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

Optimal. Leaf size=182 \[ \frac{e \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g) \sqrt{a e^2-b d e+c d^2}}-\frac{g \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{(e f-d g) \sqrt{a g^2-b f g+c f^2}} \]

[Out]

(e*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)) - (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)*Sqrt[c*f^2 - b*f*g + a*g^2])

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Rubi [A]  time = 0.492819, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{e \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g) \sqrt{a e^2-b d e+c d^2}}-\frac{g \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{(e f-d g) \sqrt{a g^2-b f g+c f^2}} \]

Antiderivative was successfully verified.

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

[Out]

(e*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)) - (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)*Sqrt[c*f^2 - b*f*g + a*g^2])

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Rubi in Sympy [A]  time = 64.1461, size = 162, normalized size = 0.89 \[ \frac{e \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (d g - e f\right ) \sqrt{a e^{2} - b d e + c d^{2}}} - \frac{g \operatorname{atanh}{\left (\frac{2 a g - b f + x \left (b g - 2 c f\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a g^{2} - b f g + c f^{2}}} \right )}}{\left (d g - e f\right ) \sqrt{a g^{2} - b f g + c f^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*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)*sqrt(a*e**2 - b*d*e + c*d**2)) - 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)*sqrt(a*g**2 - b*f*g + c*f**2))

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Mathematica [A]  time = 0.867588, size = 222, normalized size = 1.22 \[ \frac{\frac{e \log \left (2 \sqrt{a+x (b+c x)} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{a e^2-b d e+c d^2}}-\frac{e \log (d+e x)}{\sqrt{e (a e-b d)+c d^2}}+\frac{g \log (f+g x)}{\sqrt{a g^2-b f g+c f^2}}-\frac{g \log \left (2 \sqrt{a+x (b+c x)} \sqrt{a g^2-b f g+c f^2}+2 a g-b f+b g x-2 c f x\right )}{\sqrt{a g^2-b f g+c f^2}}}{d g-e f} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 327, normalized size = 1.8 \[ -{\frac{1}{dg-ef}\ln \left ({1 \left ( 2\,{\frac{a{g}^{2}-bfg+c{f}^{2}}{{g}^{2}}}+{\frac{bg-2\,cf}{g} \left ( x+{\frac{f}{g}} \right ) }+2\,\sqrt{{\frac{a{g}^{2}-bfg+c{f}^{2}}{{g}^{2}}}}\sqrt{ \left ( x+{\frac{f}{g}} \right ) ^{2}c+{\frac{bg-2\,cf}{g} \left ( x+{\frac{f}{g}} \right ) }+{\frac{a{g}^{2}-bfg+c{f}^{2}}{{g}^{2}}}} \right ) \left ( x+{\frac{f}{g}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{g}^{2}-bfg+c{f}^{2}}{{g}^{2}}}}}}}+{\frac{1}{dg-ef}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c+{\frac{be-2\,cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(d*g-e*f)/((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))+1/(d*g-e*f)/((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))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 124.009, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((d + e*x)*(f + g*x)*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)),x, algorithm="giac")

[Out]

Exception raised: TypeError