3.1.28 \(\int \frac {(e+f x)^2 (A+B x+C x^2)}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\)

Optimal. Leaf size=368 \[ -\frac {\left (a^2-b^2 x^2\right ) \left (f x \left (9 a^2 C f^2-b^2 \left (2 C e^2-4 f (3 A f+2 B e)\right )\right )+4 \left (4 a^2 f^2 (B f+2 C e)-b^2 e \left (C e^2-4 f (3 A f+B e)\right )\right )\right )}{24 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (4 A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 \left (3 a^2 C f^2+4 b^2 e (2 B f+C e)\right )\right )}{8 b^5 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (a^2-b^2 x^2\right ) (e+f x)^2 (C e-4 B f)}{12 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C \left (a^2-b^2 x^2\right ) (e+f x)^3}{4 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}} \]

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Rubi [A]  time = 0.88, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1610, 1654, 833, 780, 217, 203} \begin {gather*} -\frac {\left (a^2-b^2 x^2\right ) \left (f x \left (9 a^2 C f^2-b^2 \left (2 C e^2-4 f (3 A f+2 B e)\right )\right )+4 \left (4 a^2 f^2 (B f+2 C e)-\frac {1}{4} b^2 \left (4 C e^3-16 e f (3 A f+B e)\right )\right )\right )}{24 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (4 A \left (a^2 b^2 f^2+2 b^4 e^2\right )+4 a^2 b^2 e (2 B f+C e)+3 a^4 C f^2\right )}{8 b^5 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (a^2-b^2 x^2\right ) (e+f x)^2 (C e-4 B f)}{12 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C \left (a^2-b^2 x^2\right ) (e+f x)^3}{4 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^2*(A + B*x + C*x^2))/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]

[Out]

((C*e - 4*B*f)*(e + f*x)^2*(a^2 - b^2*x^2))/(12*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (C*(e + f*x)^3*(a^2 -
b^2*x^2))/(4*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - ((4*(4*a^2*f^2*(2*C*e + B*f) - (b^2*(4*C*e^3 - 16*e*f*(
B*e + 3*A*f)))/4) + f*(9*a^2*C*f^2 - b^2*(2*C*e^2 - 4*f*(2*B*e + 3*A*f)))*x)*(a^2 - b^2*x^2))/(24*b^4*f*Sqrt[a
+ b*x]*Sqrt[a*c - b*c*x]) + ((3*a^4*C*f^2 + 4*a^2*b^2*e*(C*e + 2*B*f) + 4*A*(2*b^4*e^2 + a^2*b^2*f^2))*Sqrt[a
^2*c - b^2*c*x^2]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(8*b^5*Sqrt[c]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x
])

Rule 203

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 780

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

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 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 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx &=\frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{\sqrt {a+b x} \sqrt {a c-b c x}}\\ &=-\frac {C (e+f x)^3 \left (a^2-b^2 x^2\right )}{4 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {(e+f x)^2 \left (-c \left (4 A b^2+3 a^2 C\right ) f^2+b^2 c f (C e-4 B f) x\right )}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{4 b^2 c f^2 \sqrt {a+b x} \sqrt {a c-b c x}}\\ &=\frac {(C e-4 B f) (e+f x)^2 \left (a^2-b^2 x^2\right )}{12 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^3 \left (a^2-b^2 x^2\right )}{4 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {(e+f x) \left (b^2 c^2 f^2 \left (12 A b^2 e+a^2 (7 C e+8 B f)\right )+b^2 c^2 f \left (9 a^2 C f^2-2 b^2 \left (C e^2-2 f (2 B e+3 A f)\right )\right ) x\right )}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{12 b^4 c^2 f^2 \sqrt {a+b x} \sqrt {a c-b c x}}\\ &=\frac {(C e-4 B f) (e+f x)^2 \left (a^2-b^2 x^2\right )}{12 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^3 \left (a^2-b^2 x^2\right )}{4 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (4 \left (4 a^2 f^2 (2 C e+B f)-\frac {1}{4} b^2 \left (4 C e^3-16 e f (B e+3 A f)\right )\right )+f \left (9 a^2 C f^2-b^2 \left (2 C e^2-4 f (2 B e+3 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{24 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (\left (3 a^4 C f^2+4 a^2 b^2 e (C e+2 B f)+4 A \left (2 b^4 e^2+a^2 b^2 f^2\right )\right ) \sqrt {a^2 c-b^2 c x^2}\right ) \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{8 b^4 \sqrt {a+b x} \sqrt {a c-b c x}}\\ &=\frac {(C e-4 B f) (e+f x)^2 \left (a^2-b^2 x^2\right )}{12 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^3 \left (a^2-b^2 x^2\right )}{4 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (4 \left (4 a^2 f^2 (2 C e+B f)-\frac {1}{4} b^2 \left (4 C e^3-16 e f (B e+3 A f)\right )\right )+f \left (9 a^2 C f^2-b^2 \left (2 C e^2-4 f (2 B e+3 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{24 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (\left (3 a^4 C f^2+4 a^2 b^2 e (C e+2 B f)+4 A \left (2 b^4 e^2+a^2 b^2 f^2\right )\right ) \sqrt {a^2 c-b^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+b^2 c x^2} \, dx,x,\frac {x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{8 b^4 \sqrt {a+b x} \sqrt {a c-b c x}}\\ &=\frac {(C e-4 B f) (e+f x)^2 \left (a^2-b^2 x^2\right )}{12 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^3 \left (a^2-b^2 x^2\right )}{4 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (4 \left (4 a^2 f^2 (2 C e+B f)-\frac {1}{4} b^2 \left (4 C e^3-16 e f (B e+3 A f)\right )\right )+f \left (9 a^2 C f^2-b^2 \left (2 C e^2-4 f (2 B e+3 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{24 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (3 a^4 C f^2+4 a^2 b^2 e (C e+2 B f)+4 A \left (2 b^4 e^2+a^2 b^2 f^2\right )\right ) \sqrt {a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{8 b^5 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x}}\\ \end {align*}

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Mathematica [A]  time = 2.68, size = 555, normalized size = 1.51 \begin {gather*} \frac {-24 \sqrt {a-b x} \sqrt {a+b x} (b e-a f) \left (\sqrt {a-b x} \sqrt {\frac {b x}{a}+1}+2 \sqrt {a} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )\right ) \left (4 a^2 C f-a b (3 B f+2 C e)+b^2 (2 A f+B e)\right )-12 \sqrt {a-b x} \sqrt {a+b x} \left (6 a^{3/2} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )+\sqrt {a-b x} (4 a+b x) \sqrt {\frac {b x}{a}+1}\right ) \left (6 a^2 C f^2-3 a b f (B f+2 C e)+b^2 \left (f (A f+2 B e)+C e^2\right )\right )-4 f \sqrt {a-b x} \sqrt {a+b x} \left (30 a^{5/2} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )+\sqrt {a-b x} \sqrt {\frac {b x}{a}+1} \left (22 a^2+9 a b x+2 b^2 x^2\right )\right ) (-4 a C f+b B f+2 b C e)-C f^2 \sqrt {a+b x} \left (210 a^{7/2} \sqrt {a-b x} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )+(a-b x) \sqrt {\frac {b x}{a}+1} \left (160 a^3+81 a^2 b x+32 a b^2 x^2+6 b^3 x^3\right )\right )-48 \sqrt {a-b x} \sqrt {\frac {b x}{a}+1} (b e-a f)^2 \tan ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {a+b x}}\right ) \left (a (a C-b B)+A b^2\right )}{24 b^5 \sqrt {\frac {b x}{a}+1} \sqrt {c (a-b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^2*(A + B*x + C*x^2))/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]

[Out]

(-24*(b*e - a*f)*(4*a^2*C*f + b^2*(B*e + 2*A*f) - a*b*(2*C*e + 3*B*f))*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b
*x]*Sqrt[1 + (b*x)/a] + 2*Sqrt[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 12*(6*a^2*C*f^2 - 3*a*b*f*(2*C*e
+ B*f) + b^2*(C*e^2 + f*(2*B*e + A*f)))*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*(4*a + b*x)*Sqrt[1 + (b*x)/
a] + 6*a^(3/2)*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 4*f*(2*b*C*e + b*B*f - 4*a*C*f)*Sqrt[a - b*x]*Sqrt[a
+ b*x]*(Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*(22*a^2 + 9*a*b*x + 2*b^2*x^2) + 30*a^(5/2)*ArcSin[Sqrt[a - b*x]/(Sqr
t[2]*Sqrt[a])]) - C*f^2*Sqrt[a + b*x]*((a - b*x)*Sqrt[1 + (b*x)/a]*(160*a^3 + 81*a^2*b*x + 32*a*b^2*x^2 + 6*b^
3*x^3) + 210*a^(7/2)*Sqrt[a - b*x]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 48*(A*b^2 + a*(-(b*B) + a*C))*(b
*e - a*f)^2*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/(24*b^5*Sqrt[c*(a - b*x)]*Sqr
t[1 + (b*x)/a])

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IntegrateAlgebraic [B]  time = 0.82, size = 1213, normalized size = 3.30 \begin {gather*} \frac {\frac {15 C f^2 (a c-b c x)^{7/2} a^4}{(a+b x)^{7/2}}-\frac {9 c C f^2 (a c-b c x)^{5/2} a^4}{(a+b x)^{5/2}}+\frac {9 c^2 C f^2 (a c-b c x)^{3/2} a^4}{(a+b x)^{3/2}}-\frac {15 c^3 C f^2 \sqrt {a c-b c x} a^4}{\sqrt {a+b x}}-\frac {24 b B f^2 (a c-b c x)^{7/2} a^3}{(a+b x)^{7/2}}-\frac {48 b C e f (a c-b c x)^{7/2} a^3}{(a+b x)^{7/2}}-\frac {40 b B c f^2 (a c-b c x)^{5/2} a^3}{(a+b x)^{5/2}}-\frac {80 b c C e f (a c-b c x)^{5/2} a^3}{(a+b x)^{5/2}}-\frac {40 b B c^2 f^2 (a c-b c x)^{3/2} a^3}{(a+b x)^{3/2}}-\frac {80 b c^2 C e f (a c-b c x)^{3/2} a^3}{(a+b x)^{3/2}}-\frac {24 b B c^3 f^2 \sqrt {a c-b c x} a^3}{\sqrt {a+b x}}-\frac {48 b c^3 C e f \sqrt {a c-b c x} a^3}{\sqrt {a+b x}}+\frac {12 b^2 C e^2 (a c-b c x)^{7/2} a^2}{(a+b x)^{7/2}}+\frac {12 A b^2 f^2 (a c-b c x)^{7/2} a^2}{(a+b x)^{7/2}}+\frac {24 b^2 B e f (a c-b c x)^{7/2} a^2}{(a+b x)^{7/2}}+\frac {12 b^2 c C e^2 (a c-b c x)^{5/2} a^2}{(a+b x)^{5/2}}+\frac {12 A b^2 c f^2 (a c-b c x)^{5/2} a^2}{(a+b x)^{5/2}}+\frac {24 b^2 B c e f (a c-b c x)^{5/2} a^2}{(a+b x)^{5/2}}-\frac {12 b^2 c^2 C e^2 (a c-b c x)^{3/2} a^2}{(a+b x)^{3/2}}-\frac {12 A b^2 c^2 f^2 (a c-b c x)^{3/2} a^2}{(a+b x)^{3/2}}-\frac {24 b^2 B c^2 e f (a c-b c x)^{3/2} a^2}{(a+b x)^{3/2}}-\frac {12 b^2 c^3 C e^2 \sqrt {a c-b c x} a^2}{\sqrt {a+b x}}-\frac {12 A b^2 c^3 f^2 \sqrt {a c-b c x} a^2}{\sqrt {a+b x}}-\frac {24 b^2 B c^3 e f \sqrt {a c-b c x} a^2}{\sqrt {a+b x}}-\frac {24 b^3 B e^2 (a c-b c x)^{7/2} a}{(a+b x)^{7/2}}-\frac {48 A b^3 e f (a c-b c x)^{7/2} a}{(a+b x)^{7/2}}-\frac {72 b^3 B c e^2 (a c-b c x)^{5/2} a}{(a+b x)^{5/2}}-\frac {144 A b^3 c e f (a c-b c x)^{5/2} a}{(a+b x)^{5/2}}-\frac {72 b^3 B c^2 e^2 (a c-b c x)^{3/2} a}{(a+b x)^{3/2}}-\frac {144 A b^3 c^2 e f (a c-b c x)^{3/2} a}{(a+b x)^{3/2}}-\frac {24 b^3 B c^3 e^2 \sqrt {a c-b c x} a}{\sqrt {a+b x}}-\frac {48 A b^3 c^3 e f \sqrt {a c-b c x} a}{\sqrt {a+b x}}}{12 b^5 \left (c+\frac {a c-b c x}{a+b x}\right )^4}+\frac {\left (-3 C f^2 a^4-4 b^2 C e^2 a^2-4 A b^2 f^2 a^2-8 b^2 B e f a^2-8 A b^4 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {a c-b c x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 b^5 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((e + f*x)^2*(A + B*x + C*x^2))/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]

[Out]

((-24*a*b^3*B*c^3*e^2*Sqrt[a*c - b*c*x])/Sqrt[a + b*x] - (12*a^2*b^2*c^3*C*e^2*Sqrt[a*c - b*c*x])/Sqrt[a + b*x
] - (48*a*A*b^3*c^3*e*f*Sqrt[a*c - b*c*x])/Sqrt[a + b*x] - (24*a^2*b^2*B*c^3*e*f*Sqrt[a*c - b*c*x])/Sqrt[a + b
*x] - (48*a^3*b*c^3*C*e*f*Sqrt[a*c - b*c*x])/Sqrt[a + b*x] - (12*a^2*A*b^2*c^3*f^2*Sqrt[a*c - b*c*x])/Sqrt[a +
b*x] - (24*a^3*b*B*c^3*f^2*Sqrt[a*c - b*c*x])/Sqrt[a + b*x] - (15*a^4*c^3*C*f^2*Sqrt[a*c - b*c*x])/Sqrt[a + b
*x] - (72*a*b^3*B*c^2*e^2*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) - (12*a^2*b^2*c^2*C*e^2*(a*c - b*c*x)^(3/2))/(a
+ b*x)^(3/2) - (144*a*A*b^3*c^2*e*f*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) - (24*a^2*b^2*B*c^2*e*f*(a*c - b*c*x
)^(3/2))/(a + b*x)^(3/2) - (80*a^3*b*c^2*C*e*f*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) - (12*a^2*A*b^2*c^2*f^2*(a
*c - b*c*x)^(3/2))/(a + b*x)^(3/2) - (40*a^3*b*B*c^2*f^2*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) + (9*a^4*c^2*C*f
^2*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) - (72*a*b^3*B*c*e^2*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) + (12*a^2*b^2
*c*C*e^2*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) - (144*a*A*b^3*c*e*f*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) + (24*
a^2*b^2*B*c*e*f*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) - (80*a^3*b*c*C*e*f*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2)
+ (12*a^2*A*b^2*c*f^2*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) - (40*a^3*b*B*c*f^2*(a*c - b*c*x)^(5/2))/(a + b*x)^
(5/2) - (9*a^4*c*C*f^2*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) - (24*a*b^3*B*e^2*(a*c - b*c*x)^(7/2))/(a + b*x)^(
7/2) + (12*a^2*b^2*C*e^2*(a*c - b*c*x)^(7/2))/(a + b*x)^(7/2) - (48*a*A*b^3*e*f*(a*c - b*c*x)^(7/2))/(a + b*x)
^(7/2) + (24*a^2*b^2*B*e*f*(a*c - b*c*x)^(7/2))/(a + b*x)^(7/2) - (48*a^3*b*C*e*f*(a*c - b*c*x)^(7/2))/(a + b*
x)^(7/2) + (12*a^2*A*b^2*f^2*(a*c - b*c*x)^(7/2))/(a + b*x)^(7/2) - (24*a^3*b*B*f^2*(a*c - b*c*x)^(7/2))/(a +
b*x)^(7/2) + (15*a^4*C*f^2*(a*c - b*c*x)^(7/2))/(a + b*x)^(7/2))/(12*b^5*(c + (a*c - b*c*x)/(a + b*x))^4) + ((
-8*A*b^4*e^2 - 4*a^2*b^2*C*e^2 - 8*a^2*b^2*B*e*f - 4*a^2*A*b^2*f^2 - 3*a^4*C*f^2)*ArcTan[Sqrt[a*c - b*c*x]/(Sq
rt[c]*Sqrt[a + b*x])])/(4*b^5*Sqrt[c])

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fricas [A]  time = 1.22, size = 482, normalized size = 1.31 \begin {gather*} \left [-\frac {3 \, {\left (8 \, B a^{2} b^{2} e f + 4 \, {\left (C a^{2} b^{2} + 2 \, A b^{4}\right )} e^{2} + {\left (3 \, C a^{4} + 4 \, A a^{2} b^{2}\right )} f^{2}\right )} \sqrt {-c} \log \left (2 \, b^{2} c x^{2} - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) + 2 \, {\left (6 \, C b^{3} f^{2} x^{3} + 24 \, B b^{3} e^{2} + 16 \, B a^{2} b f^{2} + 16 \, {\left (2 \, C a^{2} b + 3 \, A b^{3}\right )} e f + 8 \, {\left (2 \, C b^{3} e f + B b^{3} f^{2}\right )} x^{2} + 3 \, {\left (4 \, C b^{3} e^{2} + 8 \, B b^{3} e f + {\left (3 \, C a^{2} b + 4 \, A b^{3}\right )} f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{48 \, b^{5} c}, -\frac {3 \, {\left (8 \, B a^{2} b^{2} e f + 4 \, {\left (C a^{2} b^{2} + 2 \, A b^{4}\right )} e^{2} + {\left (3 \, C a^{4} + 4 \, A a^{2} b^{2}\right )} f^{2}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (6 \, C b^{3} f^{2} x^{3} + 24 \, B b^{3} e^{2} + 16 \, B a^{2} b f^{2} + 16 \, {\left (2 \, C a^{2} b + 3 \, A b^{3}\right )} e f + 8 \, {\left (2 \, C b^{3} e f + B b^{3} f^{2}\right )} x^{2} + 3 \, {\left (4 \, C b^{3} e^{2} + 8 \, B b^{3} e f + {\left (3 \, C a^{2} b + 4 \, A b^{3}\right )} f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{24 \, b^{5} c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(3*(8*B*a^2*b^2*e*f + 4*(C*a^2*b^2 + 2*A*b^4)*e^2 + (3*C*a^4 + 4*A*a^2*b^2)*f^2)*sqrt(-c)*log(2*b^2*c*x
^2 - 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(-c)*x - a^2*c) + 2*(6*C*b^3*f^2*x^3 + 24*B*b^3*e^2 + 16*B*a^2*b
*f^2 + 16*(2*C*a^2*b + 3*A*b^3)*e*f + 8*(2*C*b^3*e*f + B*b^3*f^2)*x^2 + 3*(4*C*b^3*e^2 + 8*B*b^3*e*f + (3*C*a^
2*b + 4*A*b^3)*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(b^5*c), -1/24*(3*(8*B*a^2*b^2*e*f + 4*(C*a^2*b^2 + 2
*A*b^4)*e^2 + (3*C*a^4 + 4*A*a^2*b^2)*f^2)*sqrt(c)*arctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(c)*x/(b^2*c*
x^2 - a^2*c)) + (6*C*b^3*f^2*x^3 + 24*B*b^3*e^2 + 16*B*a^2*b*f^2 + 16*(2*C*a^2*b + 3*A*b^3)*e*f + 8*(2*C*b^3*e
*f + B*b^3*f^2)*x^2 + 3*(4*C*b^3*e^2 + 8*B*b^3*e*f + (3*C*a^2*b + 4*A*b^3)*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x
+ a))/(b^5*c)]

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giac [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((f*x+e)^2*(C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.03, size = 635, normalized size = 1.73 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {-\left (b x -a \right ) c}\, \left (12 A \,a^{2} b^{2} c \,f^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+24 A \,b^{4} c \,e^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+24 B \,a^{2} b^{2} c e f \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+9 C \,a^{4} c \,f^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+12 C \,a^{2} b^{2} c \,e^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )-6 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,b^{2} f^{2} x^{3}-8 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,b^{2} f^{2} x^{2}-16 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,b^{2} e f \,x^{2}-12 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, A \,b^{2} f^{2} x -24 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,b^{2} e f x -9 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,a^{2} f^{2} x -12 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,b^{2} e^{2} x -48 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, A \,b^{2} e f -16 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,a^{2} f^{2}-24 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,b^{2} e^{2}-32 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,a^{2} e f \right )}{24 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, b^{4} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^2*(C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x)

[Out]

1/24*(b*x+a)^(1/2)*(-(b*x-a)*c)^(1/2)/c*(-6*C*x^3*b^2*f^2*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)+12*A*arctan((
b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^2*b^2*c*f^2+24*A*arctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*b
^4*c*e^2+24*B*arctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^2*b^2*c*e*f-8*B*x^2*b^2*f^2*(b^2*c)^(1/2)*(-(
b^2*x^2-a^2)*c)^(1/2)+9*C*arctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^4*c*f^2+12*C*arctan((b^2*c)^(1/2)
/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^2*b^2*c*e^2-16*C*x^2*b^2*e*f*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-12*A*(b^2*c
)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*b^2*f^2-24*B*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*b^2*e*f-9*C*(b^2*c)^(
1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^2*f^2-12*C*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*b^2*e^2-48*A*(b^2*c)^(1/
2)*(-(b^2*x^2-a^2)*c)^(1/2)*b^2*e*f-16*B*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*a^2*f^2-24*B*(b^2*c)^(1/2)*(-(
b^2*x^2-a^2)*c)^(1/2)*b^2*e^2-32*C*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*a^2*e*f)/b^4/(-(b^2*x^2-a^2)*c)^(1/2
)/(b^2*c)^(1/2)

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maxima [A]  time = 2.02, size = 317, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {-b^{2} c x^{2} + a^{2} c} C f^{2} x^{3}}{4 \, b^{2} c} + \frac {A e^{2} \arcsin \left (\frac {b x}{a}\right )}{b \sqrt {c}} + \frac {3 \, C a^{4} f^{2} \arcsin \left (\frac {b x}{a}\right )}{8 \, b^{5} \sqrt {c}} - \frac {3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} C a^{2} f^{2} x}{8 \, b^{4} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} B e^{2}}{b^{2} c} - \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} A e f}{b^{2} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (2 \, C e f + B f^{2}\right )} x^{2}}{3 \, b^{2} c} + \frac {{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} a^{2} \arcsin \left (\frac {b x}{a}\right )}{2 \, b^{3} \sqrt {c}} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (C e^{2} + 2 \, B e f + A f^{2}\right )} x}{2 \, b^{2} c} - \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} {\left (2 \, C e f + B f^{2}\right )} a^{2}}{3 \, b^{4} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*(C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="maxima")

[Out]

-1/4*sqrt(-b^2*c*x^2 + a^2*c)*C*f^2*x^3/(b^2*c) + A*e^2*arcsin(b*x/a)/(b*sqrt(c)) + 3/8*C*a^4*f^2*arcsin(b*x/a
)/(b^5*sqrt(c)) - 3/8*sqrt(-b^2*c*x^2 + a^2*c)*C*a^2*f^2*x/(b^4*c) - sqrt(-b^2*c*x^2 + a^2*c)*B*e^2/(b^2*c) -
2*sqrt(-b^2*c*x^2 + a^2*c)*A*e*f/(b^2*c) - 1/3*sqrt(-b^2*c*x^2 + a^2*c)*(2*C*e*f + B*f^2)*x^2/(b^2*c) + 1/2*(C
*e^2 + 2*B*e*f + A*f^2)*a^2*arcsin(b*x/a)/(b^3*sqrt(c)) - 1/2*sqrt(-b^2*c*x^2 + a^2*c)*(C*e^2 + 2*B*e*f + A*f^
2)*x/(b^2*c) - 2/3*sqrt(-b^2*c*x^2 + a^2*c)*(2*C*e*f + B*f^2)*a^2/(b^4*c)

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mupad [B]  time = 81.65, size = 2799, normalized size = 7.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e + f*x)^2*(A + B*x + C*x^2))/((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)

[Out]

- ((a^(1/2)*(a*c)^(1/2)*(64*B*a^2*c*f^2 + 32*B*b^2*c*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/(b^4*((a + b*
x)^(1/2) - a^(1/2))^8) + (a^(1/2)*(a*c)^(1/2)*(64*B*a^2*c^3*f^2 + 32*B*b^2*c^3*e^2)*((a*c - b*c*x)^(1/2) - (a*
c)^(1/2))^4)/(b^4*((a + b*x)^(1/2) - a^(1/2))^4) - (a^(1/2)*(a*c)^(1/2)*((128*B*a^2*c^2*f^2)/3 - 48*B*b^2*c^2*
e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/(b^4*((a + b*x)^(1/2) - a^(1/2))^6) + (4*B*a^2*e*f*((a*c - b*c*x)^
(1/2) - (a*c)^(1/2))^11)/(b^3*((a + b*x)^(1/2) - a^(1/2))^11) + (8*B*a^(1/2)*e^2*(a*c)^(1/2)*((a*c - b*c*x)^(1
/2) - (a*c)^(1/2))^10)/(b^2*((a + b*x)^(1/2) - a^(1/2))^10) + (20*B*a^2*c^4*e*f*((a*c - b*c*x)^(1/2) - (a*c)^(
1/2))^3)/(b^3*((a + b*x)^(1/2) - a^(1/2))^3) + (24*B*a^2*c^3*e*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(b^3*(
(a + b*x)^(1/2) - a^(1/2))^5) - (24*B*a^2*c^2*e*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(b^3*((a + b*x)^(1/2)
- a^(1/2))^7) + (8*B*a^(1/2)*c^4*e^2*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^2*((a + b*x)^(1/2)
- a^(1/2))^2) - (4*B*a^2*c^5*e*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^3*((a + b*x)^(1/2) - a^(1/2))) - (20
*B*a^2*c*e*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^9)/(b^3*((a + b*x)^(1/2) - a^(1/2))^9))/(((a*c - b*c*x)^(1/2)
- (a*c)^(1/2))^12/((a + b*x)^(1/2) - a^(1/2))^12 + c^6 + (6*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^10)/((a + b
*x)^(1/2) - a^(1/2))^10 + (6*c^5*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (15*c^
4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/((a + b*x)^(1/2) - a^(1/2))^4 + (20*c^3*((a*c - b*c*x)^(1/2) - (a*c)^
(1/2))^6)/((a + b*x)^(1/2) - a^(1/2))^6 + (15*c^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/((a + b*x)^(1/2) - a^
(1/2))^8) - ((2*A*a^2*f^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(b^3*((a + b*x)^(1/2) - a^(1/2))^7) + (14*A*a
^2*c^2*f^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/(b^3*((a + b*x)^(1/2) - a^(1/2))^3) - (2*A*a^2*c^3*f^2*((a*c
- b*c*x)^(1/2) - (a*c)^(1/2)))/(b^3*((a + b*x)^(1/2) - a^(1/2))) - (14*A*a^2*c*f^2*((a*c - b*c*x)^(1/2) - (a*
c)^(1/2))^5)/(b^3*((a + b*x)^(1/2) - a^(1/2))^5) + (16*A*a^(1/2)*e*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^
(1/2))^6)/(b^2*((a + b*x)^(1/2) - a^(1/2))^6) + (32*A*a^(1/2)*c*e*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(
1/2))^4)/(b^2*((a + b*x)^(1/2) - a^(1/2))^4) + (16*A*a^(1/2)*c^2*e*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^
(1/2))^2)/(b^2*((a + b*x)^(1/2) - a^(1/2))^2))/(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8/((a + b*x)^(1/2) - a^(1/
2))^8 + c^4 + (4*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^(1/2) - a^(1/2))^6 + (4*c^3*((a*c - b*c*x
)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (6*c^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/((a +
b*x)^(1/2) - a^(1/2))^4) - ((((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5*((333*C*a^4*c^5*f^2)/2 + 30*C*a^2*b^2*c^5*e
^2))/(b^5*((a + b*x)^(1/2) - a^(1/2))^5) - (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3*((23*C*a^4*c^6*f^2)/2 - 6*C*
a^2*b^2*c^6*e^2))/(b^5*((a + b*x)^(1/2) - a^(1/2))^3) - (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*((3*C*a^4*c^7*f^2
)/2 + 2*C*a^2*b^2*c^7*e^2))/(b^5*((a + b*x)^(1/2) - a^(1/2))) - (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^11*((333*
C*a^4*c^2*f^2)/2 + 30*C*a^2*b^2*c^2*e^2))/(b^5*((a + b*x)^(1/2) - a^(1/2))^11) - (((a*c - b*c*x)^(1/2) - (a*c)
^(1/2))^7*((671*C*a^4*c^4*f^2)/2 - 22*C*a^2*b^2*c^4*e^2))/(b^5*((a + b*x)^(1/2) - a^(1/2))^7) + (((a*c - b*c*x
)^(1/2) - (a*c)^(1/2))^9*((671*C*a^4*c^3*f^2)/2 - 22*C*a^2*b^2*c^3*e^2))/(b^5*((a + b*x)^(1/2) - a^(1/2))^9) +
(((23*C*a^4*c*f^2)/2 - 6*C*a^2*b^2*c*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^13)/(b^5*((a + b*x)^(1/2) - a^(
1/2))^13) + (((3*C*a^4*f^2)/2 + 2*C*a^2*b^2*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^15)/(b^5*((a + b*x)^(1/2)
- a^(1/2))^15) + (128*C*a^(5/2)*c*e*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^12)/(b^4*((a + b*x)^(1/
2) - a^(1/2))^12) + (128*C*a^(5/2)*c^5*e*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/(b^4*((a + b*x)^
(1/2) - a^(1/2))^4) + (512*C*a^(5/2)*c^4*e*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/(3*b^4*((a + b
*x)^(1/2) - a^(1/2))^6) + (256*C*a^(5/2)*c^3*e*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/(3*b^4*((a
+ b*x)^(1/2) - a^(1/2))^8) + (512*C*a^(5/2)*c^2*e*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^10)/(3*b^
4*((a + b*x)^(1/2) - a^(1/2))^10))/(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^16/((a + b*x)^(1/2) - a^(1/2))^16 + c^
8 + (8*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^14)/((a + b*x)^(1/2) - a^(1/2))^14 + (8*c^7*((a*c - b*c*x)^(1/2)
- (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (28*c^6*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/((a + b*x)^(1
/2) - a^(1/2))^4 + (56*c^5*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^(1/2) - a^(1/2))^6 + (70*c^4*((a*
c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/((a + b*x)^(1/2) - a^(1/2))^8 + (56*c^3*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))
^10)/((a + b*x)^(1/2) - a^(1/2))^10 + (28*c^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^12)/((a + b*x)^(1/2) - a^(1/
2))^12) - (2*A*atan((A*(a^2*f^2 + 2*b^2*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(c^(1/2)*(A*a^2*f^2 + 2*A*b^
2*e^2)*((a + b*x)^(1/2) - a^(1/2))))*(a^2*f^2 + 2*b^2*e^2))/(b^3*c^(1/2)) - (C*a^2*atan((C*a^2*(3*a^2*f^2 + 4*
b^2*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(c^(1/2)*(3*C*a^4*f^2 + 4*C*a^2*b^2*e^2)*((a + b*x)^(1/2) - a^(1
/2))))*(3*a^2*f^2 + 4*b^2*e^2))/(2*b^5*c^(1/2)) - (4*B*a^2*e*f*atan(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))/(c^(1/
2)*((a + b*x)^(1/2) - a^(1/2)))))/(b^3*c^(1/2))

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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((f*x+e)**2*(C*x**2+B*x+A)/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)

[Out]

Timed out

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