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

Optimal. Leaf size=451 \[ \frac{f \left (\sqrt{e^2-4 d f}+e\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{f \left (e-\sqrt{e^2-4 d f}\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 5.86835, antiderivative size = 451, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{f \left (\sqrt{e^2-4 d f}+e\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{f \left (e-\sqrt{e^2-4 d f}\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [B]  time = 6.24593, size = 994, normalized size = 2.2 \[ \frac{\sqrt{c x^2+b x+a} \log (x)}{\sqrt{a} d \sqrt{a+x (b+c x)}}-\frac{f \left (e+\sqrt{e^2-4 d f}\right ) \sqrt{c x^2+b x+a} \log \left (-e-2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{a+x (b+c x)}}-\frac{f \left (\sqrt{e^2-4 d f}-e\right ) \sqrt{c x^2+b x+a} \log \left (e+2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{a+x (b+c x)}}-\frac{\sqrt{c x^2+b x+a} \log \left (2 a+2 \sqrt{c x^2+b x+a} \sqrt{a}+b x\right )}{\sqrt{a} d \sqrt{a+x (b+c x)}}+\frac{f \left (\sqrt{e^2-4 d f}-e\right ) \sqrt{c x^2+b x+a} \log \left (-b e^2-2 c x e^2-2 c \sqrt{e^2-4 d f} x e-b \sqrt{e^2-4 d f} e+4 b d f+8 c d f x+2 b f \sqrt{e^2-4 d f} x+4 a f \sqrt{e^2-4 d f}+2 \sqrt{2} \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{a+x (b+c x)}}+\frac{f \left (e+\sqrt{e^2-4 d f}\right ) \sqrt{c x^2+b x+a} \log \left (b e^2+2 c x e^2-2 c \sqrt{e^2-4 d f} x e-b \sqrt{e^2-4 d f} e-4 b d f-8 c d f x+2 b f \sqrt{e^2-4 d f} x+4 a f \sqrt{e^2-4 d f}+2 \sqrt{2} \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.022, size = 859, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(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^(1/2)*ln((2*a+b*x+2*a^(1/2)
*(c*x^2+b*x+a)^(1/2))/x)-2*f/(-e+(-4*d*f+e^2)^(1/2))/(-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))-2*f/(e+(-4*d*f+e^2)^(1/2))/(-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))

_______________________________________________________________________________________

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

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),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

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),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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),x, algorithm="giac")

[Out]

Exception raised: TypeError