3.1552 \(\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=203 \[ \frac{\left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^3 \sqrt{a e^2-b d e+c d^2}}-\frac{2 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^3}+\frac{\sqrt{a+b x+c x^2} (-b e+4 c d+2 c e x)}{e^2 (d+e x)} \]

[Out]

((4*c*d - b*e + 2*c*e*x)*Sqrt[a + b*x + c*x^2])/(e^2*(d + e*x)) - (2*Sqrt[c]*(2*
c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e^3 + ((8*c^2
*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*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^3*Sqrt[c*d^2 - b*d*e
+ a*e^2])

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Rubi [A]  time = 0.51298, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^3 \sqrt{a e^2-b d e+c d^2}}-\frac{2 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^3}+\frac{\sqrt{a+b x+c x^2} (-b e+4 c d+2 c e x)}{e^2 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

((4*c*d - b*e + 2*c*e*x)*Sqrt[a + b*x + c*x^2])/(e^2*(d + e*x)) - (2*Sqrt[c]*(2*
c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e^3 + ((8*c^2
*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*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^3*Sqrt[c*d^2 - b*d*e
+ a*e^2])

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Rubi in Sympy [A]  time = 104.061, size = 194, normalized size = 0.96 \[ \frac{2 \sqrt{c} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{e^{3}} - \frac{\sqrt{a + b x + c x^{2}} \left (b e - 4 c d - 2 c e x\right )}{e^{2} \left (d + e x\right )} - \frac{\left (4 a c e^{2} + b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 e^{3} \sqrt{a e^{2} - b d e + c d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c)*(b*e - 2*c*d)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/e*
*3 - sqrt(a + b*x + c*x**2)*(b*e - 4*c*d - 2*c*e*x)/(e**2*(d + e*x)) - (4*a*c*e*
*2 + b**2*e**2 - 8*b*c*d*e + 8*c**2*d**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)))/(2*e**3*sqrt(a*e**2 -
b*d*e + c*d**2))

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Mathematica [A]  time = 0.450031, size = 247, normalized size = 1.22 \[ \frac{\frac{\log (d+e x) \left (4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right )}{\sqrt{e (a e-b d)+c d^2}}-\frac{\left (4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}+\frac{2 e \sqrt{a+x (b+c x)} (-b e+4 c d+2 c e x)}{d+e x}-4 \sqrt{c} (2 c d-b e) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{2 e^3} \]

Antiderivative was successfully verified.

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

[Out]

((2*e*(4*c*d - b*e + 2*c*e*x)*Sqrt[a + x*(b + c*x)])/(d + e*x) + ((8*c^2*d^2 + b
^2*e^2 + 4*c*e*(-2*b*d + a*e))*Log[d + e*x])/Sqrt[c*d^2 + e*(-(b*d) + a*e)] - 4*
Sqrt[c]*(2*c*d - b*e)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]] - ((8*c^2
*d^2 + b^2*e^2 + 4*c*e*(-2*b*d + a*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^3)

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Maple [B]  time = 0.017, size = 3088, normalized size = 15.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/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))*b*d^3*c^2-2/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))*a*c^2*d^2-2/e^3/(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^(5/2)*d^3+2*c/e^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)+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^2-2*c^2/e^4/((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))*d^2+2/e^2/(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^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^2+1/(a*e^2-b*d*e+c*d^2)*c*(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*b+1/(a*e^2-b*d*e+c*d^2)*c^(1/2)*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*b-2*c/e^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-5/2/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^2*d^2*c+2*c/e^3/((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/e/(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)*c*d-3/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)*b*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^2+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^3*d-2/e^4/(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^3*d^4-2/e/(a*e^2-b*d*e+c*d^2)*c^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*d-2/e/(a*e^2-b*d*e+c*d^2)*c^(3/2)*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*d+3/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*b+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))*a*b*c*d-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)*b+c^(1/2)/e^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))*b-2*c^(3/2)/e^3*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))*d

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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)*(2*c*x + b)/(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 22.9817, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError