3.1.14 \(\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)^3} \, dx\)

Optimal. Leaf size=248 \[ \frac {\sqrt {1-d^2 x^2} \left (A f^2-B e f+C e^2\right )}{2 f \left (d^2 e^2-f^2\right ) (e+f x)^2}+\frac {\tan ^{-1}\left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right ) \left (C \left (d^2 e^2+2 f^2\right )-d^2 \left (3 B e f-A \left (2 d^2 e^2+f^2\right )\right )\right )}{2 \left (d^2 e^2-f^2\right )^{5/2}}-\frac {\sqrt {1-d^2 x^2} \left (-3 A d^2 e f^2+B d^2 e^2 f+2 B f^3+C d^2 e^3-4 C e f^2\right )}{2 f \left (d^2 e^2-f^2\right )^2 (e+f x)} \]

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Rubi [A]  time = 0.33, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1609, 1651, 807, 725, 204} \begin {gather*} \frac {\sqrt {1-d^2 x^2} \left (A f^2-B e f+C e^2\right )}{2 f \left (d^2 e^2-f^2\right ) (e+f x)^2}-\frac {\sqrt {1-d^2 x^2} \left (-3 A d^2 e f^2+B d^2 e^2 f+2 B f^3+C d^2 e^3-4 C e f^2\right )}{2 f \left (d^2 e^2-f^2\right )^2 (e+f x)}+\frac {\tan ^{-1}\left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right ) \left (C \left (d^2 e^2+2 f^2\right )-d^2 \left (3 B e f-A \left (2 d^2 e^2+f^2\right )\right )\right )}{2 \left (d^2 e^2-f^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^3),x]

[Out]

((C*e^2 - B*e*f + A*f^2)*Sqrt[1 - d^2*x^2])/(2*f*(d^2*e^2 - f^2)*(e + f*x)^2) - ((C*d^2*e^3 + B*d^2*e^2*f - 4*
C*e*f^2 - 3*A*d^2*e*f^2 + 2*B*f^3)*Sqrt[1 - d^2*x^2])/(2*f*(d^2*e^2 - f^2)^2*(e + f*x)) + ((C*(d^2*e^2 + 2*f^2
) - d^2*(3*B*e*f - A*(2*d^2*e^2 + f^2)))*ArcTan[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(2*(d^
2*e^2 - f^2)^(5/2))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 1609

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

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-d x} \sqrt {1+d x} (e+f x)^3} \, dx &=\int \frac {A+B x+C x^2}{(e+f x)^3 \sqrt {1-d^2 x^2}} \, dx\\ &=\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {1-d^2 x^2}}{2 f \left (d^2 e^2-f^2\right ) (e+f x)^2}+\frac {\int \frac {2 \left (C e+A d^2 e-B f\right )+\left (B d^2 e+\frac {C d^2 e^2}{f}-2 C f-A d^2 f\right ) x}{(e+f x)^2 \sqrt {1-d^2 x^2}} \, dx}{2 \left (d^2 e^2-f^2\right )}\\ &=\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {1-d^2 x^2}}{2 f \left (d^2 e^2-f^2\right ) (e+f x)^2}-\frac {\left (C d^2 e^3+B d^2 e^2 f-4 C e f^2-3 A d^2 e f^2+2 B f^3\right ) \sqrt {1-d^2 x^2}}{2 f \left (d^2 e^2-f^2\right )^2 (e+f x)}+\frac {\left (C \left (d^2 e^2+2 f^2\right )-d^2 \left (3 B e f-A \left (2 d^2 e^2+f^2\right )\right )\right ) \int \frac {1}{(e+f x) \sqrt {1-d^2 x^2}} \, dx}{2 \left (d^2 e^2-f^2\right )^2}\\ &=\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {1-d^2 x^2}}{2 f \left (d^2 e^2-f^2\right ) (e+f x)^2}-\frac {\left (C d^2 e^3+B d^2 e^2 f-4 C e f^2-3 A d^2 e f^2+2 B f^3\right ) \sqrt {1-d^2 x^2}}{2 f \left (d^2 e^2-f^2\right )^2 (e+f x)}-\frac {\left (C \left (d^2 e^2+2 f^2\right )-d^2 \left (3 B e f-A \left (2 d^2 e^2+f^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-d^2 e^2+f^2-x^2} \, dx,x,\frac {f+d^2 e x}{\sqrt {1-d^2 x^2}}\right )}{2 \left (d^2 e^2-f^2\right )^2}\\ &=\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {1-d^2 x^2}}{2 f \left (d^2 e^2-f^2\right ) (e+f x)^2}-\frac {\left (C d^2 e^3+B d^2 e^2 f-4 C e f^2-3 A d^2 e f^2+2 B f^3\right ) \sqrt {1-d^2 x^2}}{2 f \left (d^2 e^2-f^2\right )^2 (e+f x)}+\frac {\left (C \left (d^2 e^2+2 f^2\right )-d^2 \left (3 B e f-A \left (2 d^2 e^2+f^2\right )\right )\right ) \tan ^{-1}\left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{2 \left (d^2 e^2-f^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.38, size = 273, normalized size = 1.10 \begin {gather*} \frac {1}{2} \left (-\frac {\log \left (\sqrt {1-d^2 x^2} \sqrt {f^2-d^2 e^2}+d^2 e x+f\right ) \left (d^2 \left (A \left (2 d^2 e^2+f^2\right )-3 B e f\right )+C \left (d^2 e^2+2 f^2\right )\right )}{\left (f^2-d^2 e^2\right )^{5/2}}+\frac {\log (e+f x) \left (d^2 \left (A \left (2 d^2 e^2+f^2\right )-3 B e f\right )+C \left (d^2 e^2+2 f^2\right )\right )}{\left (f^2-d^2 e^2\right )^{5/2}}-\frac {\sqrt {1-d^2 x^2} \left (-A d^2 e f (4 e+3 f x)+A f^3+B d^2 e^2 (2 e+f x)+B f^2 (e+2 f x)+C e \left (d^2 e^2 x-3 e f-4 f^2 x\right )\right )}{\left (f^2-d^2 e^2\right )^2 (e+f x)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^3),x]

[Out]

(-((Sqrt[1 - d^2*x^2]*(A*f^3 + B*d^2*e^2*(2*e + f*x) + B*f^2*(e + 2*f*x) - A*d^2*e*f*(4*e + 3*f*x) + C*e*(-3*e
*f + d^2*e^2*x - 4*f^2*x)))/((-(d^2*e^2) + f^2)^2*(e + f*x)^2)) + ((C*(d^2*e^2 + 2*f^2) + d^2*(-3*B*e*f + A*(2
*d^2*e^2 + f^2)))*Log[e + f*x])/(-(d^2*e^2) + f^2)^(5/2) - ((C*(d^2*e^2 + 2*f^2) + d^2*(-3*B*e*f + A*(2*d^2*e^
2 + f^2)))*Log[f + d^2*e*x + Sqrt[-(d^2*e^2) + f^2]*Sqrt[1 - d^2*x^2]])/(-(d^2*e^2) + f^2)^(5/2))/2

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IntegrateAlgebraic [B]  time = 0.00, size = 533, normalized size = 2.15 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {1-d x} \sqrt {-d e-f} \sqrt {f-d e}}{\sqrt {d x+1} (d e+f)}\right ) \left (2 A d^4 e^2 \sqrt {f-d e}+A d^2 f^2 \sqrt {f-d e}-3 B d^2 e f \sqrt {f-d e}+C d^2 e^2 \sqrt {f-d e}+2 C f^2 \sqrt {f-d e}\right )}{(-d e-f)^{5/2} (d e-f)^3}-\frac {d \sqrt {1-d x} \left (-\frac {4 A d^3 e^2 f (1-d x)}{d x+1}-4 A d^3 e^2 f+\frac {3 A d^2 e f^2 (1-d x)}{d x+1}-3 A d^2 e f^2+\frac {A d f^3 (1-d x)}{d x+1}+A d f^3+\frac {2 B d^3 e^3 (1-d x)}{d x+1}+2 B d^3 e^3-\frac {B d^2 e^2 f (1-d x)}{d x+1}+B d^2 e^2 f+\frac {B d e f^2 (1-d x)}{d x+1}+B d e f^2-\frac {2 B f^3 (1-d x)}{d x+1}+2 B f^3-\frac {C d^2 e^3 (1-d x)}{d x+1}+C d^2 e^3-\frac {3 C d e^2 f (1-d x)}{d x+1}-3 C d e^2 f+\frac {4 C e f^2 (1-d x)}{d x+1}-4 C e f^2\right )}{\sqrt {d x+1} (d e-f)^2 (d e+f)^2 \left (\frac {d e (1-d x)}{d x+1}+d e-\frac {f (1-d x)}{d x+1}+f\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^3),x]

[Out]

-((d*Sqrt[1 - d*x]*(C*d^2*e^3 + 2*B*d^3*e^3 - 3*C*d*e^2*f + B*d^2*e^2*f - 4*A*d^3*e^2*f - 4*C*e*f^2 + B*d*e*f^
2 - 3*A*d^2*e*f^2 + 2*B*f^3 + A*d*f^3 - (C*d^2*e^3*(1 - d*x))/(1 + d*x) + (2*B*d^3*e^3*(1 - d*x))/(1 + d*x) -
(3*C*d*e^2*f*(1 - d*x))/(1 + d*x) - (B*d^2*e^2*f*(1 - d*x))/(1 + d*x) - (4*A*d^3*e^2*f*(1 - d*x))/(1 + d*x) +
(4*C*e*f^2*(1 - d*x))/(1 + d*x) + (B*d*e*f^2*(1 - d*x))/(1 + d*x) + (3*A*d^2*e*f^2*(1 - d*x))/(1 + d*x) - (2*B
*f^3*(1 - d*x))/(1 + d*x) + (A*d*f^3*(1 - d*x))/(1 + d*x)))/((d*e - f)^2*(d*e + f)^2*Sqrt[1 + d*x]*(d*e + f +
(d*e*(1 - d*x))/(1 + d*x) - (f*(1 - d*x))/(1 + d*x))^2)) + ((C*d^2*e^2*Sqrt[-(d*e) + f] + 2*A*d^4*e^2*Sqrt[-(d
*e) + f] - 3*B*d^2*e*f*Sqrt[-(d*e) + f] + 2*C*f^2*Sqrt[-(d*e) + f] + A*d^2*f^2*Sqrt[-(d*e) + f])*ArcTan[(Sqrt[
-(d*e) - f]*Sqrt[-(d*e) + f]*Sqrt[1 - d*x])/((d*e + f)*Sqrt[1 + d*x])])/((-(d*e) - f)^(5/2)*(d*e - f)^3)

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fricas [B]  time = 0.85, size = 1580, normalized size = 6.37

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(f*x+e)^3/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="fricas")

[Out]

[-1/2*(2*B*d^4*e^7 - B*d^2*e^5*f^2 - (4*A*d^4 + 3*C*d^2)*e^6*f + (5*A*d^2 + 3*C)*e^4*f^3 - B*e^3*f^4 - A*e^2*f
^5 + (2*B*d^4*e^5*f^2 - B*d^2*e^3*f^4 - (4*A*d^4 + 3*C*d^2)*e^4*f^3 + (5*A*d^2 + 3*C)*e^2*f^5 - B*e*f^6 - A*f^
7)*x^2 - (3*B*d^2*e^5*f - (2*A*d^4 + C*d^2)*e^6 - (A*d^2 + 2*C)*e^4*f^2 + (3*B*d^2*e^3*f^3 - (2*A*d^4 + C*d^2)
*e^4*f^2 - (A*d^2 + 2*C)*e^2*f^4)*x^2 + 2*(3*B*d^2*e^4*f^2 - (2*A*d^4 + C*d^2)*e^5*f - (A*d^2 + 2*C)*e^3*f^3)*
x)*sqrt(-d^2*e^2 + f^2)*log((d^2*e*f*x + f^2 - sqrt(-d^2*e^2 + f^2)*(d^2*e*x + f) - (sqrt(-d^2*e^2 + f^2)*sqrt
(-d*x + 1)*f + (d^2*e^2 - f^2)*sqrt(-d*x + 1))*sqrt(d*x + 1))/(f*x + e)) + (2*B*d^4*e^7 - B*d^2*e^5*f^2 - (4*A
*d^4 + 3*C*d^2)*e^6*f + (5*A*d^2 + 3*C)*e^4*f^3 - B*e^3*f^4 - A*e^2*f^5 + (C*d^4*e^7 + B*d^4*e^6*f + B*d^2*e^4
*f^3 - (3*A*d^4 + 5*C*d^2)*e^5*f^2 + (3*A*d^2 + 4*C)*e^3*f^4 - 2*B*e^2*f^5)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) +
2*(2*B*d^4*e^6*f - B*d^2*e^4*f^3 - (4*A*d^4 + 3*C*d^2)*e^5*f^2 + (5*A*d^2 + 3*C)*e^3*f^4 - B*e^2*f^5 - A*e*f^6
)*x)/(d^6*e^10 - 3*d^4*e^8*f^2 + 3*d^2*e^6*f^4 - e^4*f^6 + (d^6*e^8*f^2 - 3*d^4*e^6*f^4 + 3*d^2*e^4*f^6 - e^2*
f^8)*x^2 + 2*(d^6*e^9*f - 3*d^4*e^7*f^3 + 3*d^2*e^5*f^5 - e^3*f^7)*x), -1/2*(2*B*d^4*e^7 - B*d^2*e^5*f^2 - (4*
A*d^4 + 3*C*d^2)*e^6*f + (5*A*d^2 + 3*C)*e^4*f^3 - B*e^3*f^4 - A*e^2*f^5 + (2*B*d^4*e^5*f^2 - B*d^2*e^3*f^4 -
(4*A*d^4 + 3*C*d^2)*e^4*f^3 + (5*A*d^2 + 3*C)*e^2*f^5 - B*e*f^6 - A*f^7)*x^2 + 2*(3*B*d^2*e^5*f - (2*A*d^4 + C
*d^2)*e^6 - (A*d^2 + 2*C)*e^4*f^2 + (3*B*d^2*e^3*f^3 - (2*A*d^4 + C*d^2)*e^4*f^2 - (A*d^2 + 2*C)*e^2*f^4)*x^2
+ 2*(3*B*d^2*e^4*f^2 - (2*A*d^4 + C*d^2)*e^5*f - (A*d^2 + 2*C)*e^3*f^3)*x)*sqrt(d^2*e^2 - f^2)*arctan(-(sqrt(d
^2*e^2 - f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1)*e - sqrt(d^2*e^2 - f^2)*(f*x + e))/((d^2*e^2 - f^2)*x)) + (2*B*d^4*
e^7 - B*d^2*e^5*f^2 - (4*A*d^4 + 3*C*d^2)*e^6*f + (5*A*d^2 + 3*C)*e^4*f^3 - B*e^3*f^4 - A*e^2*f^5 + (C*d^4*e^7
+ B*d^4*e^6*f + B*d^2*e^4*f^3 - (3*A*d^4 + 5*C*d^2)*e^5*f^2 + (3*A*d^2 + 4*C)*e^3*f^4 - 2*B*e^2*f^5)*x)*sqrt(
d*x + 1)*sqrt(-d*x + 1) + 2*(2*B*d^4*e^6*f - B*d^2*e^4*f^3 - (4*A*d^4 + 3*C*d^2)*e^5*f^2 + (5*A*d^2 + 3*C)*e^3
*f^4 - B*e^2*f^5 - A*e*f^6)*x)/(d^6*e^10 - 3*d^4*e^8*f^2 + 3*d^2*e^6*f^4 - e^4*f^6 + (d^6*e^8*f^2 - 3*d^4*e^6*
f^4 + 3*d^2*e^4*f^6 - e^2*f^8)*x^2 + 2*(d^6*e^9*f - 3*d^4*e^7*f^3 + 3*d^2*e^5*f^5 - e^3*f^7)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(f*x+e)^3/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unde
f/Unsigned Inf encountered in limit

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maple [C]  time = 0.00, size = 1449, normalized size = 5.84

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/(f*x+e)^3/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x)

[Out]

-1/2*(2*A*d^4*e^2*f^2*x^2*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))+4*A*d^4*e^
3*f*x*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))+2*A*d^4*e^4*ln(2*(d^2*e*x+(-d^
2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))+A*d^2*f^4*x^2*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*
e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))-3*B*d^2*e*f^3*x^2*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2
)*f+f)/(f*x+e))+C*d^2*e^2*f^2*x^2*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))+2*
A*d^2*e*f^3*x*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))-6*B*d^2*e^2*f^2*x*ln(2
*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))+2*C*d^2*e^3*f*x*ln(2*(d^2*e*x+(-d^2*x^2+
1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))+A*d^2*e^2*f^2*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^
2)/f^2)^(1/2)*f+f)/(f*x+e))-3*(-(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)*A*d^2*e*f^3*x-3*B*d^2*e^3*f*ln(2*(
d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))+(-(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1
/2)*B*d^2*e^2*f^2*x+C*d^2*e^4*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))+(-(d^2
*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)*C*d^2*e^3*f*x+2*C*f^4*x^2*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-
f^2)/f^2)^(1/2)*f+f)/(f*x+e))-4*(-(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)*A*d^2*e^2*f^2+2*(-(d^2*e^2-f^2)/
f^2)^(1/2)*(-d^2*x^2+1)^(1/2)*B*d^2*e^3*f+4*C*e*f^3*x*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1
/2)*f+f)/(f*x+e))+2*(-(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)*B*f^4*x+2*C*e^2*f^2*ln(2*(d^2*e*x+(-d^2*x^2+
1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))-4*(-(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)*C*e*f^3*x+(-
(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)*A*f^4+(-(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)*B*e*f^3-3*(-(d
^2*e^2-f^2)/f^2)^(1/2)*(-d^2*x^2+1)^(1/2)*C*e^2*f^2)*(d*x+1)^(1/2)*(-d*x+1)^(1/2)/(-d^2*x^2+1)^(1/2)/(d*e+f)/(
d*e-f)/(d^2*e^2-f^2)/(f*x+e)^2/(-(d^2*e^2-f^2)/f^2)^(1/2)/f*csgn(d)^2

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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)/(f*x+e)^3/(-d*x+1)^(1/2)/(d*x+1)^(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(f-d*e>0)', see `assume?` for m
ore details)Is f-d*e positive, negative or zero?

________________________________________________________________________________________

mupad [B]  time = 0.01, size = 9097, normalized size = 36.68

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2)/((e + f*x)^3*(1 - d*x)^(1/2)*(d*x + 1)^(1/2)),x)

[Out]

((12*(2*C*f^3 + C*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^2)/(((d*x + 1)^(1/2) - 1)^2*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2)
) - (24*(2*C*f^3 - C*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^4)/(((d*x + 1)^(1/2) - 1)^4*(f^4 + d^4*e^4 - 2*d^2*e^2*f
^2)) + (12*(2*C*f^3 + C*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^6)/(((d*x + 1)^(1/2) - 1)^6*(f^4 + d^4*e^4 - 2*d^2*e^
2*f^2)) - (2*((1 - d*x)^(1/2) - 1)^7*(C*d^3*e^3 + 2*C*d*e*f^2))/(((d*x + 1)^(1/2) - 1)^7*(f^4 + d^4*e^4 - 2*d^
2*e^2*f^2)) - (2*((1 - d*x)^(1/2) - 1)^3*(7*C*d^3*e^3 - 34*C*d*e*f^2))/(((d*x + 1)^(1/2) - 1)^3*(f^4 + d^4*e^4
- 2*d^2*e^2*f^2)) + (2*((1 - d*x)^(1/2) - 1)^5*(7*C*d^3*e^3 - 34*C*d*e*f^2))/(((d*x + 1)^(1/2) - 1)^5*(f^4 +
d^4*e^4 - 2*d^2*e^2*f^2)) + (2*d*e*((1 - d*x)^(1/2) - 1)*(2*C*f^2 + C*d^2*e^2))/(((d*x + 1)^(1/2) - 1)*(f^4 +
d^4*e^4 - 2*d^2*e^2*f^2)))/(d^2*e^2 + (((1 - d*x)^(1/2) - 1)^2*(16*f^2 + 4*d^2*e^2))/((d*x + 1)^(1/2) - 1)^2 +
(((1 - d*x)^(1/2) - 1)^6*(16*f^2 + 4*d^2*e^2))/((d*x + 1)^(1/2) - 1)^6 - (((1 - d*x)^(1/2) - 1)^4*(32*f^2 - 6
*d^2*e^2))/((d*x + 1)^(1/2) - 1)^4 + (d^2*e^2*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8 + (8*d*e*f*((1
- d*x)^(1/2) - 1)^3)/((d*x + 1)^(1/2) - 1)^3 - (8*d*e*f*((1 - d*x)^(1/2) - 1)^5)/((d*x + 1)^(1/2) - 1)^5 - (8*
d*e*f*((1 - d*x)^(1/2) - 1)^7)/((d*x + 1)^(1/2) - 1)^7 + (8*d*e*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1)
) + ((4*((1 - d*x)^(1/2) - 1)^2*(4*A*d^4*e^4*f - 2*A*f^5 + 7*A*d^2*e^2*f^3))/(e^2*((d*x + 1)^(1/2) - 1)^2*(f^4
+ d^4*e^4 - 2*d^2*e^2*f^2)) + (8*((1 - d*x)^(1/2) - 1)^4*(2*A*f^5 + 4*A*d^4*e^4*f - 9*A*d^2*e^2*f^3))/(e^2*((
d*x + 1)^(1/2) - 1)^4*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2)) + (4*((1 - d*x)^(1/2) - 1)^6*(4*A*d^4*e^4*f - 2*A*f^5 +
7*A*d^2*e^2*f^3))/(e^2*((d*x + 1)^(1/2) - 1)^6*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2)) + (2*f*((1 - d*x)^(1/2) - 1)^
7*(2*A*d*f^3 - 5*A*d^3*e^2*f))/(e*((d*x + 1)^(1/2) - 1)^7*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2)) - (2*f*((1 - d*x)^(
1/2) - 1)^3*(2*A*d*f^3 - 29*A*d^3*e^2*f))/(e*((d*x + 1)^(1/2) - 1)^3*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2)) + (2*f*(
(1 - d*x)^(1/2) - 1)^5*(2*A*d*f^3 - 29*A*d^3*e^2*f))/(e*((d*x + 1)^(1/2) - 1)^5*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2
)) - (2*d*f*(2*A*f^3 - 5*A*d^2*e^2*f)*((1 - d*x)^(1/2) - 1))/(e*((d*x + 1)^(1/2) - 1)*(f^4 + d^4*e^4 - 2*d^2*e
^2*f^2)))/(d^2*e^2 + (((1 - d*x)^(1/2) - 1)^2*(16*f^2 + 4*d^2*e^2))/((d*x + 1)^(1/2) - 1)^2 + (((1 - d*x)^(1/2
) - 1)^6*(16*f^2 + 4*d^2*e^2))/((d*x + 1)^(1/2) - 1)^6 - (((1 - d*x)^(1/2) - 1)^4*(32*f^2 - 6*d^2*e^2))/((d*x
+ 1)^(1/2) - 1)^4 + (d^2*e^2*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8 + (8*d*e*f*((1 - d*x)^(1/2) - 1)
^3)/((d*x + 1)^(1/2) - 1)^3 - (8*d*e*f*((1 - d*x)^(1/2) - 1)^5)/((d*x + 1)^(1/2) - 1)^5 - (8*d*e*f*((1 - d*x)^
(1/2) - 1)^7)/((d*x + 1)^(1/2) - 1)^7 + (8*d*e*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1)) - ((4*((1 - d*x
)^(1/2) - 1)^2*(2*B*f^4 + 2*B*d^4*e^4 + 5*B*d^2*e^2*f^2))/(e*((d*x + 1)^(1/2) - 1)^2*(f^4 + d^4*e^4 - 2*d^2*e^
2*f^2)) - (8*((1 - d*x)^(1/2) - 1)^4*(2*B*f^4 - 2*B*d^4*e^4 + 3*B*d^2*e^2*f^2))/(e*((d*x + 1)^(1/2) - 1)^4*(f^
4 + d^4*e^4 - 2*d^2*e^2*f^2)) + (4*((1 - d*x)^(1/2) - 1)^6*(2*B*f^4 + 2*B*d^4*e^4 + 5*B*d^2*e^2*f^2))/(e*((d*x
+ 1)^(1/2) - 1)^6*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2)) + (2*f*(11*B*d^3*e^2 + 16*B*d*f^2)*((1 - d*x)^(1/2) - 1)^3
)/(((d*x + 1)^(1/2) - 1)^3*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2)) - (2*f*(11*B*d^3*e^2 + 16*B*d*f^2)*((1 - d*x)^(1/2
) - 1)^5)/(((d*x + 1)^(1/2) - 1)^5*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2)) - (6*B*d^3*e^2*f*((1 - d*x)^(1/2) - 1)^7)/
(((d*x + 1)^(1/2) - 1)^7*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2)) + (6*B*d^3*e^2*f*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^
(1/2) - 1)*(f^4 + d^4*e^4 - 2*d^2*e^2*f^2)))/(d^2*e^2 + (((1 - d*x)^(1/2) - 1)^2*(16*f^2 + 4*d^2*e^2))/((d*x +
1)^(1/2) - 1)^2 + (((1 - d*x)^(1/2) - 1)^6*(16*f^2 + 4*d^2*e^2))/((d*x + 1)^(1/2) - 1)^6 - (((1 - d*x)^(1/2)
- 1)^4*(32*f^2 - 6*d^2*e^2))/((d*x + 1)^(1/2) - 1)^4 + (d^2*e^2*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)
^8 + (8*d*e*f*((1 - d*x)^(1/2) - 1)^3)/((d*x + 1)^(1/2) - 1)^3 - (8*d*e*f*((1 - d*x)^(1/2) - 1)^5)/((d*x + 1)^
(1/2) - 1)^5 - (8*d*e*f*((1 - d*x)^(1/2) - 1)^7)/((d*x + 1)^(1/2) - 1)^7 + (8*d*e*f*((1 - d*x)^(1/2) - 1))/((d
*x + 1)^(1/2) - 1)) + (C*atan(((C*(2*f^2 + d^2*e^2)*((4*((1 - d*x)^(1/2) - 1)^2*(8*C*d*e*f^7 + 4*C*d^7*e^7*f -
12*C*d^3*e^3*f^5))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2))
- (4*(8*C*d*e*f^7 + 4*C*d^7*e^7*f - 12*C*d^3*e^3*f^5))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*
e^6*f^2) + (C*(2*f^2 + d^2*e^2)*((4*(4*d^11*e^11 - 12*d^3*e^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^4 - 12*d^9*e^9
*f^2 + 4*d*e*f^10))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d*x)^(1/2) - 1)
^2*(4*d^11*e^11 + 52*d^3*e^3*f^8 - 88*d^5*e^5*f^6 + 72*d^7*e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^10))/(((d*x + 1
)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*((1 - d*x)^(1/
2) - 1))/((d*x + 1)^(1/2) - 1)))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2)))*1i)/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2))
- (C*(2*f^2 + d^2*e^2)*((4*(8*C*d*e*f^7 + 4*C*d^7*e^7*f - 12*C*d^3*e^3*f^5))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 +
6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) - (4*((1 - d*x)^(1/2) - 1)^2*(8*C*d*e*f^7 + 4*C*d^7*e^7*f - 12*C*d^3*e^3*f^5))/
(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (C*(2*f^2 + d^2*e^
2)*((4*(4*d^11*e^11 - 12*d^3*e^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^4 - 12*d^9*e^9*f^2 + 4*d*e*f^10))/(f^8 + d^
8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d*x)^(1/2) - 1)^2*(4*d^11*e^11 + 52*d^3*e^3*
f^8 - 88*d^5*e^5*f^6 + 72*d^7*e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^10))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8
- 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1
)))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2)))*1i)/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2)))/((8*(C^2*d^5*e^5 + 4*C^2*d^3
*e^3*f^2 + 4*C^2*d*e*f^4))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (8*((1 - d*x)^(1/
2) - 1)^2*(C^2*d^5*e^5 + 4*C^2*d^3*e^3*f^2 + 4*C^2*d*e*f^4))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e
^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (C*(2*f^2 + d^2*e^2)*((4*((1 - d*x)^(1/2) - 1)^2*(8*C*d*e*f^7 + 4*C
*d^7*e^7*f - 12*C*d^3*e^3*f^5))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^
6*e^6*f^2)) - (4*(8*C*d*e*f^7 + 4*C*d^7*e^7*f - 12*C*d^3*e^3*f^5))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*
f^4 - 4*d^6*e^6*f^2) + (C*(2*f^2 + d^2*e^2)*((4*(4*d^11*e^11 - 12*d^3*e^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^4
- 12*d^9*e^9*f^2 + 4*d*e*f^10))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d*x
)^(1/2) - 1)^2*(4*d^11*e^11 + 52*d^3*e^3*f^8 - 88*d^5*e^5*f^6 + 72*d^7*e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^10)
)/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*((
1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1)))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2))))/(2*(f + d*e)^(5/2)*(f - d*e
)^(5/2)) + (C*(2*f^2 + d^2*e^2)*((4*(8*C*d*e*f^7 + 4*C*d^7*e^7*f - 12*C*d^3*e^3*f^5))/(f^8 + d^8*e^8 - 4*d^2*e
^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) - (4*((1 - d*x)^(1/2) - 1)^2*(8*C*d*e*f^7 + 4*C*d^7*e^7*f - 12*C*d^3*e
^3*f^5))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (C*(2*f^2
+ d^2*e^2)*((4*(4*d^11*e^11 - 12*d^3*e^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^4 - 12*d^9*e^9*f^2 + 4*d*e*f^10))/
(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d*x)^(1/2) - 1)^2*(4*d^11*e^11 + 52
*d^3*e^3*f^8 - 88*d^5*e^5*f^6 + 72*d^7*e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^10))/(((d*x + 1)^(1/2) - 1)^2*(f^8
+ d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^
(1/2) - 1)))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2))))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2))))*(2*f^2 + d^2*e^2)*1i)
/((f + d*e)^(5/2)*(f - d*e)^(5/2)) + (A*d^2*atan(((A*d^2*(f^2 + 2*d^2*e^2)*((4*((1 - d*x)^(1/2) - 1)^2*(4*A*d^
3*e*f^7 + 8*A*d^9*e^7*f - 12*A*d^7*e^5*f^3))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e
^4*f^4 - 4*d^6*e^6*f^2)) - (4*(4*A*d^3*e*f^7 + 8*A*d^9*e^7*f - 12*A*d^7*e^5*f^3))/(f^8 + d^8*e^8 - 4*d^2*e^2*f
^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (A*d^2*(f^2 + 2*d^2*e^2)*((4*(4*d^11*e^11 - 12*d^3*e^3*f^8 + 8*d^5*e^5*f
^6 + 8*d^7*e^7*f^4 - 12*d^9*e^9*f^2 + 4*d*e*f^10))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*
f^2) + (4*((1 - d*x)^(1/2) - 1)^2*(4*d^11*e^11 + 52*d^3*e^3*f^8 - 88*d^5*e^5*f^6 + 72*d^7*e^7*f^4 - 28*d^9*e^9
*f^2 - 12*d*e*f^10))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2))
+ (64*d^2*e^2*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1)))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2)))*1i)/(2*(f
+ d*e)^(5/2)*(f - d*e)^(5/2)) - (A*d^2*(f^2 + 2*d^2*e^2)*((4*(4*A*d^3*e*f^7 + 8*A*d^9*e^7*f - 12*A*d^7*e^5*f^
3))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) - (4*((1 - d*x)^(1/2) - 1)^2*(4*A*d^3*e*f^
7 + 8*A*d^9*e^7*f - 12*A*d^7*e^5*f^3))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4
- 4*d^6*e^6*f^2)) + (A*d^2*(f^2 + 2*d^2*e^2)*((4*(4*d^11*e^11 - 12*d^3*e^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^
4 - 12*d^9*e^9*f^2 + 4*d*e*f^10))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d
*x)^(1/2) - 1)^2*(4*d^11*e^11 + 52*d^3*e^3*f^8 - 88*d^5*e^5*f^6 + 72*d^7*e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^1
0))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*
((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1)))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2)))*1i)/(2*(f + d*e)^(5/2)*(f
- d*e)^(5/2)))/((8*(4*A^2*d^9*e^5 + 4*A^2*d^7*e^3*f^2 + A^2*d^5*e*f^4))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4
*e^4*f^4 - 4*d^6*e^6*f^2) + (8*((1 - d*x)^(1/2) - 1)^2*(4*A^2*d^9*e^5 + 4*A^2*d^7*e^3*f^2 + A^2*d^5*e*f^4))/((
(d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (A*d^2*(f^2 + 2*d^2*
e^2)*((4*((1 - d*x)^(1/2) - 1)^2*(4*A*d^3*e*f^7 + 8*A*d^9*e^7*f - 12*A*d^7*e^5*f^3))/(((d*x + 1)^(1/2) - 1)^2*
(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) - (4*(4*A*d^3*e*f^7 + 8*A*d^9*e^7*f - 12*A*d^
7*e^5*f^3))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (A*d^2*(f^2 + 2*d^2*e^2)*((4*(4*
d^11*e^11 - 12*d^3*e^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^4 - 12*d^9*e^9*f^2 + 4*d*e*f^10))/(f^8 + d^8*e^8 - 4*
d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d*x)^(1/2) - 1)^2*(4*d^11*e^11 + 52*d^3*e^3*f^8 - 88*d
^5*e^5*f^6 + 72*d^7*e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^10))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e
^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1)))/(2*(f
+ d*e)^(5/2)*(f - d*e)^(5/2))))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2)) + (A*d^2*(f^2 + 2*d^2*e^2)*((4*(4*A*d^3*e*
f^7 + 8*A*d^9*e^7*f - 12*A*d^7*e^5*f^3))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) - (4*
((1 - d*x)^(1/2) - 1)^2*(4*A*d^3*e*f^7 + 8*A*d^9*e^7*f - 12*A*d^7*e^5*f^3))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^
8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (A*d^2*(f^2 + 2*d^2*e^2)*((4*(4*d^11*e^11 - 12*d^3*e
^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^4 - 12*d^9*e^9*f^2 + 4*d*e*f^10))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*
e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d*x)^(1/2) - 1)^2*(4*d^11*e^11 + 52*d^3*e^3*f^8 - 88*d^5*e^5*f^6 + 72*d^7*
e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^10))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f
^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1)))/(2*(f + d*e)^(5/2)*(f - d*
e)^(5/2))))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2))))*(f^2 + 2*d^2*e^2)*1i)/((f + d*e)^(5/2)*(f - d*e)^(5/2)) - (B
*d^2*e*f*atan(((B*d^2*e*f*((4*((1 - d*x)^(1/2) - 1)^2*(12*B*d^3*e^2*f^6 - 24*B*d^5*e^4*f^4 + 12*B*d^7*e^6*f^2)
)/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) - (4*(12*B*d^3*e^2
*f^6 - 24*B*d^5*e^4*f^4 + 12*B*d^7*e^6*f^2))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) +
(3*B*d^2*e*f*((4*(4*d^11*e^11 - 12*d^3*e^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^4 - 12*d^9*e^9*f^2 + 4*d*e*f^10)
)/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d*x)^(1/2) - 1)^2*(4*d^11*e^11 +
52*d^3*e^3*f^8 - 88*d^5*e^5*f^6 + 72*d^7*e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^10))/(((d*x + 1)^(1/2) - 1)^2*(f^
8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*((1 - d*x)^(1/2) - 1))/((d*x + 1
)^(1/2) - 1)))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2)))*3i)/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2)) - (B*d^2*e*f*((4*(
12*B*d^3*e^2*f^6 - 24*B*d^5*e^4*f^4 + 12*B*d^7*e^6*f^2))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^
6*e^6*f^2) - (4*((1 - d*x)^(1/2) - 1)^2*(12*B*d^3*e^2*f^6 - 24*B*d^5*e^4*f^4 + 12*B*d^7*e^6*f^2))/(((d*x + 1)^
(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (3*B*d^2*e*f*((4*(4*d^11*e^11
- 12*d^3*e^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^4 - 12*d^9*e^9*f^2 + 4*d*e*f^10))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^
6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d*x)^(1/2) - 1)^2*(4*d^11*e^11 + 52*d^3*e^3*f^8 - 88*d^5*e^5*f^6
+ 72*d^7*e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^10))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6
*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1)))/(2*(f + d*e)^(5/
2)*(f - d*e)^(5/2)))*3i)/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2)))/((72*B^2*d^5*e^3*f^2)/(f^8 + d^8*e^8 - 4*d^2*e^2
*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (3*B*d^2*e*f*((4*((1 - d*x)^(1/2) - 1)^2*(12*B*d^3*e^2*f^6 - 24*B*d^5*
e^4*f^4 + 12*B*d^7*e^6*f^2))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e
^6*f^2)) - (4*(12*B*d^3*e^2*f^6 - 24*B*d^5*e^4*f^4 + 12*B*d^7*e^6*f^2))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4
*e^4*f^4 - 4*d^6*e^6*f^2) + (3*B*d^2*e*f*((4*(4*d^11*e^11 - 12*d^3*e^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^4 - 1
2*d^9*e^9*f^2 + 4*d*e*f^10))/(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d*x)^(
1/2) - 1)^2*(4*d^11*e^11 + 52*d^3*e^3*f^8 - 88*d^5*e^5*f^6 + 72*d^7*e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^10))/(
((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*((1 -
d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1)))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2))))/(2*(f + d*e)^(5/2)*(f - d*e)^(
5/2)) + (3*B*d^2*e*f*((4*(12*B*d^3*e^2*f^6 - 24*B*d^5*e^4*f^4 + 12*B*d^7*e^6*f^2))/(f^8 + d^8*e^8 - 4*d^2*e^2*
f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) - (4*((1 - d*x)^(1/2) - 1)^2*(12*B*d^3*e^2*f^6 - 24*B*d^5*e^4*f^4 + 12*B*
d^7*e^6*f^2))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (3*B
*d^2*e*f*((4*(4*d^11*e^11 - 12*d^3*e^3*f^8 + 8*d^5*e^5*f^6 + 8*d^7*e^7*f^4 - 12*d^9*e^9*f^2 + 4*d*e*f^10))/(f^
8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2) + (4*((1 - d*x)^(1/2) - 1)^2*(4*d^11*e^11 + 52*d^
3*e^3*f^8 - 88*d^5*e^5*f^6 + 72*d^7*e^7*f^4 - 28*d^9*e^9*f^2 - 12*d*e*f^10))/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d
^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2)) + (64*d^2*e^2*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/
2) - 1)))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2))))/(2*(f + d*e)^(5/2)*(f - d*e)^(5/2)) + (72*B^2*d^5*e^3*f^2*((1
- d*x)^(1/2) - 1)^2)/(((d*x + 1)^(1/2) - 1)^2*(f^8 + d^8*e^8 - 4*d^2*e^2*f^6 + 6*d^4*e^4*f^4 - 4*d^6*e^6*f^2))
))*3i)/((f + d*e)^(5/2)*(f - 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)/(f*x+e)**3/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

Timed out

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