3.27 \(\int \frac {(e+f x)^3 (A+B x+C x^2)}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\)

Optimal. Leaf size=501 \[ -\frac {\left (a^2-b^2 x^2\right ) (e+f x)^2 \left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (4 A f+3 B e)\right )\right )}{60 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (4 A \left (3 a^2 b^2 e f^2+2 b^4 e^3\right )+a^2 \left (3 a^2 f^2 (B f+3 C e)+4 b^2 e^2 (3 B f+C e)\right )\right )}{8 b^5 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (a^2-b^2 x^2\right ) (e+f x)^3 (C e-5 B f)}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C \left (a^2-b^2 x^2\right ) (e+f x)^4}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (a^2-b^2 x^2\right ) \left (b^2 f x \left (a^2 f^2 (45 B f+71 C e)-2 b^2 e \left (3 C e^2-5 f (10 A f+3 B e)\right )\right )+4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (5 f (A f+3 B e)+13 C e^2\right )+b^4 \left (-e^2\right ) \left (3 C e^2-5 f (16 A f+3 B e)\right )\right )\right )}{120 b^6 f \sqrt {a+b x} \sqrt {a c-b c x}} \]

[Out]

-1/60*(16*a^2*C*f^2-b^2*(3*C*e^2-5*f*(4*A*f+3*B*e)))*(f*x+e)^2*(-b^2*x^2+a^2)/b^4/f/(b*x+a)^(1/2)/(-b*c*x+a*c)
^(1/2)+1/20*(-5*B*f+C*e)*(f*x+e)^3*(-b^2*x^2+a^2)/b^2/f/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-1/5*C*(f*x+e)^4*(-b^2
*x^2+a^2)/b^2/f/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-1/120*(64*a^4*C*f^4+16*a^2*b^2*f^2*(13*C*e^2+5*f*(A*f+3*B*e))
-4*b^4*e^2*(3*C*e^2-5*f*(16*A*f+3*B*e))+b^2*f*(a^2*f^2*(45*B*f+71*C*e)-2*b^2*e*(3*C*e^2-5*f*(10*A*f+3*B*e)))*x
)*(-b^2*x^2+a^2)/b^6/f/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+1/8*(4*A*(3*a^2*b^2*e*f^2+2*b^4*e^3)+a^2*(3*a^2*f^2*(B
*f+3*C*e)+4*b^2*e^2*(3*B*f+C*e)))*arctan(b*x*c^(1/2)/(-b^2*c*x^2+a^2*c)^(1/2))*(-b^2*c*x^2+a^2*c)^(1/2)/b^5/c^
(1/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)

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Rubi [A]  time = 1.28, antiderivative size = 496, normalized size of antiderivative = 0.99, number of steps used = 7, 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} \[ \frac {\left (a^2-b^2 x^2\right ) (e+f x)^2 \left (-\frac {16 a^2 C f^2}{b^2}-5 f (4 A f+3 B e)+3 C e^2\right )}{60 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (a^2-b^2 x^2\right ) \left (b^2 f x \left (a^2 f^2 (45 B f+71 C e)-b^2 \left (6 C e^3-10 e f (10 A f+3 B e)\right )\right )+4 \left (4 a^2 b^2 f^2 \left (5 f (A f+3 B e)+13 C e^2\right )+16 a^4 C f^4+b^4 \left (-e^2\right ) \left (3 C e^2-5 f (16 A f+3 B e)\right )\right )\right )}{120 b^6 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\sqrt {a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (4 A \left (3 a^2 b^2 e f^2+2 b^4 e^3\right )+4 a^2 b^2 e^2 (3 B f+C e)+3 a^4 f^2 (B f+3 C e)\right )}{8 b^5 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (a^2-b^2 x^2\right ) (e+f x)^3 (C e-5 B f)}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C \left (a^2-b^2 x^2\right ) (e+f x)^4}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*C*e^2 - (16*a^2*C*f^2)/b^2 - 5*f*(3*B*e + 4*A*f))*(e + f*x)^2*(a^2 - b^2*x^2))/(60*b^2*f*Sqrt[a + b*x]*Sqr
t[a*c - b*c*x]) + ((C*e - 5*B*f)*(e + f*x)^3*(a^2 - b^2*x^2))/(20*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (C*
(e + f*x)^4*(a^2 - b^2*x^2))/(5*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - ((4*(16*a^4*C*f^4 + 4*a^2*b^2*f^2*(13
*C*e^2 + 5*f*(3*B*e + A*f)) - b^4*e^2*(3*C*e^2 - 5*f*(3*B*e + 16*A*f))) + b^2*f*(a^2*f^2*(71*C*e + 45*B*f) - b
^2*(6*C*e^3 - 10*e*f*(3*B*e + 10*A*f)))*x)*(a^2 - b^2*x^2))/(120*b^6*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + ((3*
a^4*f^2*(3*C*e + B*f) + 4*a^2*b^2*e^2*(C*e + 3*B*f) + 4*A*(2*b^4*e^3 + 3*a^2*b^2*e*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)^3 \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)^3 \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)^4 \left (a^2-b^2 x^2\right )}{5 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)^3 \left (-c \left (5 A b^2+4 a^2 C\right ) f^2+b^2 c f (C e-5 B f) x\right )}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{5 b^2 c f^2 \sqrt {a+b x} \sqrt {a c-b c x}}\\ &=\frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 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 (b^2 c^2 f^2 \left (20 A b^2 e+a^2 (13 C e+15 B f)\right )+b^2 c^2 f \left (4 \left (5 A b^2+4 a^2 C\right ) f^2-3 b^2 e (C e-5 B f)\right ) x\right )}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{20 b^4 c^2 f^2 \sqrt {a+b x} \sqrt {a c-b c x}}\\ &=-\frac {\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) (e+f x)^2 \left (a^2-b^2 x^2\right )}{60 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 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^3 f^2 \left (32 a^4 C f^2+3 a^2 b^2 e (11 C e+25 B f)+20 A \left (3 b^4 e^2+2 a^2 b^2 f^2\right )\right )-b^4 c^3 f \left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x\right )}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{60 b^6 c^3 f^2 \sqrt {a+b x} \sqrt {a c-b c x}}\\ &=-\frac {\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) (e+f x)^2 \left (a^2-b^2 x^2\right )}{60 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (13 C e^2+5 f (3 B e+A f)\right )-b^4 e^2 \left (3 C e^2-5 f (3 B e+16 A f)\right )\right )+b^2 f \left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^6 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (\left (3 a^4 f^2 (3 C e+B f)+4 a^2 b^2 e^2 (C e+3 B f)+4 A \left (2 b^4 e^3+3 a^2 b^2 e 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 {\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) (e+f x)^2 \left (a^2-b^2 x^2\right )}{60 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (13 C e^2+5 f (3 B e+A f)\right )-b^4 e^2 \left (3 C e^2-5 f (3 B e+16 A f)\right )\right )+b^2 f \left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^6 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (\left (3 a^4 f^2 (3 C e+B f)+4 a^2 b^2 e^2 (C e+3 B f)+4 A \left (2 b^4 e^3+3 a^2 b^2 e 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 {\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) (e+f x)^2 \left (a^2-b^2 x^2\right )}{60 b^4 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {(C e-5 B f) (e+f x)^3 \left (a^2-b^2 x^2\right )}{20 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C (e+f x)^4 \left (a^2-b^2 x^2\right )}{5 b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (13 C e^2+5 f (3 B e+A f)\right )-b^4 e^2 \left (3 C e^2-5 f (3 B e+16 A f)\right )\right )+b^2 f \left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^6 f \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (3 a^4 f^2 (3 C e+B f)+4 a^2 b^2 e^2 (C e+3 B f)+4 A \left (2 b^4 e^3+3 a^2 b^2 e 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 = 4.90, size = 727, normalized size = 1.45 \[ \frac {-120 \sqrt {a-b x} \sqrt {a+b x} (b e-a f)^2 \left (\sqrt {a-b x} \sqrt {\frac {b x}{a}+1}+2 \sqrt {a} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )\right ) \left (5 a^2 C f-2 a b (2 B f+C e)+b^2 (3 A f+B e)\right )-20 f \sqrt {a-b x} \sqrt {a+b x} \left (30 a^{5/2} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )+\sqrt {a-b x} \sqrt {\frac {b x}{a}+1} \left (22 a^2+9 a b x+2 b^2 x^2\right )\right ) \left (10 a^2 C f^2-4 a b f (B f+3 C e)+b^2 \left (f (A f+3 B e)+3 C e^2\right )\right )-60 \sqrt {a-b x} \sqrt {a+b x} (b e-a f) \left (6 a^{3/2} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )+\sqrt {a-b x} (4 a+b x) \sqrt {\frac {b x}{a}+1}\right ) \left (10 a^2 C f^2-2 a b f (3 B f+4 C e)+b^2 \left (3 f (A f+B e)+C e^2\right )\right )-5 f^2 \sqrt {a-b x} \sqrt {a+b x} \left (210 a^{7/2} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )+\sqrt {a-b x} \sqrt {\frac {b x}{a}+1} \left (160 a^3+81 a^2 b x+32 a b^2 x^2+6 b^3 x^3\right )\right ) (-5 a C f+b B f+3 b C e)-3 C f^3 \sqrt {a+b x} \left (630 a^{9/2} \sqrt {a-b x} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )+(a-b x) \sqrt {\frac {b x}{a}+1} \left (488 a^4+275 a^3 b x+144 a^2 b^2 x^2+50 a b^3 x^3+8 b^4 x^4\right )\right )-240 \sqrt {a-b x} \sqrt {\frac {b x}{a}+1} (b e-a f)^3 \tan ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {a+b x}}\right ) \left (a (a C-b B)+A b^2\right )}{120 b^6 \sqrt {\frac {b x}{a}+1} \sqrt {c (a-b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-120*(b*e - a*f)^2*(5*a^2*C*f + b^2*(B*e + 3*A*f) - 2*a*b*(C*e + 2*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])]) - 60*(b*e - a*f)*(10*a^2*C*f^2 -
2*a*b*f*(4*C*e + 3*B*f) + b^2*(C*e^2 + 3*f*(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])]) - 20*f*(10*a^2*C*f^2 - 4*a*b*f*(3*C*
e + B*f) + b^2*(3*C*e^2 + f*(3*B*e + A*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]/(Sqrt[2]*Sqrt[a])]) - 5*f^2*(3*b*C*e + b*B*f - 5*a
*C*f)*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[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)*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 3*C*f^3*Sqrt[a + b*x]*((a - b*x)*Sqrt[1 + (b*x
)/a]*(488*a^4 + 275*a^3*b*x + 144*a^2*b^2*x^2 + 50*a*b^3*x^3 + 8*b^4*x^4) + 630*a^(9/2)*Sqrt[a - b*x]*ArcSin[S
qrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 240*(A*b^2 + a*(-(b*B) + a*C))*(b*e - a*f)^3*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a
]*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/(120*b^6*Sqrt[c*(a - b*x)]*Sqrt[1 + (b*x)/a])

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fricas [A]  time = 0.78, size = 700, normalized size = 1.40 \[ \left [-\frac {15 \, {\left (12 \, B a^{2} b^{3} e^{2} f + 3 \, B a^{4} b f^{3} + 4 \, {\left (C a^{2} b^{3} + 2 \, A b^{5}\right )} e^{3} + 3 \, {\left (3 \, C a^{4} b + 4 \, A a^{2} b^{3}\right )} e f^{2}\right )} \sqrt {-c} \log \left (2 \, b^{2} c x^{2} - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) + 2 \, {\left (24 \, C b^{4} f^{3} x^{4} + 120 \, B b^{4} e^{3} + 240 \, B a^{2} b^{2} e f^{2} + 120 \, {\left (2 \, C a^{2} b^{2} + 3 \, A b^{4}\right )} e^{2} f + 16 \, {\left (4 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f^{3} + 30 \, {\left (3 \, C b^{4} e f^{2} + B b^{4} f^{3}\right )} x^{3} + 8 \, {\left (15 \, C b^{4} e^{2} f + 15 \, B b^{4} e f^{2} + {\left (4 \, C a^{2} b^{2} + 5 \, A b^{4}\right )} f^{3}\right )} x^{2} + 15 \, {\left (4 \, C b^{4} e^{3} + 12 \, B b^{4} e^{2} f + 3 \, B a^{2} b^{2} f^{3} + 3 \, {\left (3 \, C a^{2} b^{2} + 4 \, A b^{4}\right )} e f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{240 \, b^{6} c}, -\frac {15 \, {\left (12 \, B a^{2} b^{3} e^{2} f + 3 \, B a^{4} b f^{3} + 4 \, {\left (C a^{2} b^{3} + 2 \, A b^{5}\right )} e^{3} + 3 \, {\left (3 \, C a^{4} b + 4 \, A a^{2} b^{3}\right )} e f^{2}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (24 \, C b^{4} f^{3} x^{4} + 120 \, B b^{4} e^{3} + 240 \, B a^{2} b^{2} e f^{2} + 120 \, {\left (2 \, C a^{2} b^{2} + 3 \, A b^{4}\right )} e^{2} f + 16 \, {\left (4 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f^{3} + 30 \, {\left (3 \, C b^{4} e f^{2} + B b^{4} f^{3}\right )} x^{3} + 8 \, {\left (15 \, C b^{4} e^{2} f + 15 \, B b^{4} e f^{2} + {\left (4 \, C a^{2} b^{2} + 5 \, A b^{4}\right )} f^{3}\right )} x^{2} + 15 \, {\left (4 \, C b^{4} e^{3} + 12 \, B b^{4} e^{2} f + 3 \, B a^{2} b^{2} f^{3} + 3 \, {\left (3 \, C a^{2} b^{2} + 4 \, A b^{4}\right )} e f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{120 \, b^{6} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/240*(15*(12*B*a^2*b^3*e^2*f + 3*B*a^4*b*f^3 + 4*(C*a^2*b^3 + 2*A*b^5)*e^3 + 3*(3*C*a^4*b + 4*A*a^2*b^3)*e*
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*(24*C*b^4*f^3*x^4
+ 120*B*b^4*e^3 + 240*B*a^2*b^2*e*f^2 + 120*(2*C*a^2*b^2 + 3*A*b^4)*e^2*f + 16*(4*C*a^4 + 5*A*a^2*b^2)*f^3 +
30*(3*C*b^4*e*f^2 + B*b^4*f^3)*x^3 + 8*(15*C*b^4*e^2*f + 15*B*b^4*e*f^2 + (4*C*a^2*b^2 + 5*A*b^4)*f^3)*x^2 + 1
5*(4*C*b^4*e^3 + 12*B*b^4*e^2*f + 3*B*a^2*b^2*f^3 + 3*(3*C*a^2*b^2 + 4*A*b^4)*e*f^2)*x)*sqrt(-b*c*x + a*c)*sqr
t(b*x + a))/(b^6*c), -1/120*(15*(12*B*a^2*b^3*e^2*f + 3*B*a^4*b*f^3 + 4*(C*a^2*b^3 + 2*A*b^5)*e^3 + 3*(3*C*a^4
*b + 4*A*a^2*b^3)*e*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)) + (2
4*C*b^4*f^3*x^4 + 120*B*b^4*e^3 + 240*B*a^2*b^2*e*f^2 + 120*(2*C*a^2*b^2 + 3*A*b^4)*e^2*f + 16*(4*C*a^4 + 5*A*
a^2*b^2)*f^3 + 30*(3*C*b^4*e*f^2 + B*b^4*f^3)*x^3 + 8*(15*C*b^4*e^2*f + 15*B*b^4*e*f^2 + (4*C*a^2*b^2 + 5*A*b^
4)*f^3)*x^2 + 15*(4*C*b^4*e^3 + 12*B*b^4*e^2*f + 3*B*a^2*b^2*f^3 + 3*(3*C*a^2*b^2 + 4*A*b^4)*e*f^2)*x)*sqrt(-b
*c*x + a*c)*sqrt(b*x + a))/(b^6*c)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*(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 [B]  time = 0.03, size = 965, normalized size = 1.93 \[ \frac {\sqrt {b x +a}\, \sqrt {-\left (b x -a \right ) c}\, \left (180 A \,a^{2} b^{4} c e \,f^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+120 A \,b^{6} c \,e^{3} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+45 B \,a^{4} b^{2} c \,f^{3} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+180 B \,a^{2} b^{4} c \,e^{2} f \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+135 C \,a^{4} b^{2} c e \,f^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+60 C \,a^{2} b^{4} c \,e^{3} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )-24 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,b^{4} f^{3} x^{4}-30 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,b^{4} f^{3} x^{3}-90 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,b^{4} e \,f^{2} x^{3}-40 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, A \,b^{4} f^{3} x^{2}-120 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,b^{4} e \,f^{2} x^{2}-32 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,a^{2} b^{2} f^{3} x^{2}-120 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,b^{4} e^{2} f \,x^{2}-180 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, A \,b^{4} e \,f^{2} x -45 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,a^{2} b^{2} f^{3} x -180 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,b^{4} e^{2} f x -135 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,a^{2} b^{2} e \,f^{2} x -60 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,b^{4} e^{3} x -80 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, A \,a^{2} b^{2} f^{3}-360 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, A \,b^{4} e^{2} f -240 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,a^{2} b^{2} e \,f^{2}-120 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,b^{4} e^{3}-64 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,a^{4} f^{3}-240 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,a^{2} b^{2} e^{2} f \right )}{120 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, b^{6} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(b*x+a)^(1/2)*(-(b*x-a)*c)^(1/2)/c*(-24*C*x^4*b^4*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-30*B*x^3*b^
4*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-90*C*x^3*b^4*e*f^2*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)+180*A*a
rctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^2*b^4*c*e*f^2+120*A*arctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^
(1/2)*x)*b^6*c*e^3-40*A*x^2*b^4*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)+45*B*arctan((b^2*c)^(1/2)/(-(b^2*x^
2-a^2)*c)^(1/2)*x)*a^4*b^2*c*f^3+180*B*arctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^2*b^4*c*e^2*f-120*B*
x^2*b^4*e*f^2*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)+135*C*arctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^
4*b^2*c*e*f^2+60*C*arctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^2*b^4*c*e^3-32*C*x^2*a^2*b^2*f^3*(b^2*c)
^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-120*C*x^2*b^4*e^2*f*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-180*A*(b^2*c)^(1/2)
*(-(b^2*x^2-a^2)*c)^(1/2)*x*b^4*e*f^2-45*B*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^2*b^2*f^3-180*B*(b^2*c)^
(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*b^4*e^2*f-135*C*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^2*b^2*e*f^2-60*C*(
b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*b^4*e^3-80*A*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*a^2*b^2*f^3-360*A*
(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*b^4*e^2*f-240*B*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*a^2*b^2*e*f^2-12
0*B*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*b^4*e^3-64*C*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*a^4*f^3-240*C*(
b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*a^2*b^2*e^2*f)/b^6/(-(b^2*x^2-a^2)*c)^(1/2)/(b^2*c)^(1/2)

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maxima [A]  time = 1.97, size = 471, normalized size = 0.94 \[ -\frac {\sqrt {-b^{2} c x^{2} + a^{2} c} C f^{3} x^{4}}{5 \, b^{2} c} - \frac {4 \, \sqrt {-b^{2} c x^{2} + a^{2} c} C a^{2} f^{3} x^{2}}{15 \, b^{4} c} + \frac {A e^{3} \arcsin \left (\frac {b x}{a}\right )}{b \sqrt {c}} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} B e^{3}}{b^{2} c} - \frac {3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} A e^{2} f}{b^{2} c} - \frac {8 \, \sqrt {-b^{2} c x^{2} + a^{2} c} C a^{4} f^{3}}{15 \, b^{6} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e f^{2} + B f^{3}\right )} x^{3}}{4 \, b^{2} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} x^{2}}{3 \, b^{2} c} + \frac {3 \, {\left (3 \, C e f^{2} + B f^{3}\right )} a^{4} \arcsin \left (\frac {b x}{a}\right )}{8 \, b^{5} \sqrt {c}} + \frac {{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} a^{2} \arcsin \left (\frac {b x}{a}\right )}{2 \, b^{3} \sqrt {c}} - \frac {3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e f^{2} + B f^{3}\right )} a^{2} x}{8 \, b^{4} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} x}{2 \, b^{2} c} - \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} a^{2}}{3 \, b^{4} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*sqrt(-b^2*c*x^2 + a^2*c)*C*f^3*x^4/(b^2*c) - 4/15*sqrt(-b^2*c*x^2 + a^2*c)*C*a^2*f^3*x^2/(b^4*c) + A*e^3*
arcsin(b*x/a)/(b*sqrt(c)) - sqrt(-b^2*c*x^2 + a^2*c)*B*e^3/(b^2*c) - 3*sqrt(-b^2*c*x^2 + a^2*c)*A*e^2*f/(b^2*c
) - 8/15*sqrt(-b^2*c*x^2 + a^2*c)*C*a^4*f^3/(b^6*c) - 1/4*sqrt(-b^2*c*x^2 + a^2*c)*(3*C*e*f^2 + B*f^3)*x^3/(b^
2*c) - 1/3*sqrt(-b^2*c*x^2 + a^2*c)*(3*C*e^2*f + 3*B*e*f^2 + A*f^3)*x^2/(b^2*c) + 3/8*(3*C*e*f^2 + B*f^3)*a^4*
arcsin(b*x/a)/(b^5*sqrt(c)) + 1/2*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*a^2*arcsin(b*x/a)/(b^3*sqrt(c)) - 3/8*sqrt(-
b^2*c*x^2 + a^2*c)*(3*C*e*f^2 + B*f^3)*a^2*x/(b^4*c) - 1/2*sqrt(-b^2*c*x^2 + a^2*c)*(C*e^3 + 3*B*e^2*f + 3*A*e
*f^2)*x/(b^2*c) - 2/3*sqrt(-b^2*c*x^2 + a^2*c)*(3*C*e^2*f + 3*B*e*f^2 + A*f^3)*a^2/(b^4*c)

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mupad [B]  time = 161.43, size = 4167, normalized size = 8.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((((23*B*a^4*c*f^3)/2 - 18*B*a^2*b^2*c*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^13)/(b^5*((a + b*x)^(1/2)
- a^(1/2))^13) + (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^15*((3*B*a^4*f^3)/2 + 6*B*a^2*b^2*e^2*f))/(b^5*((a + b*x
)^(1/2) - a^(1/2))^15) - (((3*B*a^4*c^7*f^3)/2 + 6*B*a^2*b^2*c^7*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(
b^5*((a + b*x)^(1/2) - a^(1/2))) - (((23*B*a^4*c^6*f^3)/2 - 18*B*a^2*b^2*c^6*e^2*f)*((a*c - b*c*x)^(1/2) - (a*
c)^(1/2))^3)/(b^5*((a + b*x)^(1/2) - a^(1/2))^3) + (((333*B*a^4*c^5*f^3)/2 + 90*B*a^2*b^2*c^5*e^2*f)*((a*c - b
*c*x)^(1/2) - (a*c)^(1/2))^5)/(b^5*((a + b*x)^(1/2) - a^(1/2))^5) - (((333*B*a^4*c^2*f^3)/2 + 90*B*a^2*b^2*c^2
*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^11)/(b^5*((a + b*x)^(1/2) - a^(1/2))^11) - (((671*B*a^4*c^4*f^3)/2
- 66*B*a^2*b^2*c^4*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(b^5*((a + b*x)^(1/2) - a^(1/2))^7) + (((671
*B*a^4*c^3*f^3)/2 - 66*B*a^2*b^2*c^3*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^9)/(b^5*((a + b*x)^(1/2) - a^(
1/2))^9) + (a^(1/2)*(a*c)^(1/2)*(48*B*b^2*c^5*e^3 + 192*B*a^2*c^5*e*f^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)*(160*B*b^2*c^3*e^3 + 128*B*a^2*c^3*e*f^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)*(120*B*b^2*c^4*e^3
+ 256*B*a^2*c^4*e*f^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/(b^4*((a + b*x)^(1/2) - a^(1/2))^6) + (a^(1/2)*(
a*c)^(1/2)*(120*B*b^2*c^2*e^3 + 256*B*a^2*c^2*e*f^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^10)/(b^4*((a + b*x)^(
1/2) - a^(1/2))^10) + (a^(1/2)*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^12*(48*B*b^2*c*e^3 + 192*B*a^2*
c*e*f^2))/(b^4*((a + b*x)^(1/2) - a^(1/2))^12) + (8*B*a^(1/2)*e^3*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/
2))^14)/(b^2*((a + b*x)^(1/2) - a^(1/2))^14) + (8*B*a^(1/2)*c^6*e^3*(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))^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) - ((a^(1/2)*(a*c)^(1/2)*(64*A*a^2*c^3*f^3 + 96*A*b^2*c^3*e^2*f)*((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*A*a^2*c^2*f^3)/3 - 144*A*b^2*c^2*
e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/(b^4*((a + b*x)^(1/2) - a^(1/2))^6) + (a^(1/2)*(a*c)^(1/2)*((a*c
- b*c*x)^(1/2) - (a*c)^(1/2))^8*(64*A*a^2*c*f^3 + 96*A*b^2*c*e^2*f))/(b^4*((a + b*x)^(1/2) - a^(1/2))^8) + (6
*A*a^2*e*f^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^11)/(b^3*((a + b*x)^(1/2) - a^(1/2))^11) - (6*A*a^2*c^5*e*f^2
*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^3*((a + b*x)^(1/2) - a^(1/2))) - (30*A*a^2*c*e*f^2*((a*c - b*c*x)^(1/
2) - (a*c)^(1/2))^9)/(b^3*((a + b*x)^(1/2) - a^(1/2))^9) + (24*A*a^(1/2)*e^2*f*(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) + (30*A*a^2*c^4*e*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) + (36*A*a^2*c^3*e*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) - (36*A*a^2*c^2*e*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) + (24*A*a^(1/2)*c^4*e^2*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))^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) - ((((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^19*((9*C*a^4*e*f
^2)/2 + 2*C*a^2*b^2*e^3))/(b^5*((a + b*x)^(1/2) - a^(1/2))^19) - ((2*C*a^2*b^2*c*e^3 - (87*C*a^4*c*e*f^2)/2)*(
(a*c - b*c*x)^(1/2) - (a*c)^(1/2))^17)/(b^5*((a + b*x)^(1/2) - a^(1/2))^17) - (((9*C*a^4*c^9*e*f^2)/2 + 2*C*a^
2*b^2*c^9*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^5*((a + b*x)^(1/2) - a^(1/2))) - (((87*C*a^4*c^8*e*f^2)
/2 - 2*C*a^2*b^2*c^8*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/(b^5*((a + b*x)^(1/2) - a^(1/2))^3) - ((42*C*
a^4*c^6*e*f^2 - 88*C*a^2*b^2*c^6*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(b^5*((a + b*x)^(1/2) - a^(1/2))^
7) + ((42*C*a^4*c^3*e*f^2 - 88*C*a^2*b^2*c^3*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^13)/(b^5*((a + b*x)^(1/2
) - a^(1/2))^13) + ((426*C*a^4*c^7*e*f^2 + 40*C*a^2*b^2*c^7*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(b^5*(
(a + b*x)^(1/2) - a^(1/2))^5) - ((426*C*a^4*c^2*e*f^2 + 40*C*a^2*b^2*c^2*e^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/
2))^15)/(b^5*((a + b*x)^(1/2) - a^(1/2))^15) - ((507*C*a^4*c^5*e*f^2 - 52*C*a^2*b^2*c^5*e^3)*((a*c - b*c*x)^(1
/2) - (a*c)^(1/2))^9)/(b^5*((a + b*x)^(1/2) - a^(1/2))^9) + ((507*C*a^4*c^4*e*f^2 - 52*C*a^2*b^2*c^4*e^3)*((a*
c - b*c*x)^(1/2) - (a*c)^(1/2))^11)/(b^5*((a + b*x)^(1/2) - a^(1/2))^11) + (a^(1/2)*(a*c)^(1/2)*((2048*C*a^4*c
^6*f^3)/3 + 640*C*a^2*b^2*c^6*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/(b^6*((a + b*x)^(1/2) - a^(1/2))^6
) + (a^(1/2)*(a*c)^(1/2)*((2048*C*a^4*c^2*f^3)/3 + 640*C*a^2*b^2*c^2*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)
)^14)/(b^6*((a + b*x)^(1/2) - a^(1/2))^14) - (a^(1/2)*(a*c)^(1/2)*((4096*C*a^4*c^5*f^3)/3 - 832*C*a^2*b^2*c^5*
e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/(b^6*((a + b*x)^(1/2) - a^(1/2))^8) - (a^(1/2)*(a*c)^(1/2)*((409
6*C*a^4*c^3*f^3)/3 - 832*C*a^2*b^2*c^3*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^12)/(b^6*((a + b*x)^(1/2) -
a^(1/2))^12) + (a^(1/2)*(a*c)^(1/2)*((12288*C*a^4*c^4*f^3)/5 + 768*C*a^2*b^2*c^4*e^2*f)*((a*c - b*c*x)^(1/2) -
(a*c)^(1/2))^10)/(b^6*((a + b*x)^(1/2) - a^(1/2))^10) + (192*C*a^(5/2)*c*e^2*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/
2) - (a*c)^(1/2))^16)/(b^4*((a + b*x)^(1/2) - a^(1/2))^16) + (192*C*a^(5/2)*c^7*e^2*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))/(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^20/((a +
b*x)^(1/2) - a^(1/2))^20 + c^10 + (10*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^18)/((a + b*x)^(1/2) - a^(1/2))^18
+ (10*c^9*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (45*c^8*((a*c - b*c*x)^(1/2)
- (a*c)^(1/2))^4)/((a + b*x)^(1/2) - a^(1/2))^4 + (120*c^7*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^
(1/2) - a^(1/2))^6 + (210*c^6*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/((a + b*x)^(1/2) - a^(1/2))^8 + (252*c^5*
((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^10)/((a + b*x)^(1/2) - a^(1/2))^10 + (210*c^4*((a*c - b*c*x)^(1/2) - (a*c)
^(1/2))^12)/((a + b*x)^(1/2) - a^(1/2))^12 + (120*c^3*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^14)/((a + b*x)^(1/2)
- a^(1/2))^14 + (45*c^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^16)/((a + b*x)^(1/2) - a^(1/2))^16) - (2*A*e*atan
((A*e*(3*a^2*f^2 + 2*b^2*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(c^(1/2)*(2*A*b^2*e^3 + 3*A*a^2*e*f^2)*((a
+ b*x)^(1/2) - a^(1/2))))*(3*a^2*f^2 + 2*b^2*e^2))/(b^3*c^(1/2)) - (3*B*a^2*f*atan((B*a^2*f*(a^2*f^2 + 4*b^2*e
^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(c^(1/2)*(B*a^4*f^3 + 4*B*a^2*b^2*e^2*f)*((a + b*x)^(1/2) - a^(1/2)))
)*(a^2*f^2 + 4*b^2*e^2))/(2*b^5*c^(1/2)) - (C*a^2*e*atan((C*a^2*e*(9*a^2*f^2 + 4*b^2*e^2)*((a*c - b*c*x)^(1/2)
- (a*c)^(1/2)))/(c^(1/2)*(9*C*a^4*e*f^2 + 4*C*a^2*b^2*e^3)*((a + b*x)^(1/2) - a^(1/2))))*(9*a^2*f^2 + 4*b^2*e
^2))/(2*b^5*c^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*(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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