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

[Out]

-((16*a^2*C*f^2 - b^2*(3*C*e^2 - 5*f*(3*B*e + 4*A*f)))*(e + f*x)^2*(a^2 - b^2*x^2))/(60*b^4*f*Sqrt[a + b*x]*Sq
rt[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*(1
3*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) -
2*b^2*e*(3*C*e^2 - 5*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]) + ((
4*A*(2*b^4*e^3 + 3*a^2*b^2*e*f^2) + a^2*(3*a^2*f^2*(3*C*e + B*f) + 4*b^2*e^2*(C*e + 3*B*f)))*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])

________________________________________________________________________________________

Rubi [A]  time = 1.28111, antiderivative size = 496, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, 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 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]))

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 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 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 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])

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*}

Mathematica [B]  time = 6.5421, size = 1107, normalized size = 2.21 \[ -\frac{a^4 C f^3 (a-b x) \sqrt{a+b x} \left (\frac{630 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{a-b x} \left (2-\frac{a-b x}{a}\right )^{11/2}}+\frac{4}{1-\frac{a-b x}{2 a}}+\frac{18}{\left (2-\frac{a-b x}{a}\right )^2}+\frac{42}{\left (2-\frac{a-b x}{a}\right )^3}+\frac{105}{\left (2-\frac{a-b x}{a}\right )^4}+\frac{315}{\left (2-\frac{a-b x}{a}\right )^5}\right ) \left (2-\frac{a-b x}{a}\right )^{11/2}}{40 b^6 \sqrt{c (a-b x)} \sqrt{\frac{a+b x}{a}}}-\frac{a^3 f^2 (3 b C e+b B f-5 a C f) (a-b x) \sqrt{a+b x} \left (\frac{210 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{a-b x} \left (2-\frac{a-b x}{a}\right )^{9/2}}+\frac{3}{1-\frac{a-b x}{2 a}}+\frac{14}{\left (2-\frac{a-b x}{a}\right )^2}+\frac{35}{\left (2-\frac{a-b x}{a}\right )^3}+\frac{105}{\left (2-\frac{a-b x}{a}\right )^4}\right ) \left (2-\frac{a-b x}{a}\right )^{9/2}}{24 b^6 \sqrt{c (a-b x)} \sqrt{\frac{a+b x}{a}}}-\frac{a^2 f \left (\left (3 C e^2+f (3 B e+A f)\right ) b^2-4 a f (3 C e+B f) b+10 a^2 C f^2\right ) (a-b x) \sqrt{a+b x} \left (\frac{30 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{a-b x} \left (2-\frac{a-b x}{a}\right )^{7/2}}+\frac{1}{1-\frac{a-b x}{2 a}}+\frac{5}{\left (2-\frac{a-b x}{a}\right )^2}+\frac{15}{\left (2-\frac{a-b x}{a}\right )^3}\right ) \left (2-\frac{a-b x}{a}\right )^{7/2}}{6 b^6 \sqrt{c (a-b x)} \sqrt{\frac{a+b x}{a}}}-\frac{a (b e-a f) \left (\left (C e^2+3 f (B e+A f)\right ) b^2-2 a f (4 C e+3 B f) b+10 a^2 C f^2\right ) (a-b x) \sqrt{a+b x} \left (\frac{12 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{a-b x} \left (2-\frac{a-b x}{a}\right )^{5/2}}+\frac{1}{1-\frac{a-b x}{2 a}}+\frac{6}{\left (2-\frac{a-b x}{a}\right )^2}\right ) \left (2-\frac{a-b x}{a}\right )^{5/2}}{4 b^6 \sqrt{c (a-b x)} \sqrt{\frac{a+b x}{a}}}-\frac{(b e-a f)^2 \left (5 C f a^2-2 b (C e+2 B f) a+b^2 (B e+3 A f)\right ) (a-b x) \sqrt{a+b x} \left (\frac{2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{a-b x} \left (2-\frac{a-b x}{a}\right )^{3/2}}+\frac{1}{2-\frac{a-b x}{a}}\right ) \left (2-\frac{a-b x}{a}\right )^{3/2}}{b^6 \sqrt{c (a-b x)} \sqrt{\frac{a+b x}{a}}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (b e-a f)^3 \sqrt{a-b x} \tan ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right )}{b^6 \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]

-(a^4*C*f^3*(a - b*x)*Sqrt[a + b*x]*(2 - (a - b*x)/a)^(11/2)*(315/(2 - (a - b*x)/a)^5 + 105/(2 - (a - b*x)/a)^
4 + 42/(2 - (a - b*x)/a)^3 + 18/(2 - (a - b*x)/a)^2 + 4/(1 - (a - b*x)/(2*a)) + (630*Sqrt[a]*ArcSin[Sqrt[a - b
*x]/(Sqrt[2]*Sqrt[a])])/(Sqrt[a - b*x]*(2 - (a - b*x)/a)^(11/2))))/(40*b^6*Sqrt[c*(a - b*x)]*Sqrt[(a + b*x)/a]
) - (a^3*f^2*(3*b*C*e + b*B*f - 5*a*C*f)*(a - b*x)*Sqrt[a + b*x]*(2 - (a - b*x)/a)^(9/2)*(105/(2 - (a - b*x)/a
)^4 + 35/(2 - (a - b*x)/a)^3 + 14/(2 - (a - b*x)/a)^2 + 3/(1 - (a - b*x)/(2*a)) + (210*Sqrt[a]*ArcSin[Sqrt[a -
b*x]/(Sqrt[2]*Sqrt[a])])/(Sqrt[a - b*x]*(2 - (a - b*x)/a)^(9/2))))/(24*b^6*Sqrt[c*(a - b*x)]*Sqrt[(a + b*x)/a
]) - (a^2*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)))*(a - b*x)*Sqrt[a + b*x]*(
2 - (a - b*x)/a)^(7/2)*(15/(2 - (a - b*x)/a)^3 + 5/(2 - (a - b*x)/a)^2 + (1 - (a - b*x)/(2*a))^(-1) + (30*Sqrt
[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])])/(Sqrt[a - b*x]*(2 - (a - b*x)/a)^(7/2))))/(6*b^6*Sqrt[c*(a - b*x)
]*Sqrt[(a + b*x)/a]) - (a*(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)))
*(a - b*x)*Sqrt[a + b*x]*(2 - (a - b*x)/a)^(5/2)*(6/(2 - (a - b*x)/a)^2 + (1 - (a - b*x)/(2*a))^(-1) + (12*Sqr
t[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])])/(Sqrt[a - b*x]*(2 - (a - b*x)/a)^(5/2))))/(4*b^6*Sqrt[c*(a - b*x
)]*Sqrt[(a + b*x)/a]) - ((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))*(a - b*x)*Sqrt[a
+ b*x]*(2 - (a - b*x)/a)^(3/2)*((2 - (a - b*x)/a)^(-1) + (2*Sqrt[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])])/(
Sqrt[a - b*x]*(2 - (a - b*x)/a)^(3/2))))/(b^6*Sqrt[c*(a - b*x)]*Sqrt[(a + b*x)/a]) - (2*(A*b^2 - a*(b*B - a*C)
)*(b*e - a*f)^3*Sqrt[a - b*x]*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/(b^6*Sqrt[c*(a - b*x)])

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Maple [B]  time = 0.028, size = 965, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.11801, size = 1538, normalized size = 3.07 \begin{align*} \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 ] \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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