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

Optimal. Leaf size=340 \[ \frac{e^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g)^2 \sqrt{a e^2-b d e+c d^2}}+\frac{g^2 \sqrt{a+b x+c x^2}}{(f+g x) (e f-d g) \left (a g^2-b f g+c f^2\right )}-\frac{e g \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{(e f-d g)^2 \sqrt{a g^2-b f g+c f^2}}-\frac{g (2 c f-b g) \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{2 (e f-d g) \left (a g^2-b f g+c f^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.902698, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{e^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g)^2 \sqrt{a e^2-b d e+c d^2}}+\frac{g^2 \sqrt{a+b x+c x^2}}{(f+g x) (e f-d g) \left (a g^2-b f g+c f^2\right )}-\frac{e g \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{(e f-d g)^2 \sqrt{a g^2-b f g+c f^2}}-\frac{g (2 c f-b g) \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{2 (e f-d g) \left (a g^2-b f g+c f^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 101.24, size = 304, normalized size = 0.89 \[ - \frac{e^{2} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (d g - e f\right )^{2} \sqrt{a e^{2} - b d e + c d^{2}}} + \frac{e g \operatorname{atanh}{\left (\frac{2 a g - b f + x \left (b g - 2 c f\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a g^{2} - b f g + c f^{2}}} \right )}}{\left (d g - e f\right )^{2} \sqrt{a g^{2} - b f g + c f^{2}}} - \frac{g^{2} \sqrt{a + b x + c x^{2}}}{\left (f + g x\right ) \left (d g - e f\right ) \left (a g^{2} - b f g + c f^{2}\right )} + \frac{g \left (b g - 2 c f\right ) \operatorname{atanh}{\left (\frac{2 a g - b f + x \left (b g - 2 c f\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a g^{2} - b f g + c f^{2}}} \right )}}{2 \left (d g - e f\right ) \left (a g^{2} - b f g + c f^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.58469, size = 343, normalized size = 1.01 \[ \frac{\frac{2 e^2 \log (d+e x)}{\sqrt{e (a e-b d)+c d^2}}-\frac{2 e^2 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}+\frac{2 g^2 \sqrt{a+x (b+c x)} (e f-d g)}{(f+g x) \left (g (a g-b f)+c f^2\right )}-\frac{g \log (f+g x) (g (2 a e g+b d g-3 b e f)+2 c f (2 e f-d g))}{\left (g (a g-b f)+c f^2\right )^{3/2}}+\frac{g (g (2 a e g+b d g-3 b e f)+2 c f (2 e f-d g)) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}+2 a g-b f+b g x-2 c f x\right )}{\left (g (a g-b f)+c f^2\right )^{3/2}}}{2 (e f-d g)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 788, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e/(d*g-e*f)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*
e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*
(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))-g/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)
/(x+f/g)*((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)+1/2*g
/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln((2*(a*g^2-b*f*
g+c*f^2)/g^2+(b*g-2*c*f)/g*(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*
c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))*b-1/(d*g-e*f)/(
a*g^2-b*f*g+c*f^2)/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2
+(b*g-2*c*f)/g*(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f
)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))*c*f+e/(d*g-e*f)^2/((a*g^2-b
*f*g+c*f^2)/g^2)^(1/2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2+(b*g-2*c*f)/g*(x+f/g)+2*((a
*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f
^2)/g^2)^(1/2))/(x+f/g))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 9.735, size = 1, normalized size = 0. \[ \left [\frac{e^{2} \log \left (\frac{{\left (8 \, a b d e - 8 \, a^{2} e^{2} -{\left (b^{2} + 4 \, a c\right )} d^{2} -{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} - 2 \,{\left (4 \, b c d^{2} + 4 \, a b e^{2} -{\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}} - 4 \,{\left (b c d^{3} + 3 \, a b d e^{2} - 2 \, a^{2} e^{3} -{\left (b^{2} + 2 \, a c\right )} d^{2} e +{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \,{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \sqrt{c d^{2} - b d e + a e^{2}}}, -\frac{e^{2} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e - a e^{2}}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{c x^{2} + b x + a}}\right )}{{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \sqrt{-c d^{2} + b d e - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*e^2*log(((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*
e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)*
sqrt(c*d^2 - b*d*e + a*e^2) - 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*
c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x
^2 + b*x + a))/(e^2*x^2 + 2*d*e*x + d^2))/((e^2*f^2 - 2*d*e*f*g + d^2*g^2)*sqrt(
c*d^2 - b*d*e + a*e^2)), -e^2*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(b*d - 2*
a*e + (2*c*d - b*e)*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)))/((e^2*f^
2 - 2*d*e*f*g + d^2*g^2)*sqrt(-c*d^2 + b*d*e - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError