3.2328 \(\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=153 \[ \frac{\sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.267018, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x + c*x^2]/(d + e*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 32.494, size = 138, normalized size = 0.9 \[ \frac{\left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{8 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{a + b x + c x^{2}} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{4 \left (d + e x\right )^{2} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**3,x)

[Out]

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

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Mathematica [A]  time = 0.438455, size = 178, normalized size = 1.16 \[ \frac{\left (b^2-4 a c\right ) (d+e x)^2 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )-\left (b^2-4 a c\right ) (d+e x)^2 \log (d+e x)+2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} (-2 a e+b (d-e x)+2 c d x)}{8 (d+e x)^2 \left (e (a e-b d)+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^3,x]

[Out]

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

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Maple [B]  time = 0.02, size = 3269, normalized size = 21.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)/(e*x+d)^3,x)

[Out]

-1/8/(a*e^2-b*d*e+c*d^2)^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-1/2/e^3*c^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))*d^2-1/4*e/(a*e^2-b*d*e+c*d^
2)^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-1/2/(a*e^2-b*d*e+c*d^2)^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+1/2/e/(a*e^2-b*d*e+c*d^2)^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^2-1/e^2/(a*e^2-b*d*e+c*d^2)^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+1/2/e^2*c/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*l
n((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-1/4*e/(a*e^2-b*d*e+c*d^2)^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-3/4/e/(a*e^2-b*d*e+c*d^2)^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-1/2/e*c/(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+1
/2/(a*e^2-b*d*e+c*d^2)^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+1/4/(a*e^2-b*d*
e+c*d^2)^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)/(
d/e+x)^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)^(3/2)+1/4*e
/(a*e^2-b*d*e+c*d^2)^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-1/2/e/(a*e^2-b*d*e+c*d^2)^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/e^2/(a*e^2-b*d*e+c*d^2)^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+1/4/e*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))*b+3/4/(a*e^2-b*d*e+c*d^2)^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-1/2/(a*e^2-b*d*e+c*d^2)^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+1/8*e/(a*e^2-b*d*e+c*d^
2)^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+1/2/e^3/(a*e^2-b*d*e+c*d^2)^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+5/8/e/(a*e^2-b*d*e+c*d^2)^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+1/2/(a*e^2-b*d*e+c*d^2)^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-1/2/e^2*c^(3/2)/(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))*d-1/4*e/(a*e^2-b*d*e+c*d^2)^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+1/2/e*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)

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.503865, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c d^{2} - b d e + a e^{2}} \sqrt{c x^{2} + b x + a}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )} -{\left ({\left (b^{2} - 4 \, a c\right )} e^{2} x^{2} + 2 \,{\left (b^{2} - 4 \, a c\right )} d e x +{\left (b^{2} - 4 \, a c\right )} d^{2}\right )} \log \left (\frac{{\left (8 \, a b d e - 8 \, a^{2} e^{2} -{\left (b^{2} + 4 \, a c\right )} d^{2} -{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} - 2 \,{\left (4 \, b c d^{2} + 4 \, a b e^{2} -{\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}} - 4 \,{\left (b c d^{3} + 3 \, a b d e^{2} - 2 \, a^{2} e^{3} -{\left (b^{2} + 2 \, a c\right )} d^{2} e +{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{16 \,{\left (c d^{4} - b d^{3} e + a d^{2} e^{2} +{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} x^{2} + 2 \,{\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}}}, \frac{2 \, \sqrt{-c d^{2} + b d e - a e^{2}} \sqrt{c x^{2} + b x + a}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )} +{\left ({\left (b^{2} - 4 \, a c\right )} e^{2} x^{2} + 2 \,{\left (b^{2} - 4 \, a c\right )} d e x +{\left (b^{2} - 4 \, a c\right )} d^{2}\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e - a e^{2}}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{c x^{2} + b x + a}}\right )}{8 \,{\left (c d^{4} - b d^{3} e + a d^{2} e^{2} +{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} x^{2} + 2 \,{\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} x\right )} \sqrt{-c d^{2} + b d e - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(e*x + d)^3,x, algorithm="fricas")

[Out]

[1/16*(4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d
- b*e)*x) - ((b^2 - 4*a*c)*e^2*x^2 + 2*(b^2 - 4*a*c)*d*e*x + (b^2 - 4*a*c)*d^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)))/((c*d^4 - b*d^3*e + a*d^2*e^2 + (c*d^2*e^2 -
b*d*e^3 + a*e^4)*x^2 + 2*(c*d^3*e - b*d^2*e^2 + a*d*e^3)*x)*sqrt(c*d^2 - b*d*e +
a*e^2)), 1/8*(2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e
+ (2*c*d - b*e)*x) + ((b^2 - 4*a*c)*e^2*x^2 + 2*(b^2 - 4*a*c)*d*e*x + (b^2 - 4*
a*c)*d^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))))/((c*d^4 - b*d^3*e + a*d^2*e
^2 + (c*d^2*e^2 - b*d*e^3 + a*e^4)*x^2 + 2*(c*d^3*e - b*d^2*e^2 + a*d*e^3)*x)*sq
rt(-c*d^2 + b*d*e - a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x + c x^{2}}}{\left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**3,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(d + e*x)**3, x)

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GIAC/XCAS [A]  time = 0.520994, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(e*x + d)^3,x, algorithm="giac")

[Out]

sage0*x