left (-c e^4+b f e^3+c \sqrt{e^2-4 d f} e^3-a f^2 e^2+4 c d f e^2-b f \sqrt{e^2-4 d f} e^2-3 b d f^2 e+a f^2 \sqrt{e^2-4 d f} e-2 c d f \sqrt{e^2-4 d f} e+2 a d f^3-2 c d^2 f^2+b d f^2 \sqrt{e^2-4 d f}\right ) \sqrt{a+x (b+c x)} \log \left (b e^2+2 c x e^2-2 c \sqrt{e^2-4 d f} x e-b \sqrt{e^2-4 d f} e-4 b d f-8 c d f x+2 b f \sqrt{e^2-4 d f} x+4 a f \sqrt{e^2-4 d f}+2 \sqrt{2} \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}} \]

Antiderivative was successfully verified.

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

[Out]

(-(4*c*e - b*f)/(4*c*f^2) + x/(2*f))*Sqrt[a + x*(b + c*x)] - ((-(c*e^4) + 4*c*d*
e^2*f + b*e^3*f - 2*c*d^2*f^2 - 3*b*d*e*f^2 - a*e^2*f^2 + 2*a*d*f^3 + c*e^3*Sqrt
[e^2 - 4*d*f] - 2*c*d*e*f*Sqrt[e^2 - 4*d*f] - b*e^2*f*Sqrt[e^2 - 4*d*f] + b*d*f^
2*Sqrt[e^2 - 4*d*f] + a*e*f^2*Sqrt[e^2 - 4*d*f])*Sqrt[a + x*(b + c*x)]*Log[-e +
Sqrt[e^2 - 4*d*f] - 2*f*x])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f
- b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x
+ c*x^2]) - ((c*e^4 - 4*c*d*e^2*f - b*e^3*f + 2*c*d^2*f^2 + 3*b*d*e*f^2 + a*e^2*
f^2 - 2*a*d*f^3 + c*e^3*Sqrt[e^2 - 4*d*f] - 2*c*d*e*f*Sqrt[e^2 - 4*d*f] - b*e^2*
f*Sqrt[e^2 - 4*d*f] + b*d*f^2*Sqrt[e^2 - 4*d*f] + a*e*f^2*Sqrt[e^2 - 4*d*f])*Sqr
t[a + x*(b + c*x)]*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x])/(Sqrt[2]*f^3*Sqrt[e^2 - 4
*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[
e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]) + ((8*c^2*e^2 - 8*c^2*d*f - 4*b*c*e*f - b^2
*f^2 + 4*a*c*f^2)*Sqrt[a + x*(b + c*x)]*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + b*x +
c*x^2]])/(8*c^(3/2)*f^3*Sqrt[a + b*x + c*x^2]) + ((c*e^4 - 4*c*d*e^2*f - b*e^3*
f + 2*c*d^2*f^2 + 3*b*d*e*f^2 + a*e^2*f^2 - 2*a*d*f^3 + c*e^3*Sqrt[e^2 - 4*d*f]
- 2*c*d*e*f*Sqrt[e^2 - 4*d*f] - b*e^2*f*Sqrt[e^2 - 4*d*f] + b*d*f^2*Sqrt[e^2 - 4
*d*f] + a*e*f^2*Sqrt[e^2 - 4*d*f])*Sqrt[a + x*(b + c*x)]*Log[-(b*e^2) + 4*b*d*f
- b*e*Sqrt[e^2 - 4*d*f] + 4*a*f*Sqrt[e^2 - 4*d*f] - 2*c*e^2*x + 8*c*d*f*x - 2*c*
e*Sqrt[e^2 - 4*d*f]*x + 2*b*f*Sqrt[e^2 - 4*d*f]*x + 2*Sqrt[2]*Sqrt[e^2 - 4*d*f]*
Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 -
4*d*f]]*Sqrt[a + b*x + c*x^2]])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*
d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a +
b*x + c*x^2]) + ((-(c*e^4) + 4*c*d*e^2*f + b*e^3*f - 2*c*d^2*f^2 - 3*b*d*e*f^2 -
a*e^2*f^2 + 2*a*d*f^3 + c*e^3*Sqrt[e^2 - 4*d*f] - 2*c*d*e*f*Sqrt[e^2 - 4*d*f] -
b*e^2*f*Sqrt[e^2 - 4*d*f] + b*d*f^2*Sqrt[e^2 - 4*d*f] + a*e*f^2*Sqrt[e^2 - 4*d*
f])*Sqrt[a + x*(b + c*x)]*Log[b*e^2 - 4*b*d*f - b*e*Sqrt[e^2 - 4*d*f] + 4*a*f*Sq
rt[e^2 - 4*d*f] + 2*c*e^2*x - 8*c*d*f*x - 2*c*e*Sqrt[e^2 - 4*d*f]*x + 2*b*f*Sqrt
[e^2 - 4*d*f]*x + 2*Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a
*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]])/(S
qrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e
^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])

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Maple [B]  time = 0.028, size = 14815, normalized size = 19.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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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)*x^2/(f*x^2 + e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError