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

Optimal. Leaf size=199 \[ -\frac{\sqrt{x-1} \sqrt{x+1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac{\sqrt{x-1} \sqrt{x+1} \left (-d e^2 (3 a+4 c)+b d^2 e+2 b e^3+c d^3\right )}{2 e \left (d^2-e^2\right )^2 (d+e x)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x+1} \sqrt{d+e}}{\sqrt{x-1} \sqrt{d-e}}\right ) \left (d^2 (2 a+c)+e^2 (a+2 c)-3 b d e\right )}{(d-e)^{5/2} (d+e)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.328291, antiderivative size = 242, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1610, 1651, 807, 725, 206} \[ -\frac{\left (1-x^2\right ) \left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right )}{2 e \sqrt{x-1} \sqrt{x+1} \left (d^2-e^2\right )^2 (d+e x)}+\frac{\left (1-x^2\right ) \left (a e^2-b d e+c d^2\right )}{2 e \sqrt{x-1} \sqrt{x+1} \left (d^2-e^2\right ) (d+e x)^2}-\frac{\sqrt{x^2-1} \tanh ^{-1}\left (\frac{d x+e}{\sqrt{x^2-1} \sqrt{d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{2 \sqrt{x-1} \sqrt{x+1} \left (d^2-e^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{a+b x+c x^2}{\sqrt{-1+x} \sqrt{1+x} (d+e x)^3} \, dx &=\frac{\sqrt{-1+x^2} \int \frac{a+b x+c x^2}{(d+e x)^3 \sqrt{-1+x^2}} \, dx}{\sqrt{-1+x} \sqrt{1+x}}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \left (1-x^2\right )}{2 e \left (d^2-e^2\right ) \sqrt{-1+x} \sqrt{1+x} (d+e x)^2}-\frac{\sqrt{-1+x^2} \int \frac{-2 (a d+c d-b e)-\left (b d+\frac{c d^2}{e}-a e-2 c e\right ) x}{(d+e x)^2 \sqrt{-1+x^2}} \, dx}{2 \left (d^2-e^2\right ) \sqrt{-1+x} \sqrt{1+x}}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \left (1-x^2\right )}{2 e \left (d^2-e^2\right ) \sqrt{-1+x} \sqrt{1+x} (d+e x)^2}-\frac{\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 ) \left (1-x^2\right )}{2 e \left (d^2-e^2\right )^2 \sqrt{-1+x} \sqrt{1+x} (d+e x)}-\frac{\left (\left (-2 d (a d+c d-b e)-e \left (-b d-\frac{c d^2}{e}+a e+2 c e\right )\right ) \sqrt{-1+x^2}\right ) \int \frac{1}{(d+e x) \sqrt{-1+x^2}} \, dx}{2 \left (d^2-e^2\right )^2 \sqrt{-1+x} \sqrt{1+x}}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \left (1-x^2\right )}{2 e \left (d^2-e^2\right ) \sqrt{-1+x} \sqrt{1+x} (d+e x)^2}-\frac{\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 ) \left (1-x^2\right )}{2 e \left (d^2-e^2\right )^2 \sqrt{-1+x} \sqrt{1+x} (d+e x)}+\frac{\left (\left (-2 d (a d+c d-b e)-e \left (-b d-\frac{c d^2}{e}+a e+2 c e\right )\right ) \sqrt{-1+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{d^2-e^2-x^2} \, dx,x,\frac{-e-d x}{\sqrt{-1+x^2}}\right )}{2 \left (d^2-e^2\right )^2 \sqrt{-1+x} \sqrt{1+x}}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \left (1-x^2\right )}{2 e \left (d^2-e^2\right ) \sqrt{-1+x} \sqrt{1+x} (d+e x)^2}-\frac{\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 ) \left (1-x^2\right )}{2 e \left (d^2-e^2\right )^2 \sqrt{-1+x} \sqrt{1+x} (d+e x)}-\frac{\left (3 b d e-a \left (2 d^2+e^2\right )-c \left (d^2+2 e^2\right )\right ) \sqrt{-1+x^2} \tanh ^{-1}\left (\frac{e+d x}{\sqrt{d^2-e^2} \sqrt{-1+x^2}}\right )}{2 \left (d^2-e^2\right )^{5/2} \sqrt{-1+x} \sqrt{1+x}}\\ \end{align*}

Mathematica [A]  time = 0.8129, size = 336, normalized size = 1.69 \[ \frac{\frac{\left (2 \left (2 d^2+e^2\right ) (d+e x) \tan ^{-1}\left (\frac{\sqrt{\frac{x-1}{x+1}} \sqrt{e-d}}{\sqrt{d+e}}\right )-3 d e \sqrt{x-1} \sqrt{x+1} \sqrt{e-d} \sqrt{d+e}\right ) \left (e (a e-b d)+c d^2\right )}{(e-d)^{5/2} (d+e)^{5/2} (d+e x)}-\frac{e \sqrt{x-1} \sqrt{x+1} \left (e (a e-b d)+c d^2\right )}{(d-e) (d+e) (d+e x)^2}+\frac{2 e \sqrt{x-1} \sqrt{x+1} (2 c d-b e)}{(d-e) (d+e) (d+e x)}+\frac{4 d (2 c d-b e) \tan ^{-1}\left (\frac{\sqrt{\frac{x-1}{x+1}} \sqrt{e-d}}{\sqrt{d+e}}\right )}{(e-d)^{3/2} (d+e)^{3/2}}+\frac{4 c \tan ^{-1}\left (\frac{\sqrt{\frac{x-1}{x+1}} \sqrt{e-d}}{\sqrt{d+e}}\right )}{\sqrt{e-d} \sqrt{d+e}}}{2 e^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.052, size = 1095, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-2*b*d^3*e*((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)-b*d*e^3*((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)+3*c*d^2*e^2*(
(d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)+2*ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*x^2*a*d^2*e
^2-3*ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*x^2*b*d*e^3+ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(
x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*x^2*c*d^2*e^2+4*ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*x
*a*d^3*e+2*ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*x*a*d*e^3-6*ln(-2*(-((d^2-e^2)/e^2)^(
1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*x*b*d^2*e^2+4*a*d^2*e^2*((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)-x*b*d^2*e^2*(
(d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)+ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*x^2*a*e^4+2*l
n(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*x^2*c*e^4+ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(
1/2)*e+d*x+e)/(e*x+d))*a*d^2*e^2-3*ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*b*d^3*e+2*ln(
-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*c*d^2*e^2-a*e^4*((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)
+2*ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*a*d^4+ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(
1/2)*e+d*x+e)/(e*x+d))*c*d^4-x*c*d^3*e*((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)+4*x*c*d*e^3*((d^2-e^2)/e^2)^(1/2)*(
x^2-1)^(1/2)+3*x*a*d*e^3*((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)+2*ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d
*x+e)/(e*x+d))*x*c*d^3*e+4*ln(-2*(-((d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2)*e+d*x+e)/(e*x+d))*x*c*d*e^3-2*x*b*e^4*(
(d^2-e^2)/e^2)^(1/2)*(x^2-1)^(1/2))*(1+x)^(1/2)*(-1+x)^(1/2)/(x^2-1)^(1/2)/(d-e)/(d+e)/(d^2-e^2)/(e*x+d)^2/((d
^2-e^2)/e^2)^(1/2)/e

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.24824, size = 2485, normalized size = 12.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(c*d^7 + b*d^6*e - (3*a + 5*c)*d^5*e^2 + b*d^4*e^3 + (3*a + 4*c)*d^3*e^4 - 2*b*d^2*e^5 + (c*d^5*e^2 + b*d
^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x^2 + ((2*a + c)*d^4*e^2 - 3*b*d^3*e^3
+ (a + 2*c)*d^2*e^4 + ((2*a + 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)*sqrt(d^2 - e^2)*log((d^2*x + d*e + (d^2 - e^2 + sqrt(d^2 - e^2)*d)*sqrt(x + 1)*sqrt(x
- 1) + sqrt(d^2 - e^2)*(d*x + e))/(e*x + d)) + (2*b*d^5*e^2 - (4*a + 3*c)*d^4*e^3 - b*d^3*e^4 + (5*a + 3*c)*d^
2*e^5 - b*d*e^6 - a*e^7 + (c*d^5*e^2 + b*d^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 - 2*b*e
^7)*x)*sqrt(x + 1)*sqrt(x - 1) + 2*(c*d^6*e + b*d^5*e^2 - (3*a + 5*c)*d^4*e^3 + b*d^3*e^4 + (3*a + 4*c)*d^2*e^
5 - 2*b*d*e^6)*x)/(d^8*e^2 - 3*d^6*e^4 + 3*d^4*e^6 - d^2*e^8 + (d^6*e^4 - 3*d^4*e^6 + 3*d^2*e^8 - e^10)*x^2 +
2*(d^7*e^3 - 3*d^5*e^5 + 3*d^3*e^7 - d*e^9)*x), 1/2*(c*d^7 + b*d^6*e - (3*a + 5*c)*d^5*e^2 + b*d^4*e^3 + (3*a
+ 4*c)*d^3*e^4 - 2*b*d^2*e^5 + (c*d^5*e^2 + b*d^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 -
2*b*e^7)*x^2 - 2*((2*a + c)*d^4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4 + ((2*a + 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)*sqrt(-d^2 + e^2)*arctan(-(sqrt(-d^2 +
e^2)*e*sqrt(x + 1)*sqrt(x - 1) - sqrt(-d^2 + e^2)*(e*x + d))/(d^2 - e^2)) + (2*b*d^5*e^2 - (4*a + 3*c)*d^4*e^
3 - b*d^3*e^4 + (5*a + 3*c)*d^2*e^5 - b*d*e^6 - a*e^7 + (c*d^5*e^2 + b*d^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e
^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x)*sqrt(x + 1)*sqrt(x - 1) + 2*(c*d^6*e + b*d^5*e^2 - (3*a + 5*c)*d^4*e^3 +
b*d^3*e^4 + (3*a + 4*c)*d^2*e^5 - 2*b*d*e^6)*x)/(d^8*e^2 - 3*d^6*e^4 + 3*d^4*e^6 - d^2*e^8 + (d^6*e^4 - 3*d^4*
e^6 + 3*d^2*e^8 - e^10)*x^2 + 2*(d^7*e^3 - 3*d^5*e^5 + 3*d^3*e^7 - d*e^9)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 3.45873, size = 817, normalized size = 4.11 \begin{align*} -\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 (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e + 2 \, d}{2 \, \sqrt{-d^{2} + e^{2}}}\right )}{{\left (d^{4} - 2 \, d^{2} e^{2} + e^{4}\right )} \sqrt{-d^{2} + e^{2}}} + \frac{2 \,{\left (2 \, c d^{4}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{6} e + 4 \, c d^{5}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} - 2 \, a d^{2}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{6} e^{3} - 5 \, c d^{2}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{6} e^{3} + 4 \, b d^{4}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e + 3 \, b d{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{6} e^{4} - 12 \, a d^{3}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{2} - 14 \, c d^{3}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{2} - a{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{6} e^{5} + 10 \, b d^{2}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{3} + 8 \, c d^{4}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e - 6 \, a d{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{4} - 8 \, c d{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{4} + 16 \, b d^{3}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e^{2} + 4 \, b{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{5} - 40 \, a d^{2}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e^{3} - 44 \, c d^{2}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e^{3} + 20 \, b d{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e^{4} + 8 \, c d^{3} e^{2} + 4 \, a{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e^{5} + 8 \, b d^{2} e^{3} - 24 \, a d e^{4} - 32 \, c d e^{4} + 16 \, b e^{5}\right )}}{{\left (d^{4} e^{2} - 2 \, d^{2} e^{4} + e^{6}\right )}{\left ({\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e + 4 \, d{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} + 4 \, e\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*d^2 + c*d^2 - 3*b*d*e + a*e^2 + 2*c*e^2)*arctan(1/2*((sqrt(x + 1) - sqrt(x - 1))^2*e + 2*d)/sqrt(-d^2 +
e^2))/((d^4 - 2*d^2*e^2 + e^4)*sqrt(-d^2 + e^2)) + 2*(2*c*d^4*(sqrt(x + 1) - sqrt(x - 1))^6*e + 4*c*d^5*(sqrt(
x + 1) - sqrt(x - 1))^4 - 2*a*d^2*(sqrt(x + 1) - sqrt(x - 1))^6*e^3 - 5*c*d^2*(sqrt(x + 1) - sqrt(x - 1))^6*e^
3 + 4*b*d^4*(sqrt(x + 1) - sqrt(x - 1))^4*e + 3*b*d*(sqrt(x + 1) - sqrt(x - 1))^6*e^4 - 12*a*d^3*(sqrt(x + 1)
- sqrt(x - 1))^4*e^2 - 14*c*d^3*(sqrt(x + 1) - sqrt(x - 1))^4*e^2 - a*(sqrt(x + 1) - sqrt(x - 1))^6*e^5 + 10*b
*d^2*(sqrt(x + 1) - sqrt(x - 1))^4*e^3 + 8*c*d^4*(sqrt(x + 1) - sqrt(x - 1))^2*e - 6*a*d*(sqrt(x + 1) - sqrt(x
- 1))^4*e^4 - 8*c*d*(sqrt(x + 1) - sqrt(x - 1))^4*e^4 + 16*b*d^3*(sqrt(x + 1) - sqrt(x - 1))^2*e^2 + 4*b*(sqr
t(x + 1) - sqrt(x - 1))^4*e^5 - 40*a*d^2*(sqrt(x + 1) - sqrt(x - 1))^2*e^3 - 44*c*d^2*(sqrt(x + 1) - sqrt(x -
1))^2*e^3 + 20*b*d*(sqrt(x + 1) - sqrt(x - 1))^2*e^4 + 8*c*d^3*e^2 + 4*a*(sqrt(x + 1) - sqrt(x - 1))^2*e^5 + 8
*b*d^2*e^3 - 24*a*d*e^4 - 32*c*d*e^4 + 16*b*e^5)/((d^4*e^2 - 2*d^2*e^4 + e^6)*((sqrt(x + 1) - sqrt(x - 1))^4*e
+ 4*d*(sqrt(x + 1) - sqrt(x - 1))^2 + 4*e)^2)