3.676 \(\int \frac{a+b x+c x^2}{(d+e x)^3 \sqrt{-1+x^2}} \, dx\)

Optimal. Leaf size=195 \[ -\frac{\sqrt{x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}-\frac{\tanh ^{-1}\left (\frac{d x+e}{\sqrt{x^2-1} \sqrt{d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{2 \left (d^2-e^2\right )^{5/2}}+\frac{\sqrt{x^2-1} \left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right )}{2 e \left (d^2-e^2\right )^2 (d+e x)} \]

[Out]

-((c*d^2 - b*d*e + a*e^2)*Sqrt[-1 + x^2])/(2*e*(d^2 - e^2)*(d + e*x)^2) + ((c*(d
^3 - 4*d*e^2) - e*(3*a*d*e - b*(d^2 + 2*e^2)))*Sqrt[-1 + x^2])/(2*e*(d^2 - e^2)^
2*(d + e*x)) - ((3*b*d*e - a*(2*d^2 + e^2) - c*(d^2 + 2*e^2))*ArcTanh[(e + d*x)/
(Sqrt[d^2 - e^2]*Sqrt[-1 + x^2])])/(2*(d^2 - e^2)^(5/2))

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Rubi [A]  time = 0.46931, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{\sqrt{x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}-\frac{\tanh ^{-1}\left (\frac{d x+e}{\sqrt{x^2-1} \sqrt{d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{2 \left (d^2-e^2\right )^{5/2}}+\frac{\sqrt{x^2-1} \left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right )}{2 e \left (d^2-e^2\right )^2 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-((c*d^2 - b*d*e + a*e^2)*Sqrt[-1 + x^2])/(2*e*(d^2 - e^2)*(d + e*x)^2) + ((c*(d
^3 - 4*d*e^2) - e*(3*a*d*e - b*(d^2 + 2*e^2)))*Sqrt[-1 + x^2])/(2*e*(d^2 - e^2)^
2*(d + e*x)) - ((3*b*d*e - a*(2*d^2 + e^2) - c*(d^2 + 2*e^2))*ArcTanh[(e + d*x)/
(Sqrt[d^2 - e^2]*Sqrt[-1 + x^2])])/(2*(d^2 - e^2)^(5/2))

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Rubi in Sympy [A]  time = 39.1502, size = 170, normalized size = 0.87 \[ - \frac{\left (2 a d^{2} + a e^{2} - 3 b d e + c d^{2} + 2 c e^{2}\right ) \operatorname{atanh}{\left (\frac{- d x - e}{\sqrt{d^{2} - e^{2}} \sqrt{x^{2} - 1}} \right )}}{2 \left (d^{2} - e^{2}\right )^{\frac{5}{2}}} + \frac{\sqrt{x^{2} - 1} \left (- 3 a d e^{2} + b d^{2} e + 2 b e^{3} + c d^{3} - 4 c d e^{2}\right )}{2 e \left (d + e x\right ) \left (d^{2} - e^{2}\right )^{2}} - \frac{\sqrt{x^{2} - 1} \left (a e^{2} - b d e + c d^{2}\right )}{2 e \left (d + e x\right )^{2} \left (d^{2} - e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a*d**2 + a*e**2 - 3*b*d*e + c*d**2 + 2*c*e**2)*atanh((-d*x - e)/(sqrt(d**2 -
e**2)*sqrt(x**2 - 1)))/(2*(d**2 - e**2)**(5/2)) + sqrt(x**2 - 1)*(-3*a*d*e**2 +
b*d**2*e + 2*b*e**3 + c*d**3 - 4*c*d*e**2)/(2*e*(d + e*x)*(d**2 - e**2)**2) - s
qrt(x**2 - 1)*(a*e**2 - b*d*e + c*d**2)/(2*e*(d + e*x)**2*(d**2 - e**2))

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Mathematica [A]  time = 0.437397, size = 240, normalized size = 1.23 \[ \frac{1}{2} \left (-\frac{\log \left (-\sqrt{x^2-1} \sqrt{d^2-e^2}+d x+e\right ) \left (a \left (2 d^2+e^2\right )-3 b d e+c \left (d^2+2 e^2\right )\right )}{(d-e)^2 (d+e)^2 \sqrt{d^2-e^2}}+\frac{\log (d+e x) \left (a \left (2 d^2+e^2\right )-3 b d e+c \left (d^2+2 e^2\right )\right )}{(d-e)^2 (d+e)^2 \sqrt{d^2-e^2}}+\frac{\sqrt{x^2-1} \left (a e \left (-4 d^2-3 d e x+e^2\right )+b \left (2 d^3+d^2 e x+d e^2+2 e^3 x\right )+c d \left (d^2 x-3 d e-4 e^2 x\right )\right )}{\left (d^2-e^2\right )^2 (d+e x)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[-1 + x^2]*(a*e*(-4*d^2 + e^2 - 3*d*e*x) + c*d*(-3*d*e + d^2*x - 4*e^2*x)
+ b*(2*d^3 + d*e^2 + d^2*e*x + 2*e^3*x)))/((d^2 - e^2)^2*(d + e*x)^2) + ((-3*b*d
*e + a*(2*d^2 + e^2) + c*(d^2 + 2*e^2))*Log[d + e*x])/((d - e)^2*(d + e)^2*Sqrt[
d^2 - e^2]) - ((-3*b*d*e + a*(2*d^2 + e^2) + c*(d^2 + 2*e^2))*Log[e + d*x - Sqrt
[d^2 - e^2]*Sqrt[-1 + x^2]])/((d - e)^2*(d + e)^2*Sqrt[d^2 - e^2]))/2

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Maple [B]  time = 0.045, size = 1407, normalized size = 7.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/e^3/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)
^(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2))/(x+d/e))-1/e/(d^2-e^2)/(x+
d/e)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*b+2/e^2/(d^2-e^2)/(x+d/e)*((x
+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*c*d-3/2/e^2*d/(d^2-e^2)/((d^2-e^2)/e^
2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*
d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2))/(x+d/e))*b+5/2/e^3*d^2/(d^2-e^2)/((d^2-e^2)/e^
2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*
d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2))/(x+d/e))*c-1/2/e/(d^2-e^2)/(x+d/e)^2*((x+d/e)^
2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*a+1/2/e^2/(d^2-e^2)/(x+d/e)^2*((x+d/e)^2-2*
d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*b*d-1/2/e^3/(d^2-e^2)/(x+d/e)^2*((x+d/e)^2-2*d/
e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*c*d^2-3/2*d/(d^2-e^2)^2/(x+d/e)*((x+d/e)^2-2*d/e*
(x+d/e)+(d^2-e^2)/e^2)^(1/2)*a+3/2/e*d^2/(d^2-e^2)^2/(x+d/e)*((x+d/e)^2-2*d/e*(x
+d/e)+(d^2-e^2)/e^2)^(1/2)*b-3/2/e^2*d^3/(d^2-e^2)^2/(x+d/e)*((x+d/e)^2-2*d/e*(x
+d/e)+(d^2-e^2)/e^2)^(1/2)*c-3/2/e*d^2/(d^2-e^2)^2/((d^2-e^2)/e^2)^(1/2)*ln((2*(
d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2
-e^2)/e^2)^(1/2))/(x+d/e))*a+3/2/e^2*d^3/(d^2-e^2)^2/((d^2-e^2)/e^2)^(1/2)*ln((2
*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+(d
^2-e^2)/e^2)^(1/2))/(x+d/e))*b-3/2/e^3*d^4/(d^2-e^2)^2/((d^2-e^2)/e^2)^(1/2)*ln(
(2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+
(d^2-e^2)/e^2)^(1/2))/(x+d/e))*c+1/2/e/(d^2-e^2)/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^
2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e
^2)/e^2)^(1/2))/(x+d/e))*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((c*x^2 + b*x + a)/((e*x + d)^3*sqrt(x^2 - 1)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.309867, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/((e*x + d)^3*sqrt(x^2 - 1)),x, algorithm="fricas")

[Out]

[1/2*((2*b*d^3*e^2 - (4*a + 3*c)*d^2*e^3 + b*d*e^4 + a*e^5 + 2*(2*c*d^4*e - (2*a
+ 5*c)*d^2*e^3 + 3*b*d*e^4 - a*e^5)*x^2 + (2*c*d^5 + 2*b*d^4*e - (6*a + 7*c)*d^
3*e^2 + 5*b*d^2*e^3 - (3*a + 4*c)*d*e^4 + 2*b*e^5)*x)*sqrt(d^2 - e^2)*sqrt(x^2 -
1) + ((2*a + c)*d^4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4 - 2*((2*a + c)*d^2*e^
4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^4 - 4*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2
*c)*d*e^5)*x^3 - (2*(2*a + c)*d^4*e^2 - 6*b*d^3*e^3 + 3*c*d^2*e^4 + 3*b*d*e^5 -
(a + 2*c)*e^6)*x^2 + 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5)*x + 2
*(((2*a + c)*d^2*e^4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^3 + 2*((2*a + c)*d^3*e^3 - 3
*b*d^2*e^4 + (a + 2*c)*d*e^5)*x^2 + ((2*a + c)*d^4*e^2 - 3*b*d^3*e^3 + (a + 2*c)
*d^2*e^4)*x)*sqrt(x^2 - 1))*log((d^3 - d*e^2 + (d^2*e - e^3)*x + (e^2*x^2 + d*e*
x + d^2 - e^2)*sqrt(d^2 - e^2) - (d^2*e - e^3 + (e^2*x + d*e)*sqrt(d^2 - e^2))*s
qrt(x^2 - 1))/(e*x^2 + d*x - (e*x + d)*sqrt(x^2 - 1))) + (c*d^5 + b*d^4*e - (3*a
+ 4*c)*d^3*e^2 + 2*b*d^2*e^3 - 2*(2*c*d^4*e - (2*a + 5*c)*d^2*e^3 + 3*b*d*e^4 -
a*e^5)*x^3 - (2*c*d^5 + 2*b*d^4*e - (6*a + 7*c)*d^3*e^2 + 5*b*d^2*e^3 - (3*a +
4*c)*d*e^4 + 2*b*e^5)*x^2 + 2*(c*d^4*e - b*d^3*e^2 + (a - c)*d^2*e^3 + b*d*e^4 -
a*e^5)*x)*sqrt(d^2 - e^2))/(2*((d^4*e^4 - 2*d^2*e^6 + e^8)*x^3 + 2*(d^5*e^3 - 2
*d^3*e^5 + d*e^7)*x^2 + (d^6*e^2 - 2*d^4*e^4 + d^2*e^6)*x)*sqrt(d^2 - e^2)*sqrt(
x^2 - 1) + (d^6*e^2 - 2*d^4*e^4 + d^2*e^6 - 2*(d^4*e^4 - 2*d^2*e^6 + e^8)*x^4 -
4*(d^5*e^3 - 2*d^3*e^5 + d*e^7)*x^3 - (2*d^6*e^2 - 5*d^4*e^4 + 4*d^2*e^6 - e^8)*
x^2 + 2*(d^5*e^3 - 2*d^3*e^5 + d*e^7)*x)*sqrt(d^2 - e^2)), 1/2*((2*b*d^3*e^2 - (
4*a + 3*c)*d^2*e^3 + b*d*e^4 + a*e^5 + 2*(2*c*d^4*e - (2*a + 5*c)*d^2*e^3 + 3*b*
d*e^4 - a*e^5)*x^2 + (2*c*d^5 + 2*b*d^4*e - (6*a + 7*c)*d^3*e^2 + 5*b*d^2*e^3 -
(3*a + 4*c)*d*e^4 + 2*b*e^5)*x)*sqrt(-d^2 + e^2)*sqrt(x^2 - 1) + 2*((2*a + c)*d^
4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4 - 2*((2*a + c)*d^2*e^4 - 3*b*d*e^5 + (a
+ 2*c)*e^6)*x^4 - 4*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5)*x^3 - (2
*(2*a + c)*d^4*e^2 - 6*b*d^3*e^3 + 3*c*d^2*e^4 + 3*b*d*e^5 - (a + 2*c)*e^6)*x^2
+ 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5)*x + 2*(((2*a + c)*d^2*e^
4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^3 + 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2
*c)*d*e^5)*x^2 + ((2*a + c)*d^4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4)*x)*sqrt(x
^2 - 1))*arctan(-(sqrt(-d^2 + e^2)*sqrt(x^2 - 1)*e - sqrt(-d^2 + e^2)*(e*x + d))
/(d^2 - e^2)) + (c*d^5 + b*d^4*e - (3*a + 4*c)*d^3*e^2 + 2*b*d^2*e^3 - 2*(2*c*d^
4*e - (2*a + 5*c)*d^2*e^3 + 3*b*d*e^4 - a*e^5)*x^3 - (2*c*d^5 + 2*b*d^4*e - (6*a
+ 7*c)*d^3*e^2 + 5*b*d^2*e^3 - (3*a + 4*c)*d*e^4 + 2*b*e^5)*x^2 + 2*(c*d^4*e -
b*d^3*e^2 + (a - c)*d^2*e^3 + b*d*e^4 - a*e^5)*x)*sqrt(-d^2 + e^2))/(2*((d^4*e^4
- 2*d^2*e^6 + e^8)*x^3 + 2*(d^5*e^3 - 2*d^3*e^5 + d*e^7)*x^2 + (d^6*e^2 - 2*d^4
*e^4 + d^2*e^6)*x)*sqrt(-d^2 + e^2)*sqrt(x^2 - 1) + (d^6*e^2 - 2*d^4*e^4 + d^2*e
^6 - 2*(d^4*e^4 - 2*d^2*e^6 + e^8)*x^4 - 4*(d^5*e^3 - 2*d^3*e^5 + d*e^7)*x^3 - (
2*d^6*e^2 - 5*d^4*e^4 + 4*d^2*e^6 - e^8)*x^2 + 2*(d^5*e^3 - 2*d^3*e^5 + d*e^7)*x
)*sqrt(-d^2 + e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + b x + c x^{2}}{\sqrt{\left (x - 1\right ) \left (x + 1\right )} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x + c*x**2)/(sqrt((x - 1)*(x + 1))*(d + e*x)**3), x)

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GIAC/XCAS [A]  time = 0.2748, size = 724, normalized size = 3.71 \[ \frac{{\left (2 \, a d^{2} + c d^{2} - 3 \, b d e + a e^{2} + 2 \, c e^{2}\right )} \arctan \left (-\frac{{\left (x - \sqrt{x^{2} - 1}\right )} e + d}{\sqrt{-d^{2} + e^{2}}}\right )}{{\left (d^{4} - 2 \, d^{2} e^{2} + e^{4}\right )} \sqrt{-d^{2} + e^{2}}} + \frac{2 \, c d^{4}{\left (x - \sqrt{x^{2} - 1}\right )}^{3} e + 2 \, c d^{5}{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 2 \, b d^{4}{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e - 2 \, a d^{2}{\left (x - \sqrt{x^{2} - 1}\right )}^{3} e^{3} - 5 \, c d^{2}{\left (x - \sqrt{x^{2} - 1}\right )}^{3} e^{3} - 6 \, a d^{3}{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{2} - 7 \, c d^{3}{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{2} + 2 \, c d^{4}{\left (x - \sqrt{x^{2} - 1}\right )} e + 3 \, b d{\left (x - \sqrt{x^{2} - 1}\right )}^{3} e^{4} + 5 \, b d^{2}{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{3} + 4 \, b d^{3}{\left (x - \sqrt{x^{2} - 1}\right )} e^{2} - a{\left (x - \sqrt{x^{2} - 1}\right )}^{3} e^{5} - 3 \, a d{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{4} - 4 \, c d{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{4} - 10 \, a d^{2}{\left (x - \sqrt{x^{2} - 1}\right )} e^{3} - 11 \, c d^{2}{\left (x - \sqrt{x^{2} - 1}\right )} e^{3} + c d^{3} e^{2} + 2 \, b{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{5} + 5 \, b d{\left (x - \sqrt{x^{2} - 1}\right )} e^{4} + b d^{2} e^{3} + a{\left (x - \sqrt{x^{2} - 1}\right )} e^{5} - 3 \, a d e^{4} - 4 \, c d e^{4} + 2 \, b e^{5}}{{\left (d^{4} e^{2} - 2 \, d^{2} e^{4} + e^{6}\right )}{\left ({\left (x - \sqrt{x^{2} - 1}\right )}^{2} e + 2 \, d{\left (x - \sqrt{x^{2} - 1}\right )} + e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/((e*x + d)^3*sqrt(x^2 - 1)),x, algorithm="giac")

[Out]

(2*a*d^2 + c*d^2 - 3*b*d*e + a*e^2 + 2*c*e^2)*arctan(-((x - sqrt(x^2 - 1))*e + d
)/sqrt(-d^2 + e^2))/((d^4 - 2*d^2*e^2 + e^4)*sqrt(-d^2 + e^2)) + (2*c*d^4*(x - s
qrt(x^2 - 1))^3*e + 2*c*d^5*(x - sqrt(x^2 - 1))^2 + 2*b*d^4*(x - sqrt(x^2 - 1))^
2*e - 2*a*d^2*(x - sqrt(x^2 - 1))^3*e^3 - 5*c*d^2*(x - sqrt(x^2 - 1))^3*e^3 - 6*
a*d^3*(x - sqrt(x^2 - 1))^2*e^2 - 7*c*d^3*(x - sqrt(x^2 - 1))^2*e^2 + 2*c*d^4*(x
- sqrt(x^2 - 1))*e + 3*b*d*(x - sqrt(x^2 - 1))^3*e^4 + 5*b*d^2*(x - sqrt(x^2 -
1))^2*e^3 + 4*b*d^3*(x - sqrt(x^2 - 1))*e^2 - a*(x - sqrt(x^2 - 1))^3*e^5 - 3*a*
d*(x - sqrt(x^2 - 1))^2*e^4 - 4*c*d*(x - sqrt(x^2 - 1))^2*e^4 - 10*a*d^2*(x - sq
rt(x^2 - 1))*e^3 - 11*c*d^2*(x - sqrt(x^2 - 1))*e^3 + c*d^3*e^2 + 2*b*(x - sqrt(
x^2 - 1))^2*e^5 + 5*b*d*(x - sqrt(x^2 - 1))*e^4 + b*d^2*e^3 + a*(x - sqrt(x^2 -
1))*e^5 - 3*a*d*e^4 - 4*c*d*e^4 + 2*b*e^5)/((d^4*e^2 - 2*d^2*e^4 + e^6)*((x - sq
rt(x^2 - 1))^2*e + 2*d*(x - sqrt(x^2 - 1)) + e)^2)