3.2327 \(\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=160 \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^2 \sqrt{a e^2-b d e+c d^2}}-\frac{\sqrt{a+b x+c x^2}}{e (d+e x)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^2} \]

[Out]

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

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Rubi [A]  time = 0.395706, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^2 \sqrt{a e^2-b d e+c d^2}}-\frac{\sqrt{a+b x+c x^2}}{e (d+e x)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.6389, size = 143, normalized size = 0.89 \[ \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{e^{2}} - \frac{\sqrt{a + b x + c x^{2}}}{e \left (d + e x\right )} - \frac{\left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 e^{2} \sqrt{a e^{2} - b d e + c d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/e**2 - sqrt(a + b*
x + c*x**2)/(e*(d + e*x)) - (b*e - 2*c*d)*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)))/(2*e**2*sqrt(a*e**2 -
b*d*e + c*d**2))

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Mathematica [A]  time = 0.741681, size = 186, normalized size = 1.16 \[ \frac{-\frac{(2 c d-b e) \log (d+e x)}{\sqrt{e (a e-b d)+c d^2}}+\frac{(2 c d-b e) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}-\frac{2 e \sqrt{a+x (b+c x)}}{d+e x}+2 \sqrt{c} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{2 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 1519, normalized size = 9.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*e^2-b*d*e+c*d^2)/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c
*d^2)/e^2)^(3/2)+1/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2
-b*d*e+c*d^2)/e^2)^(1/2)*b-1/e/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d
/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d-1/e/(a*e^2-b*d*e+c*d^2)*ln((1/2*(b*e-2*
c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)
/e^2)^(1/2))*c^(1/2)*d*b+1/e^2/(a*e^2-b*d*e+c*d^2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+
x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c
^(3/2)*d^2-1/2/(a*e^2-b*d*e+c*d^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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e
+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*a*b+1/e/(a*
e^2-b*d*e+c*d^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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*a*c*d+1/2/e/(a*e^2-b*d*e+c*d^
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*(
d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e
^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^2*d-3/2/e^2/(a*e^2-b*d*e+c*d^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*(d/e+x)+2*((a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d
^2)/e^2)^(1/2))/(d/e+x))*b*d^2*c+1/e^3/(a*e^2-b*d*e+c*d^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*(d/e+x)+2*((a*e^2-b*d*e+c
*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/
2))/(d/e+x))*c^2*d^3+c/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x+c^(1/2)/(a*e^2-b*d*e+c*d^2)*ln((1/2*(b*e-2*c*d)/e
+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2))*a

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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)/(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.9901, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError