3.119 \(\int \frac{1}{x^2 \sqrt{a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx\)

Optimal. Leaf size=543 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{3/2} d}-\frac{f \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (e-\sqrt{e^2-4 d f}\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{f \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{e \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d^2}-\frac{\sqrt{a+b x+c x^2}}{a d x} \]

[Out]

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

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Rubi [A]  time = 10.059, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{3/2} d}-\frac{f \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (e-\sqrt{e^2-4 d f}\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{f \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{e \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d^2}-\frac{\sqrt{a+b x+c x^2}}{a d x} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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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(1/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 [A]  time = 1.87223, size = 866, normalized size = 1.59 \[ \frac{-\frac{2 \sqrt{a+x (b+c x)} d}{a x}-\frac{(b d+2 a e) \log (x)}{a^{3/2}}+\frac{\sqrt{2} f \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) \log \left (-e-2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{e^2-4 d f} \sqrt{c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )+f \left (2 a f+b \left (\sqrt{e^2-4 d f}-e\right )\right )}}+\frac{\sqrt{2} f \left (-e^2+\sqrt{e^2-4 d f} e+2 d f\right ) \log \left (e+2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{e^2-4 d f} \sqrt{c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt{e^2-4 d f}\right )\right )}}+\frac{(b d+2 a e) \log \left (2 a+2 \sqrt{a+x (b+c x)} \sqrt{a}+b x\right )}{a^{3/2}}-\frac{\sqrt{2} f \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) \log \left (2 c x e^2-2 c \sqrt{e^2-4 d f} x e-8 c d f x+b \left (e^2-\sqrt{e^2-4 d f} e-4 d f+2 f \sqrt{e^2-4 d f} x\right )+4 a f \sqrt{e^2-4 d f}+2 \sqrt{2} \sqrt{e^2-4 d f} \sqrt{f \left (-e b+\sqrt{e^2-4 d f} b+2 a f\right )+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{a+x (b+c x)}\right )}{\sqrt{e^2-4 d f} \sqrt{c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )+f \left (2 a f+b \left (\sqrt{e^2-4 d f}-e\right )\right )}}-\frac{\sqrt{2} f \left (-e^2+\sqrt{e^2-4 d f} e+2 d f\right ) \log \left (-2 c x e^2-2 c \sqrt{e^2-4 d f} x e+8 c d f x-b \left (e^2+\sqrt{e^2-4 d f} e-2 f \left (2 d+\sqrt{e^2-4 d f} x\right )\right )+4 a f \sqrt{e^2-4 d f}+2 \sqrt{2} \sqrt{e^2-4 d f} \sqrt{c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt{e^2-4 d f}\right )\right )} \sqrt{a+x (b+c x)}\right )}{\sqrt{e^2-4 d f} \sqrt{c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt{e^2-4 d f}\right )\right )}}}{2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 983, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*f/(-e+(-4*d*f+e^2)^(1/2))/(e+(-4*d*f+e^2)^(1/2))/a/x*(c*x^2+b*x+a)^(1/2)-2*f/(
-e+(-4*d*f+e^2)^(1/2))/(e+(-4*d*f+e^2)^(1/2))*b/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c
*x^2+b*x+a)^(1/2))/x)-4*f^2/(-e+(-4*d*f+e^2)^(1/2))^2/(-4*d*f+e^2)^(1/2)*2^(1/2)
/(((-4*d*f+e^2)^(1/2)*b*f-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+e^2*c)/f^
2)^(1/2)*ln((((-4*d*f+e^2)^(1/2)*b*f-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*
f+e^2*c)/f^2+(c*(-4*d*f+e^2)^(1/2)+b*f-c*e)/f*(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)+
1/2*2^(1/2)*(((-4*d*f+e^2)^(1/2)*b*f-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*
f+e^2*c)/f^2)^(1/2)*(4*(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)^2*c+4*(c*(-4*d*f+e^2)^(
1/2)+b*f-c*e)/f*(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1/2)*b*f-(-4*
d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+e^2*c)/f^2)^(1/2))/(x-1/2*(-e+(-4*d*f+e
^2)^(1/2))/f))+4*f^2/(e+(-4*d*f+e^2)^(1/2))^2/(-4*d*f+e^2)^(1/2)*2^(1/2)/((-(-4*
d*f+e^2)^(1/2)*b*f+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+e^2*c)/f^2)^(1/2
)*ln(((-(-4*d*f+e^2)^(1/2)*b*f+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+e^2*
c)/f^2+1/f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*
2^(1/2)*((-(-4*d*f+e^2)^(1/2)*b*f+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+e
^2*c)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c+4/f*(-c*(-4*d*f+e^2)^(1
/2)+b*f-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*(-(-4*d*f+e^2)^(1/2)*b*f+(-4*d*f
+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+e^2*c)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^
(1/2))/f))+16*f^2*e/(-e+(-4*d*f+e^2)^(1/2))^2/(e+(-4*d*f+e^2)^(1/2))^2/a^(1/2)*l
n((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x + a}{\left (f x^{2} + e x + d\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x + a)*(f*x^2 + e*x + d)*x^2),x, algorithm="maxima")

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

[Out]

Exception raised: TypeError