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

Optimal. Leaf size=591 \[ -\frac {\sqrt {a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt {a c-b c x} \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (2 A f+B e)\right )\right )}{70 b^4 f}+\frac {x \sqrt {a+b x} \sqrt {a c-b c x} \left (A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+a^2 \left (a^2 f^2 (B f+3 C e)+2 b^2 e^2 (3 B f+C e)\right )\right )}{16 b^4}+\frac {a^2 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+a^2 \left (a^2 f^2 (B f+3 C e)+2 b^2 e^2 (3 B f+C e)\right )\right )}{16 b^5 \sqrt {a^2 c-b^2 c x^2}}+\frac {\sqrt {a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^3 \sqrt {a c-b c x} (3 C e-7 B f)}{42 b^2 f}-\frac {C \sqrt {a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^4 \sqrt {a c-b c x}}{7 b^2 f}-\frac {\sqrt {a+b x} \left (a^2-b^2 x^2\right ) \sqrt {a c-b c x} \left (3 b^2 f x \left (a^2 f^2 (35 B f+41 C e)-2 b^2 e \left (3 C e^2-7 f (7 A f+B e)\right )\right )+8 \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (7 f (A f+3 B e)+15 C e^2\right )+b^4 \left (-e^2\right ) \left (3 C e^2-7 f (12 A f+B e)\right )\right )\right )}{840 b^6 f} \]

[Out]

1/16*(A*(6*a^2*b^2*e*f^2+8*b^4*e^3)+a^2*(a^2*f^2*(B*f+3*C*e)+2*b^2*e^2*(3*B*f+C*e)))*x*(b*x+a)^(1/2)*(-b*c*x+a
*c)^(1/2)/b^4-1/70*(8*a^2*C*f^2-b^2*(3*C*e^2-7*f*(2*A*f+B*e)))*(f*x+e)^2*(-b^2*x^2+a^2)*(b*x+a)^(1/2)*(-b*c*x+
a*c)^(1/2)/b^4/f+1/42*(-7*B*f+3*C*e)*(f*x+e)^3*(-b^2*x^2+a^2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^2/f-1/7*C*(f*
x+e)^4*(-b^2*x^2+a^2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^2/f-1/840*(64*a^4*C*f^4+16*a^2*b^2*f^2*(15*C*e^2+7*f*
(A*f+3*B*e))-8*b^4*e^2*(3*C*e^2-7*f*(12*A*f+B*e))+3*b^2*f*(a^2*f^2*(35*B*f+41*C*e)-2*b^2*e*(3*C*e^2-7*f*(7*A*f
+B*e)))*x)*(-b^2*x^2+a^2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^6/f+1/16*a^2*(A*(6*a^2*b^2*e*f^2+8*b^4*e^3)+a^2*(
a^2*f^2*(B*f+3*C*e)+2*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))*c^(1/2)*(b*x+a)^(1/2)
*(-b*c*x+a*c)^(1/2)/b^5/(-b^2*c*x^2+a^2*c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.52, antiderivative size = 584, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1610, 1654, 833, 780, 195, 217, 203} \[ \frac {\sqrt {a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt {a c-b c x} \left (-\frac {8 a^2 C f^2}{b^2}-7 f (2 A f+B e)+3 C e^2\right )}{70 b^2 f}-\frac {\sqrt {a+b x} \left (a^2-b^2 x^2\right ) \sqrt {a c-b c x} \left (3 b^2 f x \left (a^2 f^2 (35 B f+41 C e)-b^2 \left (6 C e^3-14 e f (7 A f+B e)\right )\right )+8 \left (2 a^2 b^2 f^2 \left (7 f (A f+3 B e)+15 C e^2\right )+8 a^4 C f^4+b^4 \left (-\left (3 C e^4-7 e^2 f (12 A f+B e)\right )\right )\right )\right )}{840 b^6 f}+\frac {a^2 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+2 a^2 b^2 e^2 (3 B f+C e)+a^4 f^2 (B f+3 C e)\right )}{16 b^5 \sqrt {a^2 c-b^2 c x^2}}+\frac {x \sqrt {a+b x} \sqrt {a c-b c x} \left (A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+2 a^2 b^2 e^2 (3 B f+C e)+a^4 f^2 (B f+3 C e)\right )}{16 b^4}+\frac {\sqrt {a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^3 \sqrt {a c-b c x} (3 C e-7 B f)}{42 b^2 f}-\frac {C \sqrt {a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^4 \sqrt {a c-b c x}}{7 b^2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^4*f^2*(3*C*e + B*f) + 2*a^2*b^2*e^2*(C*e + 3*B*f) + A*(8*b^4*e^3 + 6*a^2*b^2*e*f^2))*x*Sqrt[a + b*x]*Sqrt[
a*c - b*c*x])/(16*b^4) + ((3*C*e^2 - (8*a^2*C*f^2)/b^2 - 7*f*(B*e + 2*A*f))*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e
+ f*x)^2*(a^2 - b^2*x^2))/(70*b^2*f) + ((3*C*e - 7*B*f)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^3*(a^2 - b^
2*x^2))/(42*b^2*f) - (C*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^4*(a^2 - b^2*x^2))/(7*b^2*f) - (Sqrt[a + b*x
]*Sqrt[a*c - b*c*x]*(8*(8*a^4*C*f^4 + 2*a^2*b^2*f^2*(15*C*e^2 + 7*f*(3*B*e + A*f)) - b^4*(3*C*e^4 - 7*e^2*f*(B
*e + 12*A*f))) + 3*b^2*f*(a^2*f^2*(41*C*e + 35*B*f) - b^2*(6*C*e^3 - 14*e*f*(B*e + 7*A*f)))*x)*(a^2 - b^2*x^2)
)/(840*b^6*f) + (a^2*Sqrt[c]*(a^4*f^2*(3*C*e + B*f) + 2*a^2*b^2*e^2*(C*e + 3*B*f) + A*(8*b^4*e^3 + 6*a^2*b^2*e
*f^2))*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(16*b^5*Sqrt[a^2*c - b^2
*c*x^2])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx &=\frac {\left (\sqrt {a+b x} \sqrt {a c-b c x}\right ) \int (e+f x)^3 \sqrt {a^2 c-b^2 c x^2} \left (A+B x+C x^2\right ) \, dx}{\sqrt {a^2 c-b^2 c x^2}}\\ &=-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac {\left (\sqrt {a+b x} \sqrt {a c-b c x}\right ) \int (e+f x)^3 \left (-c \left (7 A b^2+4 a^2 C\right ) f^2+b^2 c f (3 C e-7 B f) x\right ) \sqrt {a^2 c-b^2 c x^2} \, dx}{7 b^2 c f^2 \sqrt {a^2 c-b^2 c x^2}}\\ &=\frac {(3 C e-7 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}+\frac {\left (\sqrt {a+b x} \sqrt {a c-b c x}\right ) \int (e+f x)^2 \left (3 b^2 c^2 f^2 \left (14 A b^2 e+a^2 (5 C e+7 B f)\right )+3 b^2 c^2 f \left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) x\right ) \sqrt {a^2 c-b^2 c x^2} \, dx}{42 b^4 c^2 f^2 \sqrt {a^2 c-b^2 c x^2}}\\ &=-\frac {\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{70 b^4 f}+\frac {(3 C e-7 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac {\left (\sqrt {a+b x} \sqrt {a c-b c x}\right ) \int (e+f x) \left (-3 b^2 c^3 f^2 \left (16 a^4 C f^2+a^2 b^2 e (19 C e+49 B f)+14 A \left (5 b^4 e^2+2 a^2 b^2 f^2\right )\right )-3 b^4 c^3 f \left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x\right ) \sqrt {a^2 c-b^2 c x^2} \, dx}{210 b^6 c^3 f^2 \sqrt {a^2 c-b^2 c x^2}}\\ &=-\frac {\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{70 b^4 f}+\frac {(3 C e-7 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (8 \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (15 C e^2+7 f (3 B e+A f)\right )-b^4 \left (3 C e^4-7 e^2 f (B e+12 A f)\right )\right )+3 b^2 f \left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{840 b^6 f}+\frac {\left (\left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x}\right ) \int \sqrt {a^2 c-b^2 c x^2} \, dx}{8 b^4 \sqrt {a^2 c-b^2 c x^2}}\\ &=\frac {\left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) x \sqrt {a+b x} \sqrt {a c-b c x}}{16 b^4}-\frac {\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{70 b^4 f}+\frac {(3 C e-7 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (8 \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (15 C e^2+7 f (3 B e+A f)\right )-b^4 \left (3 C e^4-7 e^2 f (B e+12 A f)\right )\right )+3 b^2 f \left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{840 b^6 f}+\frac {\left (a^2 c \left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x}\right ) \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{16 b^4 \sqrt {a^2 c-b^2 c x^2}}\\ &=\frac {\left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) x \sqrt {a+b x} \sqrt {a c-b c x}}{16 b^4}-\frac {\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{70 b^4 f}+\frac {(3 C e-7 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (8 \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (15 C e^2+7 f (3 B e+A f)\right )-b^4 \left (3 C e^4-7 e^2 f (B e+12 A f)\right )\right )+3 b^2 f \left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{840 b^6 f}+\frac {\left (a^2 c \left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x}\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 )}{16 b^4 \sqrt {a^2 c-b^2 c x^2}}\\ &=\frac {\left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) x \sqrt {a+b x} \sqrt {a c-b c x}}{16 b^4}-\frac {\left (8 a^2 C f^2-b^2 \left (3 C e^2-7 f (B e+2 A f)\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{70 b^4 f}+\frac {(3 C e-7 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{42 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^4 \left (a^2-b^2 x^2\right )}{7 b^2 f}-\frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (8 \left (8 a^4 C f^4+2 a^2 b^2 f^2 \left (15 C e^2+7 f (3 B e+A f)\right )-b^4 \left (3 C e^4-7 e^2 f (B e+12 A f)\right )\right )+3 b^2 f \left (a^2 f^2 (41 C e+35 B f)-b^2 \left (6 C e^3-14 e f (B e+7 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{840 b^6 f}+\frac {a^2 \sqrt {c} \left (a^4 f^2 (3 C e+B f)+2 a^2 b^2 e^2 (C e+3 B f)+A \left (8 b^4 e^3+6 a^2 b^2 e f^2\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{16 b^5 \sqrt {a^2 c-b^2 c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.46, size = 427, normalized size = 0.72 \[ \frac {\sqrt {c (a-b x)} \left (210 a^{5/2} b \sqrt {a-b x} \sqrt {\frac {b x}{a}+1} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right ) \left (a^4 f^2 (B f+3 C e)+A \left (6 a^2 b^2 e f^2+8 b^4 e^3\right )+2 a^2 b^2 e^2 (3 B f+C e)\right )+\left (a^2-b^2 x^2\right ) \left (128 a^6 C f^3+a^4 b^2 f \left (7 f (32 A f+96 B e+15 B f x)+C \left (672 e^2+315 e f x+64 f^2 x^2\right )\right )+2 a^2 b^4 \left (7 A f \left (120 e^2+45 e f x+8 f^2 x^2\right )+7 B \left (40 e^3+45 e^2 f x+24 e f^2 x^2+5 f^3 x^3\right )+3 C x \left (35 e^3+56 e^2 f x+35 e f^2 x^2+8 f^3 x^3\right )\right )-4 b^6 x \left (21 A \left (10 e^3+20 e^2 f x+15 e f^2 x^2+4 f^3 x^3\right )+x \left (7 B \left (20 e^3+45 e^2 f x+36 e f^2 x^2+10 f^3 x^3\right )+3 C x \left (35 e^3+84 e^2 f x+70 e f^2 x^2+20 f^3 x^3\right )\right )\right )\right )\right )}{1680 b^6 (b x-a) \sqrt {a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c*(a - b*x)]*((a^2 - b^2*x^2)*(128*a^6*C*f^3 + a^4*b^2*f*(7*f*(96*B*e + 32*A*f + 15*B*f*x) + C*(672*e^2
+ 315*e*f*x + 64*f^2*x^2)) + 2*a^2*b^4*(7*A*f*(120*e^2 + 45*e*f*x + 8*f^2*x^2) + 7*B*(40*e^3 + 45*e^2*f*x + 24
*e*f^2*x^2 + 5*f^3*x^3) + 3*C*x*(35*e^3 + 56*e^2*f*x + 35*e*f^2*x^2 + 8*f^3*x^3)) - 4*b^6*x*(21*A*(10*e^3 + 20
*e^2*f*x + 15*e*f^2*x^2 + 4*f^3*x^3) + x*(7*B*(20*e^3 + 45*e^2*f*x + 36*e*f^2*x^2 + 10*f^3*x^3) + 3*C*x*(35*e^
3 + 84*e^2*f*x + 70*e*f^2*x^2 + 20*f^3*x^3)))) + 210*a^(5/2)*b*(a^4*f^2*(3*C*e + B*f) + 2*a^2*b^2*e^2*(C*e + 3
*B*f) + A*(8*b^4*e^3 + 6*a^2*b^2*e*f^2))*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a]
)]))/(1680*b^6*(-a + b*x)*Sqrt[a + b*x])

________________________________________________________________________________________

fricas [A]  time = 1.04, size = 1001, normalized size = 1.69 \[ \left [\frac {105 \, {\left (6 \, B a^{4} b^{3} e^{2} f + B a^{6} b f^{3} + 2 \, {\left (C a^{4} b^{3} + 4 \, A a^{2} b^{5}\right )} e^{3} + 3 \, {\left (C a^{6} b + 2 \, A a^{4} 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 (240 \, C b^{6} f^{3} x^{6} - 560 \, B a^{2} b^{4} e^{3} - 672 \, B a^{4} b^{2} e f^{2} + 280 \, {\left (3 \, C b^{6} e f^{2} + B b^{6} f^{3}\right )} x^{5} + 48 \, {\left (21 \, C b^{6} e^{2} f + 21 \, B b^{6} e f^{2} - {\left (C a^{2} b^{4} - 7 \, A b^{6}\right )} f^{3}\right )} x^{4} - 336 \, {\left (2 \, C a^{4} b^{2} + 5 \, A a^{2} b^{4}\right )} e^{2} f - 32 \, {\left (4 \, C a^{6} + 7 \, A a^{4} b^{2}\right )} f^{3} + 70 \, {\left (6 \, C b^{6} e^{3} + 18 \, B b^{6} e^{2} f - B a^{2} b^{4} f^{3} - 3 \, {\left (C a^{2} b^{4} - 6 \, A b^{6}\right )} e f^{2}\right )} x^{3} + 16 \, {\left (35 \, B b^{6} e^{3} - 21 \, B a^{2} b^{4} e f^{2} - 21 \, {\left (C a^{2} b^{4} - 5 \, A b^{6}\right )} e^{2} f - {\left (4 \, C a^{4} b^{2} + 7 \, A a^{2} b^{4}\right )} f^{3}\right )} x^{2} - 105 \, {\left (6 \, B a^{2} b^{4} e^{2} f + B a^{4} b^{2} f^{3} + 2 \, {\left (C a^{2} b^{4} - 4 \, A b^{6}\right )} e^{3} + 3 \, {\left (C a^{4} b^{2} + 2 \, A a^{2} b^{4}\right )} e f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{3360 \, b^{6}}, -\frac {105 \, {\left (6 \, B a^{4} b^{3} e^{2} f + B a^{6} b f^{3} + 2 \, {\left (C a^{4} b^{3} + 4 \, A a^{2} b^{5}\right )} e^{3} + 3 \, {\left (C a^{6} b + 2 \, A a^{4} 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 (240 \, C b^{6} f^{3} x^{6} - 560 \, B a^{2} b^{4} e^{3} - 672 \, B a^{4} b^{2} e f^{2} + 280 \, {\left (3 \, C b^{6} e f^{2} + B b^{6} f^{3}\right )} x^{5} + 48 \, {\left (21 \, C b^{6} e^{2} f + 21 \, B b^{6} e f^{2} - {\left (C a^{2} b^{4} - 7 \, A b^{6}\right )} f^{3}\right )} x^{4} - 336 \, {\left (2 \, C a^{4} b^{2} + 5 \, A a^{2} b^{4}\right )} e^{2} f - 32 \, {\left (4 \, C a^{6} + 7 \, A a^{4} b^{2}\right )} f^{3} + 70 \, {\left (6 \, C b^{6} e^{3} + 18 \, B b^{6} e^{2} f - B a^{2} b^{4} f^{3} - 3 \, {\left (C a^{2} b^{4} - 6 \, A b^{6}\right )} e f^{2}\right )} x^{3} + 16 \, {\left (35 \, B b^{6} e^{3} - 21 \, B a^{2} b^{4} e f^{2} - 21 \, {\left (C a^{2} b^{4} - 5 \, A b^{6}\right )} e^{2} f - {\left (4 \, C a^{4} b^{2} + 7 \, A a^{2} b^{4}\right )} f^{3}\right )} x^{2} - 105 \, {\left (6 \, B a^{2} b^{4} e^{2} f + B a^{4} b^{2} f^{3} + 2 \, {\left (C a^{2} b^{4} - 4 \, A b^{6}\right )} e^{3} + 3 \, {\left (C a^{4} b^{2} + 2 \, A a^{2} b^{4}\right )} e f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{1680 \, b^{6}}\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]

[1/3360*(105*(6*B*a^4*b^3*e^2*f + B*a^6*b*f^3 + 2*(C*a^4*b^3 + 4*A*a^2*b^5)*e^3 + 3*(C*a^6*b + 2*A*a^4*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*(240*C*b^6*f^3*x^
6 - 560*B*a^2*b^4*e^3 - 672*B*a^4*b^2*e*f^2 + 280*(3*C*b^6*e*f^2 + B*b^6*f^3)*x^5 + 48*(21*C*b^6*e^2*f + 21*B*
b^6*e*f^2 - (C*a^2*b^4 - 7*A*b^6)*f^3)*x^4 - 336*(2*C*a^4*b^2 + 5*A*a^2*b^4)*e^2*f - 32*(4*C*a^6 + 7*A*a^4*b^2
)*f^3 + 70*(6*C*b^6*e^3 + 18*B*b^6*e^2*f - B*a^2*b^4*f^3 - 3*(C*a^2*b^4 - 6*A*b^6)*e*f^2)*x^3 + 16*(35*B*b^6*e
^3 - 21*B*a^2*b^4*e*f^2 - 21*(C*a^2*b^4 - 5*A*b^6)*e^2*f - (4*C*a^4*b^2 + 7*A*a^2*b^4)*f^3)*x^2 - 105*(6*B*a^2
*b^4*e^2*f + B*a^4*b^2*f^3 + 2*(C*a^2*b^4 - 4*A*b^6)*e^3 + 3*(C*a^4*b^2 + 2*A*a^2*b^4)*e*f^2)*x)*sqrt(-b*c*x +
a*c)*sqrt(b*x + a))/b^6, -1/1680*(105*(6*B*a^4*b^3*e^2*f + B*a^6*b*f^3 + 2*(C*a^4*b^3 + 4*A*a^2*b^5)*e^3 + 3*
(C*a^6*b + 2*A*a^4*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)
) - (240*C*b^6*f^3*x^6 - 560*B*a^2*b^4*e^3 - 672*B*a^4*b^2*e*f^2 + 280*(3*C*b^6*e*f^2 + B*b^6*f^3)*x^5 + 48*(2
1*C*b^6*e^2*f + 21*B*b^6*e*f^2 - (C*a^2*b^4 - 7*A*b^6)*f^3)*x^4 - 336*(2*C*a^4*b^2 + 5*A*a^2*b^4)*e^2*f - 32*(
4*C*a^6 + 7*A*a^4*b^2)*f^3 + 70*(6*C*b^6*e^3 + 18*B*b^6*e^2*f - B*a^2*b^4*f^3 - 3*(C*a^2*b^4 - 6*A*b^6)*e*f^2)
*x^3 + 16*(35*B*b^6*e^3 - 21*B*a^2*b^4*e*f^2 - 21*(C*a^2*b^4 - 5*A*b^6)*e^2*f - (4*C*a^4*b^2 + 7*A*a^2*b^4)*f^
3)*x^2 - 105*(6*B*a^2*b^4*e^2*f + B*a^4*b^2*f^3 + 2*(C*a^2*b^4 - 4*A*b^6)*e^3 + 3*(C*a^4*b^2 + 2*A*a^2*b^4)*e*
f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^6]

________________________________________________________________________________________

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

________________________________________________________________________________________

maple [B]  time = 0.04, size = 1446, normalized size = 2.45 \[ \text {result too large to display} \]

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]

1/1680*(b*x+a)^(1/2)*(-c*(b*x-a))^(1/2)*(-630*B*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^2*b^4*e^2*f+105*B*a
rctan((b^2*c)^(1/2)*x/(-(b^2*x^2-a^2)*c)^(1/2))*a^6*b^2*c*f^3+240*C*x^6*b^6*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*
c)^(1/2)+280*B*x^5*b^6*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)+336*A*x^4*b^6*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a
^2)*c)^(1/2)+420*C*x^3*b^6*e^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)+560*B*x^2*b^6*e^3*(b^2*c)^(1/2)*(-(b^2*x
^2-a^2)*c)^(1/2)-224*A*a^4*b^2*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-560*B*a^2*b^4*e^3*(b^2*c)^(1/2)*(-(b
^2*x^2-a^2)*c)^(1/2)+210*C*arctan((b^2*c)^(1/2)*x/(-(b^2*x^2-a^2)*c)^(1/2))*a^4*b^4*c*e^3+840*A*(b^2*c)^(1/2)*
(-(b^2*x^2-a^2)*c)^(1/2)*x*b^6*e^3+840*A*arctan((b^2*c)^(1/2)*x/(-(b^2*x^2-a^2)*c)^(1/2))*a^2*b^6*c*e^3-128*C*
a^6*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-112*A*x^2*a^2*b^4*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)+16
80*A*x^2*b^6*e^2*f*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-64*C*x^2*a^4*b^2*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c
)^(1/2)-1680*A*a^2*b^4*e^2*f*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-672*B*a^4*b^2*e*f^2*(b^2*c)^(1/2)*(-(b^2*x
^2-a^2)*c)^(1/2)-672*C*a^4*b^2*e^2*f*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)+630*A*arctan((b^2*c)^(1/2)*x/(-(b^
2*x^2-a^2)*c)^(1/2))*a^4*b^4*c*e*f^2+630*B*arctan((b^2*c)^(1/2)*x/(-(b^2*x^2-a^2)*c)^(1/2))*a^4*b^4*c*e^2*f-10
5*B*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^4*b^2*f^3-210*C*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^2*b^
4*e^3+1008*C*x^4*b^6*e^2*f*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)+1260*A*x^3*b^6*e*f^2*(b^2*c)^(1/2)*(-(b^2*x^
2-a^2)*c)^(1/2)-70*B*x^3*a^2*b^4*f^3*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)+1260*B*x^3*b^6*e^2*f*(b^2*c)^(1/2)
*(-(b^2*x^2-a^2)*c)^(1/2)-315*C*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^4*b^2*e*f^2-630*A*(b^2*c)^(1/2)*(-(
b^2*x^2-a^2)*c)^(1/2)*x*a^2*b^4*e*f^2-210*C*x^3*a^2*b^4*e*f^2*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-336*B*x^2
*a^2*b^4*e*f^2*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-336*C*x^2*a^2*b^4*e^2*f*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)
^(1/2)+315*C*arctan((b^2*c)^(1/2)*x/(-(b^2*x^2-a^2)*c)^(1/2))*a^6*b^2*c*e*f^2+840*C*x^5*b^6*e*f^2*(b^2*c)^(1/2
)*(-(b^2*x^2-a^2)*c)^(1/2)+1008*B*x^4*b^6*e*f^2*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)-48*C*x^4*a^2*b^4*f^3*(b
^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2))/(-(b^2*x^2-a^2)*c)^(1/2)/b^6/(b^2*c)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.46, size = 584, normalized size = 0.99 \[ -\frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C f^{3} x^{4}}{7 \, b^{2} c} + \frac {A a^{2} \sqrt {c} e^{3} \arcsin \left (\frac {b x}{a}\right )}{2 \, b} + \frac {1}{2} \, \sqrt {-b^{2} c x^{2} + a^{2} c} A e^{3} x - \frac {4 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C a^{2} f^{3} x^{2}}{35 \, b^{4} c} + \frac {{\left (3 \, C e f^{2} + B f^{3}\right )} a^{6} \sqrt {c} \arcsin \left (\frac {b x}{a}\right )}{16 \, b^{5}} + \frac {{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} a^{4} \sqrt {c} \arcsin \left (\frac {b x}{a}\right )}{8 \, b^{3}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} B e^{3}}{3 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} A e^{2} f}{b^{2} c} - \frac {8 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C a^{4} f^{3}}{105 \, b^{6} c} + \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e f^{2} + B f^{3}\right )} a^{4} x}{16 \, b^{4}} + \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} a^{2} x}{8 \, b^{2}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (3 \, C e f^{2} + B f^{3}\right )} x^{3}}{6 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} x^{2}}{5 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (3 \, C e f^{2} + B f^{3}\right )} a^{2} x}{8 \, b^{4} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} x}{4 \, b^{2} c} - \frac {2 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} a^{2}}{15 \, b^{4} c} \]

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]

-1/7*(-b^2*c*x^2 + a^2*c)^(3/2)*C*f^3*x^4/(b^2*c) + 1/2*A*a^2*sqrt(c)*e^3*arcsin(b*x/a)/b + 1/2*sqrt(-b^2*c*x^
2 + a^2*c)*A*e^3*x - 4/35*(-b^2*c*x^2 + a^2*c)^(3/2)*C*a^2*f^3*x^2/(b^4*c) + 1/16*(3*C*e*f^2 + B*f^3)*a^6*sqrt
(c)*arcsin(b*x/a)/b^5 + 1/8*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*a^4*sqrt(c)*arcsin(b*x/a)/b^3 - 1/3*(-b^2*c*x^2 +
a^2*c)^(3/2)*B*e^3/(b^2*c) - (-b^2*c*x^2 + a^2*c)^(3/2)*A*e^2*f/(b^2*c) - 8/105*(-b^2*c*x^2 + a^2*c)^(3/2)*C*a
^4*f^3/(b^6*c) + 1/16*sqrt(-b^2*c*x^2 + a^2*c)*(3*C*e*f^2 + B*f^3)*a^4*x/b^4 + 1/8*sqrt(-b^2*c*x^2 + a^2*c)*(C
*e^3 + 3*B*e^2*f + 3*A*e*f^2)*a^2*x/b^2 - 1/6*(-b^2*c*x^2 + a^2*c)^(3/2)*(3*C*e*f^2 + B*f^3)*x^3/(b^2*c) - 1/5
*(-b^2*c*x^2 + a^2*c)^(3/2)*(3*C*e^2*f + 3*B*e*f^2 + A*f^3)*x^2/(b^2*c) - 1/8*(-b^2*c*x^2 + a^2*c)^(3/2)*(3*C*
e*f^2 + B*f^3)*a^2*x/(b^4*c) - 1/4*(-b^2*c*x^2 + a^2*c)^(3/2)*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*x/(b^2*c) - 2/15
*(-b^2*c*x^2 + a^2*c)^(3/2)*(3*C*e^2*f + 3*B*e*f^2 + A*f^3)*a^2/(b^4*c)

________________________________________________________________________________________

mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

________________________________________________________________________________________

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

________________________________________________________________________________________