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

Optimal. Leaf size=451 \[ -\frac {\sqrt {a+b x} \left (a^2-b^2 x^2\right ) \sqrt {a c-b c x} \left (3 f x \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (5 A f+2 B e)\right )\right )+8 \left (2 a^2 f^2 (B f+2 C e)-b^2 e \left (C e^2-2 f (5 A f+B e)\right )\right )\right )}{120 b^4 f}+\frac {x \sqrt {a+b x} \sqrt {a c-b c x} \left (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+a^2 \left (a^2 C f^2+2 b^2 e (2 B f+C e)\right )\right )}{16 b^4}+\frac {a^2 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+a^2 \left (a^2 C f^2+2 b^2 e (2 B f+C e)\right )\right )}{16 b^5 \sqrt {a^2 c-b^2 c x^2}}+\frac {\sqrt {a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt {a c-b c x} (C e-2 B f)}{10 b^2 f}-\frac {C \sqrt {a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^3 \sqrt {a c-b c x}}{6 b^2 f} \]

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Rubi [A]  time = 1.01, antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} -\frac {\sqrt {a+b x} \left (a^2-b^2 x^2\right ) \sqrt {a c-b c x} \left (3 f x \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (5 A f+2 B e)\right )\right )+8 \left (2 a^2 f^2 (B f+2 C e)-\frac {1}{8} b^2 \left (8 C e^3-16 e f (5 A f+B e)\right )\right )\right )}{120 b^4 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 (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+2 a^2 b^2 e (2 B f+C e)+a^4 C f^2\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 (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+2 a^2 b^2 e (2 B f+C e)+a^4 C f^2\right )}{16 b^4}+\frac {\sqrt {a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt {a c-b c x} (C e-2 B f)}{10 b^2 f}-\frac {C \sqrt {a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^3 \sqrt {a c-b c x}}{6 b^2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a^4*C*f^2 + 2*a^2*b^2*e*(C*e + 2*B*f) + 2*A*(4*b^4*e^2 + a^2*b^2*f^2))*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(1
6*b^4) + ((C*e - 2*B*f)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^2*(a^2 - b^2*x^2))/(10*b^2*f) - (C*Sqrt[a +
b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^3*(a^2 - b^2*x^2))/(6*b^2*f) - (Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(8*(2*a^2*f^2
*(2*C*e + B*f) - (b^2*(8*C*e^3 - 16*e*f*(B*e + 5*A*f)))/8) + 3*f*(5*a^2*C*f^2 - b^2*(2*C*e^2 - 2*f*(2*B*e + 5*
A*f)))*x)*(a^2 - b^2*x^2))/(120*b^4*f) + (a^2*Sqrt[c]*(a^4*C*f^2 + 2*a^2*b^2*e*(C*e + 2*B*f) + 2*A*(4*b^4*e^2
+ a^2*b^2*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)^2 \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)^2 \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)^3 \left (a^2-b^2 x^2\right )}{6 b^2 f}-\frac {\left (\sqrt {a+b x} \sqrt {a c-b c x}\right ) \int (e+f x)^2 \left (-3 c \left (2 A b^2+a^2 C\right ) f^2+3 b^2 c f (C e-2 B f) x\right ) \sqrt {a^2 c-b^2 c x^2} \, dx}{6 b^2 c f^2 \sqrt {a^2 c-b^2 c x^2}}\\ &=\frac {(C e-2 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{10 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{6 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^2 f^2 \left (10 A b^2 e+a^2 (3 C e+4 B f)\right )+3 b^2 c^2 f \left (5 \left (2 A b^2+a^2 C\right ) f^2-2 b^2 e (C e-2 B f)\right ) x\right ) \sqrt {a^2 c-b^2 c x^2} \, dx}{30 b^4 c^2 f^2 \sqrt {a^2 c-b^2 c x^2}}\\ &=\frac {(C e-2 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{10 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{6 b^2 f}-\frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (8 \left (2 a^2 f^2 (2 C e+B f)-\frac {1}{8} b^2 \left (8 C e^3-16 e f (B e+5 A f)\right )\right )+3 f \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (2 B e+5 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^4 f}+\frac {\left (\left (a^4 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 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 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 f^2\right )\right ) x \sqrt {a+b x} \sqrt {a c-b c x}}{16 b^4}+\frac {(C e-2 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{10 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{6 b^2 f}-\frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (8 \left (2 a^2 f^2 (2 C e+B f)-\frac {1}{8} b^2 \left (8 C e^3-16 e f (B e+5 A f)\right )\right )+3 f \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (2 B e+5 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^4 f}+\frac {\left (a^2 c \left (a^4 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 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 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 f^2\right )\right ) x \sqrt {a+b x} \sqrt {a c-b c x}}{16 b^4}+\frac {(C e-2 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{10 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{6 b^2 f}-\frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (8 \left (2 a^2 f^2 (2 C e+B f)-\frac {1}{8} b^2 \left (8 C e^3-16 e f (B e+5 A f)\right )\right )+3 f \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (2 B e+5 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^4 f}+\frac {\left (a^2 c \left (a^4 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 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 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 f^2\right )\right ) x \sqrt {a+b x} \sqrt {a c-b c x}}{16 b^4}+\frac {(C e-2 B f) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2 \left (a^2-b^2 x^2\right )}{10 b^2 f}-\frac {C \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3 \left (a^2-b^2 x^2\right )}{6 b^2 f}-\frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (8 \left (2 a^2 f^2 (2 C e+B f)-\frac {1}{8} b^2 \left (8 C e^3-16 e f (B e+5 A f)\right )\right )+3 f \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (2 B e+5 A f)\right )\right ) x\right ) \left (a^2-b^2 x^2\right )}{120 b^4 f}+\frac {a^2 \sqrt {c} \left (a^4 C f^2+2 a^2 b^2 e (C e+2 B f)+2 A \left (4 b^4 e^2+a^2 b^2 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*}

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Mathematica [A]  time = 1.02, size = 311, normalized size = 0.69 \begin {gather*} \frac {\sqrt {c (a-b x)} \left (b \left (a^2-b^2 x^2\right ) \left (a^4 f (32 B f+64 C e+15 C f x)+2 a^2 b^2 \left (5 A f (16 e+3 f x)+B \left (40 e^2+30 e f x+8 f^2 x^2\right )+C x \left (15 e^2+16 e f x+5 f^2 x^2\right )\right )-4 b^4 x \left (5 A \left (6 e^2+8 e f x+3 f^2 x^2\right )+x \left (2 B \left (10 e^2+15 e f x+6 f^2 x^2\right )+C x \left (15 e^2+24 e f x+10 f^2 x^2\right )\right )\right )\right )+30 a^{5/2} \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 C f^2+2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+2 a^2 b^2 e (2 B f+C e)\right )\right )}{240 b^5 (b x-a) \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c*(a - b*x)]*(b*(a^2 - b^2*x^2)*(a^4*f*(64*C*e + 32*B*f + 15*C*f*x) + 2*a^2*b^2*(5*A*f*(16*e + 3*f*x) +
C*x*(15*e^2 + 16*e*f*x + 5*f^2*x^2) + B*(40*e^2 + 30*e*f*x + 8*f^2*x^2)) - 4*b^4*x*(5*A*(6*e^2 + 8*e*f*x + 3*f
^2*x^2) + x*(2*B*(10*e^2 + 15*e*f*x + 6*f^2*x^2) + C*x*(15*e^2 + 24*e*f*x + 10*f^2*x^2)))) + 30*a^(5/2)*(a^4*C
*f^2 + 2*a^2*b^2*e*(C*e + 2*B*f) + 2*A*(4*b^4*e^2 + a^2*b^2*f^2))*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcSin[Sqrt[
a - b*x]/(Sqrt[2]*Sqrt[a])]))/(240*b^5*(-a + b*x)*Sqrt[a + b*x])

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IntegrateAlgebraic [B]  time = 1.29, size = 1792, normalized size = 3.97

result too large to display

Antiderivative was successfully verified.

[In]

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

[Out]

((120*a^2*A*b^4*c^6*e^2*Sqrt[a*c - b*c*x])/Sqrt[a + b*x] + (30*a^4*b^2*c^6*C*e^2*Sqrt[a*c - b*c*x])/Sqrt[a + b
*x] + (60*a^4*b^2*B*c^6*e*f*Sqrt[a*c - b*c*x])/Sqrt[a + b*x] + (30*a^4*A*b^2*c^6*f^2*Sqrt[a*c - b*c*x])/Sqrt[a
+ b*x] + (15*a^6*c^6*C*f^2*Sqrt[a*c - b*c*x])/Sqrt[a + b*x] + (360*a^2*A*b^4*c^5*e^2*(a*c - b*c*x)^(3/2))/(a
+ b*x)^(3/2) - (320*a^3*b^3*B*c^5*e^2*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) - (150*a^4*b^2*c^5*C*e^2*(a*c - b*c
*x)^(3/2))/(a + b*x)^(3/2) - (640*a^3*A*b^3*c^5*e*f*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) - (300*a^4*b^2*B*c^5*
e*f*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) - (640*a^5*b*c^5*C*e*f*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) - (150*a^
4*A*b^2*c^5*f^2*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) - (320*a^5*b*B*c^5*f^2*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/
2) - (235*a^6*c^5*C*f^2*(a*c - b*c*x)^(3/2))/(a + b*x)^(3/2) + (240*a^2*A*b^4*c^4*e^2*(a*c - b*c*x)^(5/2))/(a
+ b*x)^(5/2) - (960*a^3*b^3*B*c^4*e^2*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) - (180*a^4*b^2*c^4*C*e^2*(a*c - b*c
*x)^(5/2))/(a + b*x)^(5/2) - (1920*a^3*A*b^3*c^4*e*f*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) - (360*a^4*b^2*B*c^4
*e*f*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) - (384*a^5*b*c^4*C*e*f*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) - (180*a
^4*A*b^2*c^4*f^2*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) - (192*a^5*b*B*c^4*f^2*(a*c - b*c*x)^(5/2))/(a + b*x)^(5
/2) + (390*a^6*c^4*C*f^2*(a*c - b*c*x)^(5/2))/(a + b*x)^(5/2) - (240*a^2*A*b^4*c^3*e^2*(a*c - b*c*x)^(7/2))/(a
+ b*x)^(7/2) - (960*a^3*b^3*B*c^3*e^2*(a*c - b*c*x)^(7/2))/(a + b*x)^(7/2) + (180*a^4*b^2*c^3*C*e^2*(a*c - b*
c*x)^(7/2))/(a + b*x)^(7/2) - (1920*a^3*A*b^3*c^3*e*f*(a*c - b*c*x)^(7/2))/(a + b*x)^(7/2) + (360*a^4*b^2*B*c^
3*e*f*(a*c - b*c*x)^(7/2))/(a + b*x)^(7/2) - (384*a^5*b*c^3*C*e*f*(a*c - b*c*x)^(7/2))/(a + b*x)^(7/2) + (180*
a^4*A*b^2*c^3*f^2*(a*c - b*c*x)^(7/2))/(a + b*x)^(7/2) - (192*a^5*b*B*c^3*f^2*(a*c - b*c*x)^(7/2))/(a + b*x)^(
7/2) - (390*a^6*c^3*C*f^2*(a*c - b*c*x)^(7/2))/(a + b*x)^(7/2) - (360*a^2*A*b^4*c^2*e^2*(a*c - b*c*x)^(9/2))/(
a + b*x)^(9/2) - (320*a^3*b^3*B*c^2*e^2*(a*c - b*c*x)^(9/2))/(a + b*x)^(9/2) + (150*a^4*b^2*c^2*C*e^2*(a*c - b
*c*x)^(9/2))/(a + b*x)^(9/2) - (640*a^3*A*b^3*c^2*e*f*(a*c - b*c*x)^(9/2))/(a + b*x)^(9/2) + (300*a^4*b^2*B*c^
2*e*f*(a*c - b*c*x)^(9/2))/(a + b*x)^(9/2) - (640*a^5*b*c^2*C*e*f*(a*c - b*c*x)^(9/2))/(a + b*x)^(9/2) + (150*
a^4*A*b^2*c^2*f^2*(a*c - b*c*x)^(9/2))/(a + b*x)^(9/2) - (320*a^5*b*B*c^2*f^2*(a*c - b*c*x)^(9/2))/(a + b*x)^(
9/2) + (235*a^6*c^2*C*f^2*(a*c - b*c*x)^(9/2))/(a + b*x)^(9/2) - (120*a^2*A*b^4*c*e^2*(a*c - b*c*x)^(11/2))/(a
+ b*x)^(11/2) - (30*a^4*b^2*c*C*e^2*(a*c - b*c*x)^(11/2))/(a + b*x)^(11/2) - (60*a^4*b^2*B*c*e*f*(a*c - b*c*x
)^(11/2))/(a + b*x)^(11/2) - (30*a^4*A*b^2*c*f^2*(a*c - b*c*x)^(11/2))/(a + b*x)^(11/2) - (15*a^6*c*C*f^2*(a*c
- b*c*x)^(11/2))/(a + b*x)^(11/2))/(120*b^5*(c + (a*c - b*c*x)/(a + b*x))^6) + ((-8*a^2*A*b^4*Sqrt[c]*e^2 - 2
*a^4*b^2*Sqrt[c]*C*e^2 - 4*a^4*b^2*B*Sqrt[c]*e*f - 2*a^4*A*b^2*Sqrt[c]*f^2 - a^6*Sqrt[c]*C*f^2)*ArcTan[Sqrt[a*
c - b*c*x]/(Sqrt[c]*Sqrt[a + b*x])])/(8*b^5)

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fricas [A]  time = 0.98, size = 703, normalized size = 1.56 \begin {gather*} \left [\frac {15 \, {\left (4 \, B a^{4} b^{2} e f + 2 \, {\left (C a^{4} b^{2} + 4 \, A a^{2} b^{4}\right )} e^{2} + {\left (C a^{6} + 2 \, A a^{4} 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 (40 \, C b^{5} f^{2} x^{5} - 80 \, B a^{2} b^{3} e^{2} - 32 \, B a^{4} b f^{2} + 48 \, {\left (2 \, C b^{5} e f + B b^{5} f^{2}\right )} x^{4} + 10 \, {\left (6 \, C b^{5} e^{2} + 12 \, B b^{5} e f - {\left (C a^{2} b^{3} - 6 \, A b^{5}\right )} f^{2}\right )} x^{3} - 32 \, {\left (2 \, C a^{4} b + 5 \, A a^{2} b^{3}\right )} e f + 16 \, {\left (5 \, B b^{5} e^{2} - B a^{2} b^{3} f^{2} - 2 \, {\left (C a^{2} b^{3} - 5 \, A b^{5}\right )} e f\right )} x^{2} - 15 \, {\left (4 \, B a^{2} b^{3} e f + 2 \, {\left (C a^{2} b^{3} - 4 \, A b^{5}\right )} e^{2} + {\left (C a^{4} b + 2 \, A a^{2} b^{3}\right )} f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{480 \, b^{5}}, -\frac {15 \, {\left (4 \, B a^{4} b^{2} e f + 2 \, {\left (C a^{4} b^{2} + 4 \, A a^{2} b^{4}\right )} e^{2} + {\left (C a^{6} + 2 \, A a^{4} 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 (40 \, C b^{5} f^{2} x^{5} - 80 \, B a^{2} b^{3} e^{2} - 32 \, B a^{4} b f^{2} + 48 \, {\left (2 \, C b^{5} e f + B b^{5} f^{2}\right )} x^{4} + 10 \, {\left (6 \, C b^{5} e^{2} + 12 \, B b^{5} e f - {\left (C a^{2} b^{3} - 6 \, A b^{5}\right )} f^{2}\right )} x^{3} - 32 \, {\left (2 \, C a^{4} b + 5 \, A a^{2} b^{3}\right )} e f + 16 \, {\left (5 \, B b^{5} e^{2} - B a^{2} b^{3} f^{2} - 2 \, {\left (C a^{2} b^{3} - 5 \, A b^{5}\right )} e f\right )} x^{2} - 15 \, {\left (4 \, B a^{2} b^{3} e f + 2 \, {\left (C a^{2} b^{3} - 4 \, A b^{5}\right )} e^{2} + {\left (C a^{4} b + 2 \, A a^{2} b^{3}\right )} f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{240 \, b^{5}}\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]

[1/480*(15*(4*B*a^4*b^2*e*f + 2*(C*a^4*b^2 + 4*A*a^2*b^4)*e^2 + (C*a^6 + 2*A*a^4*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*(40*C*b^5*f^2*x^5 - 80*B*a^2*b^3*e^2 - 32
*B*a^4*b*f^2 + 48*(2*C*b^5*e*f + B*b^5*f^2)*x^4 + 10*(6*C*b^5*e^2 + 12*B*b^5*e*f - (C*a^2*b^3 - 6*A*b^5)*f^2)*
x^3 - 32*(2*C*a^4*b + 5*A*a^2*b^3)*e*f + 16*(5*B*b^5*e^2 - B*a^2*b^3*f^2 - 2*(C*a^2*b^3 - 5*A*b^5)*e*f)*x^2 -
15*(4*B*a^2*b^3*e*f + 2*(C*a^2*b^3 - 4*A*b^5)*e^2 + (C*a^4*b + 2*A*a^2*b^3)*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*
x + a))/b^5, -1/240*(15*(4*B*a^4*b^2*e*f + 2*(C*a^4*b^2 + 4*A*a^2*b^4)*e^2 + (C*a^6 + 2*A*a^4*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)) - (40*C*b^5*f^2*x^5 - 80*B*a^2*b^3*
e^2 - 32*B*a^4*b*f^2 + 48*(2*C*b^5*e*f + B*b^5*f^2)*x^4 + 10*(6*C*b^5*e^2 + 12*B*b^5*e*f - (C*a^2*b^3 - 6*A*b^
5)*f^2)*x^3 - 32*(2*C*a^4*b + 5*A*a^2*b^3)*e*f + 16*(5*B*b^5*e^2 - B*a^2*b^3*f^2 - 2*(C*a^2*b^3 - 5*A*b^5)*e*f
)*x^2 - 15*(4*B*a^2*b^3*e*f + 2*(C*a^2*b^3 - 4*A*b^5)*e^2 + (C*a^4*b + 2*A*a^2*b^3)*f^2)*x)*sqrt(-b*c*x + a*c)
*sqrt(b*x + a))/b^5]

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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 [B]  time = 0.02, size = 987, normalized size = 2.19 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {-\left (b x -a \right ) c}\, \left (40 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, C \,b^{4} f^{2} x^{5}+30 A \,a^{4} b^{2} c \,f^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+120 A \,a^{2} b^{4} c \,e^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+60 B \,a^{4} b^{2} c e f \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+48 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, B \,b^{4} f^{2} x^{4}+15 C \,a^{6} c \,f^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+30 C \,a^{4} b^{2} c \,e^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )+96 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, C \,b^{4} e f \,x^{4}+60 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, A \,b^{4} f^{2} x^{3}+120 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, B \,b^{4} e f \,x^{3}-10 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, C \,a^{2} b^{2} f^{2} x^{3}+60 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, C \,b^{4} e^{2} x^{3}+160 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, A \,b^{4} e f \,x^{2}-16 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, B \,a^{2} b^{2} f^{2} x^{2}+80 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, B \,b^{4} e^{2} x^{2}-32 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, C \,a^{2} b^{2} e f \,x^{2}-30 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, A \,a^{2} b^{2} f^{2} x +120 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, A \,b^{4} e^{2} x -60 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, B \,a^{2} b^{2} e f x -15 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,a^{4} f^{2} x -30 \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,a^{2} b^{2} e^{2} x -160 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, A \,a^{2} b^{2} e f -32 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, B \,a^{4} f^{2}-80 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, B \,a^{2} b^{2} e^{2}-64 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, C \,a^{4} e f \right )}{240 \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, \sqrt {b^{2} c}\, b^{4}} \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]

1/240*(b*x+a)^(1/2)*(-(b*x-a)*c)^(1/2)*(40*C*x^5*b^4*f^2*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)+48*B*x^4*b^4*f
^2*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)+60*A*x^3*b^4*f^2*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)+60*C*x^3*b^4
*e^2*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)+80*B*x^2*b^4*e^2*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)-80*B*a^2*b
^2*e^2*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)-60*B*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^2*b^2*e*f+15*C*a
rctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^6*c*f^2-32*B*a^4*f^2*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)-
64*C*a^4*e*f*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)+30*A*arctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^4*
b^2*c*f^2+120*A*arctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^2*b^4*c*e^2+30*C*arctan((b^2*c)^(1/2)/(-(b^
2*x^2-a^2)*c)^(1/2)*x)*a^4*b^2*c*e^2+120*A*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*b^4*e^2-15*C*(b^2*c)^(1/2)
*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^4*f^2-32*C*x^2*a^2*b^2*e*f*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)-30*A*(b^2*c)^(
1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^2*b^2*f^2-30*C*(b^2*c)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*x*a^2*b^2*e^2-10*C*x^3
*a^2*b^2*f^2*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)+160*A*x^2*b^4*e*f*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)-1
6*B*x^2*a^2*b^2*f^2*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2)-160*A*a^2*b^2*e*f*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^
(1/2)+60*B*arctan((b^2*c)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)*x)*a^4*b^2*c*e*f+96*C*x^4*b^4*e*f*(-(b^2*x^2-a^2)*c)^
(1/2)*(b^2*c)^(1/2)+120*B*x^3*b^4*e*f*(-(b^2*x^2-a^2)*c)^(1/2)*(b^2*c)^(1/2))/(-(b^2*x^2-a^2)*c)^(1/2)/b^4/(b^
2*c)^(1/2)

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maxima [A]  time = 2.07, size = 417, normalized size = 0.92 \begin {gather*} \frac {A a^{2} \sqrt {c} e^{2} \arcsin \left (\frac {b x}{a}\right )}{2 \, b} + \frac {C a^{6} \sqrt {c} f^{2} \arcsin \left (\frac {b x}{a}\right )}{16 \, b^{5}} + \frac {1}{2} \, \sqrt {-b^{2} c x^{2} + a^{2} c} A e^{2} x + \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} C a^{4} f^{2} x}{16 \, b^{4}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C f^{2} x^{3}}{6 \, b^{2} c} + \frac {{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} a^{4} \sqrt {c} \arcsin \left (\frac {b x}{a}\right )}{8 \, b^{3}} + \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (C e^{2} + 2 \, B e f + A f^{2}\right )} a^{2} x}{8 \, b^{2}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C a^{2} f^{2} x}{8 \, b^{4} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} B e^{2}}{3 \, b^{2} c} - \frac {2 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} A e f}{3 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (2 \, C e f + B f^{2}\right )} x^{2}}{5 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (C e^{2} + 2 \, B e f + A f^{2}\right )} x}{4 \, b^{2} c} - \frac {2 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (2 \, C e f + B f^{2}\right )} a^{2}}{15 \, 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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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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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