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

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

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Rubi [A]  time = 0.33, 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} \begin {gather*} -\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}} \end {gather*}

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 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])

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 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 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]

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*}

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Mathematica [A]  time = 0.76, size = 343, normalized size = 1.72 \begin {gather*} \frac {-(d+e x) \left (3 d e \sqrt {x-1} \sqrt {x+1} \sqrt {d-e} \sqrt {d+e}-2 \left (2 d^2+e^2\right ) (d+e x) \tanh ^{-1}\left (\frac {\sqrt {\frac {x-1}{x+1}} \sqrt {d-e}}{\sqrt {d+e}}\right )\right ) \left (e (a e-b d)+c d^2\right )-e \sqrt {x-1} \sqrt {x+1} (d-e)^{3/2} (d+e)^{3/2} \left (e (a e-b d)+c d^2\right )+2 e \sqrt {x-1} \sqrt {x+1} (d-e)^{3/2} (d+e)^{3/2} (d+e x) (2 c d-b e)-4 d (d-e) (d+e) (d+e x)^2 (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {\frac {x-1}{x+1}} \sqrt {d-e}}{\sqrt {d+e}}\right )+4 c (d-e)^2 (d+e)^2 (d+e x)^2 \tanh ^{-1}\left (\frac {\sqrt {\frac {x-1}{x+1}} \sqrt {d-e}}{\sqrt {d+e}}\right )}{2 e^2 (d-e)^{5/2} (d+e)^{5/2} (d+e x)^2} \end {gather*}

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]

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

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IntegrateAlgebraic [B]  time = 0.67, size = 546, normalized size = 2.74 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {x-1} \sqrt {-d-e} \sqrt {d-e}}{\sqrt {x+1} (d+e)}\right ) \left (2 a d^2+a e^2-3 b d e+c d^2+2 c e^2\right )}{\sqrt {-d-e} (d-e)^{5/2} (d+e)^2}+\frac {-\frac {4 a d^2 e \sqrt {x-1}}{\sqrt {x+1}}+\frac {4 a d^2 e (x-1)^{3/2}}{(x+1)^{3/2}}-\frac {3 a d e^2 \sqrt {x-1}}{\sqrt {x+1}}-\frac {3 a d e^2 (x-1)^{3/2}}{(x+1)^{3/2}}+\frac {a e^3 \sqrt {x-1}}{\sqrt {x+1}}-\frac {a e^3 (x-1)^{3/2}}{(x+1)^{3/2}}+\frac {2 b d^3 \sqrt {x-1}}{\sqrt {x+1}}-\frac {2 b d^3 (x-1)^{3/2}}{(x+1)^{3/2}}+\frac {b d^2 e \sqrt {x-1}}{\sqrt {x+1}}+\frac {b d^2 e (x-1)^{3/2}}{(x+1)^{3/2}}+\frac {b d e^2 \sqrt {x-1}}{\sqrt {x+1}}-\frac {b d e^2 (x-1)^{3/2}}{(x+1)^{3/2}}+\frac {2 b e^3 \sqrt {x-1}}{\sqrt {x+1}}+\frac {2 b e^3 (x-1)^{3/2}}{(x+1)^{3/2}}+\frac {c d^3 \sqrt {x-1}}{\sqrt {x+1}}+\frac {c d^3 (x-1)^{3/2}}{(x+1)^{3/2}}-\frac {3 c d^2 e \sqrt {x-1}}{\sqrt {x+1}}+\frac {3 c d^2 e (x-1)^{3/2}}{(x+1)^{3/2}}-\frac {4 c d e^2 \sqrt {x-1}}{\sqrt {x+1}}-\frac {4 c d e^2 (x-1)^{3/2}}{(x+1)^{3/2}}}{(d-e)^2 (d+e)^2 \left (\frac {d (x-1)}{x+1}-d-\frac {e (x-1)}{x+1}-e\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.00, size = 1186, normalized size = 5.96

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/(-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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giac [B]  time = 3.24, size = 605, normalized size = 3.04 \begin {gather*} -\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 {gather*}

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)

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maple [B]  time = 0.05, size = 1095, normalized size = 5.50 \begin {gather*} -\frac {\left (2 a \,d^{2} e^{2} x^{2} \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+a \,e^{4} x^{2} \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )-3 b d \,e^{3} x^{2} \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+c \,d^{2} e^{2} x^{2} \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+2 c \,e^{4} x^{2} \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+4 a \,d^{3} e x \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+2 a d \,e^{3} x \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )-6 b \,d^{2} e^{2} x \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+2 c \,d^{3} e x \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+4 c d \,e^{3} x \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+2 a \,d^{4} \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+a \,d^{2} e^{2} \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+3 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, a d \,e^{3} x -3 b \,d^{3} e \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )-\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, b \,d^{2} e^{2} x -2 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, b \,e^{4} x +c \,d^{4} \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )-\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, c \,d^{3} e x +2 c \,d^{2} e^{2} \ln \left (-\frac {2 \left (d x -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, e +e \right )}{e x +d}\right )+4 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, c d \,e^{3} x +4 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, a \,d^{2} e^{2}-\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, a \,e^{4}-2 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, b \,d^{3} e -\sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, b d \,e^{3}+3 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {x^{2}-1}\, c \,d^{2} e^{2}\right ) \sqrt {x +1}\, \sqrt {x -1}}{2 \sqrt {x^{2}-1}\, \left (d -e \right ) \left (d +e \right ) \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \left (d^{2}-e^{2}\right ) \left (e x +d \right )^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-d>0)', see `assume?` for mor
e details)Is e-d positive, negative or zero?

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mupad [B]  time = 66.85, size = 7235, normalized size = 36.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((((x - 1)^(1/2) - 1i)^2*(2*c*e^3 + c*d^2*e)*12i)/(d^2*((x + 1)^(1/2) - 1)^2*(d^4 + e^4 - 2*d^2*e^2)) - (2*(7*
c*d^4 + 14*c*d^2*e^2)*((x - 1)^(1/2) - 1i))/(7*d^3*((x + 1)^(1/2) - 1)*(d^4 + e^4 - 2*d^2*e^2)) + (((x - 1)^(1
/2) - 1i)^4*(2*c*e^3 - c*d^2*e)*24i)/(d^2*((x + 1)^(1/2) - 1)^4*(d^4 + e^4 - 2*d^2*e^2)) - (2*(21*c*d^4 - 102*
c*d^2*e^2)*((x - 1)^(1/2) - 1i)^5)/(3*d^3*((x + 1)^(1/2) - 1)^5*(d^4 + e^4 - 2*d^2*e^2)) - (2*(35*c*d^4 - 170*
c*d^2*e^2)*((x - 1)^(1/2) - 1i)^3)/(5*d^3*((x + 1)^(1/2) - 1)^3*(d^4 + e^4 - 2*d^2*e^2)) + (c*((x - 1)^(1/2) -
1i)^7*(d^2*1i + e^2*2i)*2i)/(d*((x + 1)^(1/2) - 1)^7*(d^4 + e^4 - 2*d^2*e^2)) + (12*c*e*((x - 1)^(1/2) - 1i)^
6*(d^2*1i + e^2*2i))/(d^2*((x + 1)^(1/2) - 1)^6*(d^4 + e^4 - 2*d^2*e^2)))/(((x - 1)^(1/2) - 1i)^8/((x + 1)^(1/
2) - 1)^8 - (e*((x - 1)^(1/2) - 1i)*8i)/(d*((x + 1)^(1/2) - 1)) + (e*((x - 1)^(1/2) - 1i)^3*8i)/(d*((x + 1)^(1
/2) - 1)^3) + (e*((x - 1)^(1/2) - 1i)^5*8i)/(d*((x + 1)^(1/2) - 1)^5) - (e*((x - 1)^(1/2) - 1i)^7*8i)/(d*((x +
1)^(1/2) - 1)^7) - (((x - 1)^(1/2) - 1i)^2*(4*d^2 + 16*e^2))/(d^2*((x + 1)^(1/2) - 1)^2) - (((x - 1)^(1/2) -
1i)^6*(4*d^2 + 16*e^2))/(d^2*((x + 1)^(1/2) - 1)^6) + (((x - 1)^(1/2) - 1i)^4*(6*d^2 - 32*e^2))/(d^2*((x + 1)^
(1/2) - 1)^4) + 1) - ((2*((x - 1)^(1/2) - 1i)^3*(16*b*e^3 + 11*b*d^2*e))/(d^2*((x + 1)^(1/2) - 1)^3*(d^4 + e^4
- 2*d^2*e^2)) - (6*b*e*((x - 1)^(1/2) - 1i)^7)/(((x + 1)^(1/2) - 1)^7*(d^4 + e^4 - 2*d^2*e^2)) - (6*b*e*((x -
1)^(1/2) - 1i))/(((x + 1)^(1/2) - 1)*(d^4 + e^4 - 2*d^2*e^2)) + (((x - 1)^(1/2) - 1i)^4*(2*b*e^4 - 2*b*d^4 +
3*b*d^2*e^2)*8i)/(d^3*((x + 1)^(1/2) - 1)^4*(d^4 + e^4 - 2*d^2*e^2)) + (b*((x - 1)^(1/2) - 1i)^2*(2*d^4 + 2*e^
4 + 5*d^2*e^2)*4i)/(d^3*((x + 1)^(1/2) - 1)^2*(d^4 + e^4 - 2*d^2*e^2)) + (b*((x - 1)^(1/2) - 1i)^6*(2*d^4 + 2*
e^4 + 5*d^2*e^2)*4i)/(d^3*((x + 1)^(1/2) - 1)^6*(d^4 + e^4 - 2*d^2*e^2)) + (2*b*e*((x - 1)^(1/2) - 1i)^5*(11*d
^2 + 16*e^2))/(d^2*((x + 1)^(1/2) - 1)^5*(d^4 + e^4 - 2*d^2*e^2)))/(((x - 1)^(1/2) - 1i)^8/((x + 1)^(1/2) - 1)
^8 - (e*((x - 1)^(1/2) - 1i)*8i)/(d*((x + 1)^(1/2) - 1)) + (e*((x - 1)^(1/2) - 1i)^3*8i)/(d*((x + 1)^(1/2) - 1
)^3) + (e*((x - 1)^(1/2) - 1i)^5*8i)/(d*((x + 1)^(1/2) - 1)^5) - (e*((x - 1)^(1/2) - 1i)^7*8i)/(d*((x + 1)^(1/
2) - 1)^7) - (((x - 1)^(1/2) - 1i)^2*(4*d^2 + 16*e^2))/(d^2*((x + 1)^(1/2) - 1)^2) - (((x - 1)^(1/2) - 1i)^6*(
4*d^2 + 16*e^2))/(d^2*((x + 1)^(1/2) - 1)^6) + (((x - 1)^(1/2) - 1i)^4*(6*d^2 - 32*e^2))/(d^2*((x + 1)^(1/2) -
1)^4) + 1) + ((2*(2*a*e^4 - 5*a*d^2*e^2)*((x - 1)^(1/2) - 1i))/(d^3*((x + 1)^(1/2) - 1)*(d^4 + e^4 - 2*d^2*e^
2)) - (((x - 1)^(1/2) - 1i)^4*(2*a*e^5 - 9*a*d^2*e^3 + 4*a*d^4*e)*8i)/(d^4*((x + 1)^(1/2) - 1)^4*(d^4 + e^4 -
2*d^2*e^2)) + (2*(2*a*e^4 - 5*a*d^2*e^2)*((x - 1)^(1/2) - 1i)^7)/(d^3*((x + 1)^(1/2) - 1)^7*(d^4 + e^4 - 2*d^2
*e^2)) - (2*(2*a*e^4 - 29*a*d^2*e^2)*((x - 1)^(1/2) - 1i)^3)/(d^3*((x + 1)^(1/2) - 1)^3*(d^4 + e^4 - 2*d^2*e^2
)) - (2*(2*a*e^4 - 29*a*d^2*e^2)*((x - 1)^(1/2) - 1i)^5)/(d^3*((x + 1)^(1/2) - 1)^5*(d^4 + e^4 - 2*d^2*e^2)) +
(e*((x - 1)^(1/2) - 1i)^2*(4*a*d^4 - 2*a*e^4 + 7*a*d^2*e^2)*4i)/(d^4*((x + 1)^(1/2) - 1)^2*(d^4 + e^4 - 2*d^2
*e^2)) + (e*((x - 1)^(1/2) - 1i)^6*(4*a*d^4 - 2*a*e^4 + 7*a*d^2*e^2)*4i)/(d^4*((x + 1)^(1/2) - 1)^6*(d^4 + e^4
- 2*d^2*e^2)))/(((x - 1)^(1/2) - 1i)^8/((x + 1)^(1/2) - 1)^8 - (e*((x - 1)^(1/2) - 1i)*8i)/(d*((x + 1)^(1/2)
- 1)) + (e*((x - 1)^(1/2) - 1i)^3*8i)/(d*((x + 1)^(1/2) - 1)^3) + (e*((x - 1)^(1/2) - 1i)^5*8i)/(d*((x + 1)^(1
/2) - 1)^5) - (e*((x - 1)^(1/2) - 1i)^7*8i)/(d*((x + 1)^(1/2) - 1)^7) - (((x - 1)^(1/2) - 1i)^2*(4*d^2 + 16*e^
2))/(d^2*((x + 1)^(1/2) - 1)^2) - (((x - 1)^(1/2) - 1i)^6*(4*d^2 + 16*e^2))/(d^2*((x + 1)^(1/2) - 1)^6) + (((x
- 1)^(1/2) - 1i)^4*(6*d^2 - 32*e^2))/(d^2*((x + 1)^(1/2) - 1)^4) + 1) - (c*atan(((c*(d^2 + 2*e^2)*((4*(c*e^7*
8i - c*d^2*e^5*12i + c*d^6*e*4i))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1
i)^2*(c*e^7*8i - c*d^2*e^5*12i + c*d^6*e*4i))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 -
4*d^8*e^2)) - (c*(d^2 + 2*e^2)*((e*((x - 1)^(1/2) - 1i)*64i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 + 4*e^10 -
12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8*e^2))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((
x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52*d^2*e^8 - 88*d^4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1)^(1/2) -
1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2))))/(2*(d + e)^(5/2)*(d - e)^(5/2)))*1i)/(2*(d + e)^
(5/2)*(d - e)^(5/2)) + (c*(d^2 + 2*e^2)*((4*(c*e^7*8i - c*d^2*e^5*12i + c*d^6*e*4i))/(d^10 + d^2*e^8 - 4*d^4*e
^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(c*e^7*8i - c*d^2*e^5*12i + c*d^6*e*4i))/(((x + 1)^(1/
2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2)) + (c*(d^2 + 2*e^2)*((e*((x - 1)^(1/2) - 1i)*64
i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 + 4*e^10 - 12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8*e^2))/(d^10 + d
^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52*d^2*e^8 - 88*d^
4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2))
))/(2*(d + e)^(5/2)*(d - e)^(5/2)))*1i)/(2*(d + e)^(5/2)*(d - e)^(5/2)))/((8*(c^2*d^4 + 4*c^2*e^4 + 4*c^2*d^2*
e^2))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) - (8*((x - 1)^(1/2) - 1i)^2*(c^2*d^4 + 4*c^2*e^4 +
4*c^2*d^2*e^2))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2)) - (c*(d^2 + 2*e^2
)*((4*(c*e^7*8i - c*d^2*e^5*12i + c*d^6*e*4i))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x -
1)^(1/2) - 1i)^2*(c*e^7*8i - c*d^2*e^5*12i + c*d^6*e*4i))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6
+ 6*d^6*e^4 - 4*d^8*e^2)) - (c*(d^2 + 2*e^2)*((e*((x - 1)^(1/2) - 1i)*64i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^1
0 + 4*e^10 - 12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8*e^2))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8
*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52*d^2*e^8 - 88*d^4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x
+ 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2))))/(2*(d + e)^(5/2)*(d - e)^(5/2))))/(
2*(d + e)^(5/2)*(d - e)^(5/2)) + (c*(d^2 + 2*e^2)*((4*(c*e^7*8i - c*d^2*e^5*12i + c*d^6*e*4i))/(d^10 + d^2*e^8
- 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(c*e^7*8i - c*d^2*e^5*12i + c*d^6*e*4i))/(((
x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2)) + (c*(d^2 + 2*e^2)*((e*((x - 1)^(1/2
) - 1i)*64i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 + 4*e^10 - 12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8*e^2))
/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52*d^2*e
^8 - 88*d^4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4
*d^8*e^2))))/(2*(d + e)^(5/2)*(d - e)^(5/2))))/(2*(d + e)^(5/2)*(d - e)^(5/2))))*(d^2 + 2*e^2)*1i)/((d + e)^(5
/2)*(d - e)^(5/2)) - (a*atan(((a*(2*d^2 + e^2)*((4*(a*e^7*4i - a*d^4*e^3*12i + a*d^6*e*8i))/(d^10 + d^2*e^8 -
4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(a*e^7*4i - a*d^4*e^3*12i + a*d^6*e*8i))/(((x +
1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2)) - (a*(2*d^2 + e^2)*((e*((x - 1)^(1/2) -
1i)*64i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 + 4*e^10 - 12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8*e^2))/(d
^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52*d^2*e^8
- 88*d^4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^
8*e^2))))/(2*(d + e)^(5/2)*(d - e)^(5/2)))*1i)/(2*(d + e)^(5/2)*(d - e)^(5/2)) + (a*(2*d^2 + e^2)*((4*(a*e^7*4
i - a*d^4*e^3*12i + a*d^6*e*8i))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i
)^2*(a*e^7*4i - a*d^4*e^3*12i + a*d^6*e*8i))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 -
4*d^8*e^2)) + (a*(2*d^2 + e^2)*((e*((x - 1)^(1/2) - 1i)*64i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 + 4*e^10 - 1
2*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8*e^2))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x
- 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52*d^2*e^8 - 88*d^4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1)^(1/2) -
1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2))))/(2*(d + e)^(5/2)*(d - e)^(5/2)))*1i)/(2*(d + e)^(
5/2)*(d - e)^(5/2)))/((8*(4*a^2*d^4 + a^2*e^4 + 4*a^2*d^2*e^2))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^
8*e^2) - (8*((x - 1)^(1/2) - 1i)^2*(4*a^2*d^4 + a^2*e^4 + 4*a^2*d^2*e^2))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e
^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2)) - (a*(2*d^2 + e^2)*((4*(a*e^7*4i - a*d^4*e^3*12i + a*d^6*e*8i))/(d^10
+ d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(a*e^7*4i - a*d^4*e^3*12i + a*d^6*
e*8i))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2)) - (a*(2*d^2 + e^2)*((e*((x
- 1)^(1/2) - 1i)*64i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 + 4*e^10 - 12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12
*d^8*e^2))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10
+ 52*d^2*e^8 - 88*d^4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d
^6*e^4 - 4*d^8*e^2))))/(2*(d + e)^(5/2)*(d - e)^(5/2))))/(2*(d + e)^(5/2)*(d - e)^(5/2)) + (a*(2*d^2 + e^2)*((
4*(a*e^7*4i - a*d^4*e^3*12i + a*d^6*e*8i))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^
(1/2) - 1i)^2*(a*e^7*4i - a*d^4*e^3*12i + a*d^6*e*8i))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*
d^6*e^4 - 4*d^8*e^2)) + (a*(2*d^2 + e^2)*((e*((x - 1)^(1/2) - 1i)*64i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 +
4*e^10 - 12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8*e^2))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2
) + (4*((x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52*d^2*e^8 - 88*d^4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1
)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2))))/(2*(d + e)^(5/2)*(d - e)^(5/2))))/(2*(d
+ e)^(5/2)*(d - e)^(5/2))))*(2*d^2 + e^2)*1i)/((d + e)^(5/2)*(d - e)^(5/2)) + (b*d*e*atan(((b*d*e*((4*(b*d^5*
e^2*12i - b*d^3*e^4*24i + b*d*e^6*12i))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/
2) - 1i)^2*(b*d^5*e^2*12i - b*d^3*e^4*24i + b*d*e^6*12i))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 +
6*d^6*e^4 - 4*d^8*e^2)) - (3*b*d*e*((e*((x - 1)^(1/2) - 1i)*64i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 + 4*e^1
0 - 12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8*e^2))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (
4*((x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52*d^2*e^8 - 88*d^4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1)^(1/
2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2))))/(2*(d + e)^(5/2)*(d - e)^(5/2)))*3i)/(2*(d +
e)^(5/2)*(d - e)^(5/2)) + (b*d*e*((4*(b*d^5*e^2*12i - b*d^3*e^4*24i + b*d*e^6*12i))/(d^10 + d^2*e^8 - 4*d^4*e
^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(b*d^5*e^2*12i - b*d^3*e^4*24i + b*d*e^6*12i))/(((x +
1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2)) + (3*b*d*e*((e*((x - 1)^(1/2) - 1i)*64i)
/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 + 4*e^10 - 12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8*e^2))/(d^10 + d^2
*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52*d^2*e^8 - 88*d^4*
e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2))))
/(2*(d + e)^(5/2)*(d - e)^(5/2)))*3i)/(2*(d + e)^(5/2)*(d - e)^(5/2)))/((72*b^2*d^2*e^2)/(d^10 + d^2*e^8 - 4*d
^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) - (72*b^2*d^2*e^2*((x - 1)^(1/2) - 1i)^2)/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e
^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2)) - (3*b*d*e*((4*(b*d^5*e^2*12i - b*d^3*e^4*24i + b*d*e^6*12i))/(d^10 +
d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(b*d^5*e^2*12i - b*d^3*e^4*24i + b*d
*e^6*12i))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2)) - (3*b*d*e*((e*((x - 1
)^(1/2) - 1i)*64i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 + 4*e^10 - 12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8
*e^2))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52
*d^2*e^8 - 88*d^4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e
^4 - 4*d^8*e^2))))/(2*(d + e)^(5/2)*(d - e)^(5/2))))/(2*(d + e)^(5/2)*(d - e)^(5/2)) + (3*b*d*e*((4*(b*d^5*e^2
*12i - b*d^3*e^4*24i + b*d*e^6*12i))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*((x - 1)^(1/2)
- 1i)^2*(b*d^5*e^2*12i - b*d^3*e^4*24i + b*d*e^6*12i))/(((x + 1)^(1/2) - 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*
d^6*e^4 - 4*d^8*e^2)) + (3*b*d*e*((e*((x - 1)^(1/2) - 1i)*64i)/(d*((x + 1)^(1/2) - 1)) - (4*(4*d^10 + 4*e^10 -
12*d^2*e^8 + 8*d^4*e^6 + 8*d^6*e^4 - 12*d^8*e^2))/(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2) + (4*(
(x - 1)^(1/2) - 1i)^2*(4*d^10 - 12*e^10 + 52*d^2*e^8 - 88*d^4*e^6 + 72*d^6*e^4 - 28*d^8*e^2))/(((x + 1)^(1/2)
- 1)^2*(d^10 + d^2*e^8 - 4*d^4*e^6 + 6*d^6*e^4 - 4*d^8*e^2))))/(2*(d + e)^(5/2)*(d - e)^(5/2))))/(2*(d + e)^(5
/2)*(d - e)^(5/2))))*3i)/((d + e)^(5/2)*(d - e)^(5/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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