3.1.8 \(\int \frac {(e+f x)^3 (A+B x+C x^2)}{\sqrt {1-d x} \sqrt {1+d x}} \, dx\)

Optimal. Leaf size=340 \[ -\frac {\sqrt {1-d^2 x^2} (e+f x)^2 \left (4 f^2 \left (5 A d^2+4 C\right )-3 d^2 e (C e-5 B f)\right )}{60 d^4 f}+\frac {\sqrt {1-d^2 x^2} \left (d^2 f x \left (-100 A d^2 e f^2-30 B d^2 e^2 f-45 B f^3+6 C d^2 e^3-71 C e f^2\right )+4 \left (C \left (3 d^4 e^4-52 d^2 e^2 f^2-16 f^4\right )-5 d^2 f \left (4 A f \left (4 d^2 e^2+f^2\right )+3 B \left (d^2 e^3+4 e f^2\right )\right )\right )\right )}{120 d^6 f}+\frac {\sin ^{-1}(d x) \left (8 A d^4 e^3+12 A d^2 e f^2+12 B d^2 e^2 f+3 B f^3+4 C d^2 e^3+9 C e f^2\right )}{8 d^5}+\frac {\sqrt {1-d^2 x^2} (e+f x)^3 (C e-5 B f)}{20 d^2 f}-\frac {C \sqrt {1-d^2 x^2} (e+f x)^4}{5 d^2 f} \]

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Rubi [A]  time = 0.63, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1609, 1654, 833, 780, 216} \begin {gather*} -\frac {\sqrt {1-d^2 x^2} (e+f x)^2 \left (5 f (4 A f+3 B e)-C \left (3 e^2-\frac {16 f^2}{d^2}\right )\right )}{60 d^2 f}+\frac {\sqrt {1-d^2 x^2} \left (d^2 f x \left (-100 A d^2 e f^2-30 B d^2 e^2 f-45 B f^3+6 C d^2 e^3-71 C e f^2\right )+4 \left (C \left (-52 d^2 e^2 f^2+3 d^4 e^4-16 f^4\right )-5 d^2 f \left (4 A f \left (4 d^2 e^2+f^2\right )+3 B \left (d^2 e^3+4 e f^2\right )\right )\right )\right )}{120 d^6 f}+\frac {\sin ^{-1}(d x) \left (8 A d^4 e^3+12 A d^2 e f^2+12 B d^2 e^2 f+3 B f^3+4 C d^2 e^3+9 C e f^2\right )}{8 d^5}+\frac {\sqrt {1-d^2 x^2} (e+f x)^3 (C e-5 B f)}{20 d^2 f}-\frac {C \sqrt {1-d^2 x^2} (e+f x)^4}{5 d^2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*(A + B*x + C*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]

[Out]

-((5*f*(3*B*e + 4*A*f) - C*(3*e^2 - (16*f^2)/d^2))*(e + f*x)^2*Sqrt[1 - d^2*x^2])/(60*d^2*f) + ((C*e - 5*B*f)*
(e + f*x)^3*Sqrt[1 - d^2*x^2])/(20*d^2*f) - (C*(e + f*x)^4*Sqrt[1 - d^2*x^2])/(5*d^2*f) + ((4*(C*(3*d^4*e^4 -
52*d^2*e^2*f^2 - 16*f^4) - 5*d^2*f*(4*A*f*(4*d^2*e^2 + f^2) + 3*B*(d^2*e^3 + 4*e*f^2))) + d^2*f*(6*C*d^2*e^3 -
30*B*d^2*e^2*f - 71*C*e*f^2 - 100*A*d^2*e*f^2 - 45*B*f^3)*x)*Sqrt[1 - d^2*x^2])/(120*d^6*f) + ((4*C*d^2*e^3 +
8*A*d^4*e^3 + 12*B*d^2*e^2*f + 9*C*e*f^2 + 12*A*d^2*e*f^2 + 3*B*f^3)*ArcSin[d*x])/(8*d^5)

Rule 216

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

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 1609

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[P
x*(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d,
0] && EqQ[m, n] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

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 {1-d x} \sqrt {1+d x}} \, dx &=\int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {C (e+f x)^4 \sqrt {1-d^2 x^2}}{5 d^2 f}-\frac {\int \frac {(e+f x)^3 \left (-\left (\left (4 C+5 A d^2\right ) f^2\right )+d^2 f (C e-5 B f) x\right )}{\sqrt {1-d^2 x^2}} \, dx}{5 d^2 f^2}\\ &=\frac {(C e-5 B f) (e+f x)^3 \sqrt {1-d^2 x^2}}{20 d^2 f}-\frac {C (e+f x)^4 \sqrt {1-d^2 x^2}}{5 d^2 f}+\frac {\int \frac {(e+f x)^2 \left (d^2 f^2 \left (13 C e+20 A d^2 e+15 B f\right )+d^2 f \left (4 \left (4 C+5 A d^2\right ) f^2-3 d^2 e (C e-5 B f)\right ) x\right )}{\sqrt {1-d^2 x^2}} \, dx}{20 d^4 f^2}\\ &=-\frac {\left (4 \left (4 C+5 A d^2\right ) f^2-3 d^2 e (C e-5 B f)\right ) (e+f x)^2 \sqrt {1-d^2 x^2}}{60 d^4 f}+\frac {(C e-5 B f) (e+f x)^3 \sqrt {1-d^2 x^2}}{20 d^2 f}-\frac {C (e+f x)^4 \sqrt {1-d^2 x^2}}{5 d^2 f}-\frac {\int \frac {(e+f x) \left (-d^2 f^2 \left (33 C d^2 e^2+60 A d^4 e^2+75 B d^2 e f+32 C f^2+40 A d^2 f^2\right )+d^4 f \left (6 C d^2 e^3-30 B d^2 e^2 f-71 C e f^2-100 A d^2 e f^2-45 B f^3\right ) x\right )}{\sqrt {1-d^2 x^2}} \, dx}{60 d^6 f^2}\\ &=-\frac {\left (4 \left (4 C+5 A d^2\right ) f^2-3 d^2 e (C e-5 B f)\right ) (e+f x)^2 \sqrt {1-d^2 x^2}}{60 d^4 f}+\frac {(C e-5 B f) (e+f x)^3 \sqrt {1-d^2 x^2}}{20 d^2 f}-\frac {C (e+f x)^4 \sqrt {1-d^2 x^2}}{5 d^2 f}+\frac {\left (4 \left (C \left (3 d^4 e^4-52 d^2 e^2 f^2-16 f^4\right )-5 d^2 f \left (4 A f \left (4 d^2 e^2+f^2\right )+3 B \left (d^2 e^3+4 e f^2\right )\right )\right )+d^2 f \left (6 C d^2 e^3-30 B d^2 e^2 f-71 C e f^2-100 A d^2 e f^2-45 B f^3\right ) x\right ) \sqrt {1-d^2 x^2}}{120 d^6 f}+\frac {\left (4 C d^2 e^3+8 A d^4 e^3+12 B d^2 e^2 f+9 C e f^2+12 A d^2 e f^2+3 B f^3\right ) \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{8 d^4}\\ &=-\frac {\left (4 \left (4 C+5 A d^2\right ) f^2-3 d^2 e (C e-5 B f)\right ) (e+f x)^2 \sqrt {1-d^2 x^2}}{60 d^4 f}+\frac {(C e-5 B f) (e+f x)^3 \sqrt {1-d^2 x^2}}{20 d^2 f}-\frac {C (e+f x)^4 \sqrt {1-d^2 x^2}}{5 d^2 f}+\frac {\left (4 \left (C \left (3 d^4 e^4-52 d^2 e^2 f^2-16 f^4\right )-5 d^2 f \left (4 A f \left (4 d^2 e^2+f^2\right )+3 B \left (d^2 e^3+4 e f^2\right )\right )\right )+d^2 f \left (6 C d^2 e^3-30 B d^2 e^2 f-71 C e f^2-100 A d^2 e f^2-45 B f^3\right ) x\right ) \sqrt {1-d^2 x^2}}{120 d^6 f}+\frac {\left (4 C d^2 e^3+8 A d^4 e^3+12 B d^2 e^2 f+9 C e f^2+12 A d^2 e f^2+3 B f^3\right ) \sin ^{-1}(d x)}{8 d^5}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 241, normalized size = 0.71 \begin {gather*} \frac {15 d \sin ^{-1}(d x) \left (8 A d^4 e^3+12 A d^2 e f^2+12 B d^2 e^2 f+3 B f^3+4 C d^2 e^3+9 C e f^2\right )-\sqrt {1-d^2 x^2} \left (20 A d^2 f \left (d^2 \left (18 e^2+9 e f x+2 f^2 x^2\right )+4 f^2\right )+15 B \left (2 d^4 \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )+d^2 f^2 (16 e+3 f x)\right )+C \left (6 d^4 x \left (10 e^3+20 e^2 f x+15 e f^2 x^2+4 f^3 x^3\right )+d^2 f \left (240 e^2+135 e f x+32 f^2 x^2\right )+64 f^3\right )\right )}{120 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^3*(A + B*x + C*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]

[Out]

(-(Sqrt[1 - d^2*x^2]*(20*A*d^2*f*(4*f^2 + d^2*(18*e^2 + 9*e*f*x + 2*f^2*x^2)) + 15*B*(d^2*f^2*(16*e + 3*f*x) +
2*d^4*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)) + C*(64*f^3 + d^2*f*(240*e^2 + 135*e*f*x + 32*f^2*x^2) + 6
*d^4*x*(10*e^3 + 20*e^2*f*x + 15*e*f^2*x^2 + 4*f^3*x^3)))) + 15*d*(4*C*d^2*e^3 + 8*A*d^4*e^3 + 12*B*d^2*e^2*f
+ 9*C*e*f^2 + 12*A*d^2*e*f^2 + 3*B*f^3)*ArcSin[d*x])/(120*d^6)

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IntegrateAlgebraic [B]  time = 0.77, size = 1135, normalized size = 3.34 \begin {gather*} \frac {\left (-8 A e^3 d^4-4 C e^3 d^2-12 A e f^2 d^2-12 B e^2 f d^2-3 B f^3-9 C e f^2\right ) \tan ^{-1}\left (\frac {\sqrt {1-d x}}{\sqrt {d x+1}}\right )}{4 d^5}-\frac {\sqrt {1-d x} \left (120 B e^3 d^4+360 A e^2 f d^4+\frac {480 B e^3 (1-d x) d^4}{d x+1}+\frac {1440 A e^2 f (1-d x) d^4}{d x+1}+\frac {720 B e^3 (1-d x)^2 d^4}{(d x+1)^2}+\frac {2160 A e^2 f (1-d x)^2 d^4}{(d x+1)^2}+\frac {480 B e^3 (1-d x)^3 d^4}{(d x+1)^3}+\frac {1440 A e^2 f (1-d x)^3 d^4}{(d x+1)^3}+\frac {120 B e^3 (1-d x)^4 d^4}{(d x+1)^4}+\frac {360 A e^2 f (1-d x)^4 d^4}{(d x+1)^4}+60 C e^3 d^3+180 A e f^2 d^3+180 B e^2 f d^3+\frac {120 C e^3 (1-d x) d^3}{d x+1}+\frac {360 A e f^2 (1-d x) d^3}{d x+1}+\frac {360 B e^2 f (1-d x) d^3}{d x+1}-\frac {120 C e^3 (1-d x)^3 d^3}{(d x+1)^3}-\frac {360 A e f^2 (1-d x)^3 d^3}{(d x+1)^3}-\frac {360 B e^2 f (1-d x)^3 d^3}{(d x+1)^3}-\frac {60 C e^3 (1-d x)^4 d^3}{(d x+1)^4}-\frac {180 A e f^2 (1-d x)^4 d^3}{(d x+1)^4}-\frac {180 B e^2 f (1-d x)^4 d^3}{(d x+1)^4}+120 A f^3 d^2+360 B e f^2 d^2+360 C e^2 f d^2+\frac {320 A f^3 (1-d x) d^2}{d x+1}+\frac {960 B e f^2 (1-d x) d^2}{d x+1}+\frac {960 C e^2 f (1-d x) d^2}{d x+1}+\frac {400 A f^3 (1-d x)^2 d^2}{(d x+1)^2}+\frac {1200 B e f^2 (1-d x)^2 d^2}{(d x+1)^2}+\frac {1200 C e^2 f (1-d x)^2 d^2}{(d x+1)^2}+\frac {320 A f^3 (1-d x)^3 d^2}{(d x+1)^3}+\frac {960 B e f^2 (1-d x)^3 d^2}{(d x+1)^3}+\frac {960 C e^2 f (1-d x)^3 d^2}{(d x+1)^3}+\frac {120 A f^3 (1-d x)^4 d^2}{(d x+1)^4}+\frac {360 B e f^2 (1-d x)^4 d^2}{(d x+1)^4}+\frac {360 C e^2 f (1-d x)^4 d^2}{(d x+1)^4}+75 B f^3 d+225 C e f^2 d+\frac {30 B f^3 (1-d x) d}{d x+1}+\frac {90 C e f^2 (1-d x) d}{d x+1}-\frac {30 B f^3 (1-d x)^3 d}{(d x+1)^3}-\frac {90 C e f^2 (1-d x)^3 d}{(d x+1)^3}-\frac {75 B f^3 (1-d x)^4 d}{(d x+1)^4}-\frac {225 C e f^2 (1-d x)^4 d}{(d x+1)^4}+120 C f^3+\frac {160 C f^3 (1-d x)}{d x+1}+\frac {464 C f^3 (1-d x)^2}{(d x+1)^2}+\frac {160 C f^3 (1-d x)^3}{(d x+1)^3}+\frac {120 C f^3 (1-d x)^4}{(d x+1)^4}\right )}{60 d^6 \sqrt {d x+1} \left (\frac {1-d x}{d x+1}+1\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((e + f*x)^3*(A + B*x + C*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]

[Out]

-1/60*(Sqrt[1 - d*x]*(60*C*d^3*e^3 + 120*B*d^4*e^3 + 360*C*d^2*e^2*f + 180*B*d^3*e^2*f + 360*A*d^4*e^2*f + 225
*C*d*e*f^2 + 360*B*d^2*e*f^2 + 180*A*d^3*e*f^2 + 120*C*f^3 + 75*B*d*f^3 + 120*A*d^2*f^3 - (60*C*d^3*e^3*(1 - d
*x)^4)/(1 + d*x)^4 + (120*B*d^4*e^3*(1 - d*x)^4)/(1 + d*x)^4 + (360*C*d^2*e^2*f*(1 - d*x)^4)/(1 + d*x)^4 - (18
0*B*d^3*e^2*f*(1 - d*x)^4)/(1 + d*x)^4 + (360*A*d^4*e^2*f*(1 - d*x)^4)/(1 + d*x)^4 - (225*C*d*e*f^2*(1 - d*x)^
4)/(1 + d*x)^4 + (360*B*d^2*e*f^2*(1 - d*x)^4)/(1 + d*x)^4 - (180*A*d^3*e*f^2*(1 - d*x)^4)/(1 + d*x)^4 + (120*
C*f^3*(1 - d*x)^4)/(1 + d*x)^4 - (75*B*d*f^3*(1 - d*x)^4)/(1 + d*x)^4 + (120*A*d^2*f^3*(1 - d*x)^4)/(1 + d*x)^
4 - (120*C*d^3*e^3*(1 - d*x)^3)/(1 + d*x)^3 + (480*B*d^4*e^3*(1 - d*x)^3)/(1 + d*x)^3 + (960*C*d^2*e^2*f*(1 -
d*x)^3)/(1 + d*x)^3 - (360*B*d^3*e^2*f*(1 - d*x)^3)/(1 + d*x)^3 + (1440*A*d^4*e^2*f*(1 - d*x)^3)/(1 + d*x)^3 -
(90*C*d*e*f^2*(1 - d*x)^3)/(1 + d*x)^3 + (960*B*d^2*e*f^2*(1 - d*x)^3)/(1 + d*x)^3 - (360*A*d^3*e*f^2*(1 - d*
x)^3)/(1 + d*x)^3 + (160*C*f^3*(1 - d*x)^3)/(1 + d*x)^3 - (30*B*d*f^3*(1 - d*x)^3)/(1 + d*x)^3 + (320*A*d^2*f^
3*(1 - d*x)^3)/(1 + d*x)^3 + (720*B*d^4*e^3*(1 - d*x)^2)/(1 + d*x)^2 + (1200*C*d^2*e^2*f*(1 - d*x)^2)/(1 + d*x
)^2 + (2160*A*d^4*e^2*f*(1 - d*x)^2)/(1 + d*x)^2 + (1200*B*d^2*e*f^2*(1 - d*x)^2)/(1 + d*x)^2 + (464*C*f^3*(1
- d*x)^2)/(1 + d*x)^2 + (400*A*d^2*f^3*(1 - d*x)^2)/(1 + d*x)^2 + (120*C*d^3*e^3*(1 - d*x))/(1 + d*x) + (480*B
*d^4*e^3*(1 - d*x))/(1 + d*x) + (960*C*d^2*e^2*f*(1 - d*x))/(1 + d*x) + (360*B*d^3*e^2*f*(1 - d*x))/(1 + d*x)
+ (1440*A*d^4*e^2*f*(1 - d*x))/(1 + d*x) + (90*C*d*e*f^2*(1 - d*x))/(1 + d*x) + (960*B*d^2*e*f^2*(1 - d*x))/(1
+ d*x) + (360*A*d^3*e*f^2*(1 - d*x))/(1 + d*x) + (160*C*f^3*(1 - d*x))/(1 + d*x) + (30*B*d*f^3*(1 - d*x))/(1
+ d*x) + (320*A*d^2*f^3*(1 - d*x))/(1 + d*x)))/(d^6*Sqrt[1 + d*x]*(1 + (1 - d*x)/(1 + d*x))^5) + ((-4*C*d^2*e^
3 - 8*A*d^4*e^3 - 12*B*d^2*e^2*f - 9*C*e*f^2 - 12*A*d^2*e*f^2 - 3*B*f^3)*ArcTan[Sqrt[1 - d*x]/Sqrt[1 + d*x]])/
(4*d^5)

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fricas [A]  time = 1.23, size = 286, normalized size = 0.84 \begin {gather*} -\frac {{\left (24 \, C d^{4} f^{3} x^{4} + 120 \, B d^{4} e^{3} + 240 \, B d^{2} e f^{2} + 120 \, {\left (3 \, A d^{4} + 2 \, C d^{2}\right )} e^{2} f + 16 \, {\left (5 \, A d^{2} + 4 \, C\right )} f^{3} + 30 \, {\left (3 \, C d^{4} e f^{2} + B d^{4} f^{3}\right )} x^{3} + 8 \, {\left (15 \, C d^{4} e^{2} f + 15 \, B d^{4} e f^{2} + {\left (5 \, A d^{4} + 4 \, C d^{2}\right )} f^{3}\right )} x^{2} + 15 \, {\left (4 \, C d^{4} e^{3} + 12 \, B d^{4} e^{2} f + 3 \, B d^{2} f^{3} + 3 \, {\left (4 \, A d^{4} + 3 \, C d^{2}\right )} e f^{2}\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 30 \, {\left (12 \, B d^{3} e^{2} f + 3 \, B d f^{3} + 4 \, {\left (2 \, A d^{5} + C d^{3}\right )} e^{3} + 3 \, {\left (4 \, A d^{3} + 3 \, C d\right )} e f^{2}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{120 \, d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*((24*C*d^4*f^3*x^4 + 120*B*d^4*e^3 + 240*B*d^2*e*f^2 + 120*(3*A*d^4 + 2*C*d^2)*e^2*f + 16*(5*A*d^2 + 4*
C)*f^3 + 30*(3*C*d^4*e*f^2 + B*d^4*f^3)*x^3 + 8*(15*C*d^4*e^2*f + 15*B*d^4*e*f^2 + (5*A*d^4 + 4*C*d^2)*f^3)*x^
2 + 15*(4*C*d^4*e^3 + 12*B*d^4*e^2*f + 3*B*d^2*f^3 + 3*(4*A*d^4 + 3*C*d^2)*e*f^2)*x)*sqrt(d*x + 1)*sqrt(-d*x +
1) + 30*(12*B*d^3*e^2*f + 3*B*d*f^3 + 4*(2*A*d^5 + C*d^3)*e^3 + 3*(4*A*d^3 + 3*C*d)*e*f^2)*arctan((sqrt(d*x +
1)*sqrt(-d*x + 1) - 1)/(d*x)))/d^6

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giac [A]  time = 1.82, size = 427, normalized size = 1.26 \begin {gather*} -\frac {{\left ({\left (2 \, {\left (d x + 1\right )} {\left (3 \, {\left (d x + 1\right )} {\left (\frac {4 \, {\left (d x + 1\right )} C f^{3}}{d^{5}} + \frac {5 \, B d^{26} f^{3} + 15 \, C d^{26} f^{2} e - 16 \, C d^{25} f^{3}}{d^{30}}\right )} + \frac {20 \, A d^{27} f^{3} + 60 \, B d^{27} f^{2} e - 45 \, B d^{26} f^{3} + 60 \, C d^{27} f e^{2} - 135 \, C d^{26} f^{2} e + 88 \, C d^{25} f^{3}}{d^{30}}\right )} + \frac {5 \, {\left (36 \, A d^{28} f^{2} e - 16 \, A d^{27} f^{3} + 36 \, B d^{28} f e^{2} - 48 \, B d^{27} f^{2} e + 27 \, B d^{26} f^{3} + 12 \, C d^{28} e^{3} - 48 \, C d^{27} f e^{2} + 81 \, C d^{26} f^{2} e - 32 \, C d^{25} f^{3}\right )}}{d^{30}}\right )} {\left (d x + 1\right )} + \frac {15 \, {\left (24 \, A d^{29} f e^{2} - 12 \, A d^{28} f^{2} e + 8 \, A d^{27} f^{3} + 8 \, B d^{29} e^{3} - 12 \, B d^{28} f e^{2} + 24 \, B d^{27} f^{2} e - 5 \, B d^{26} f^{3} - 4 \, C d^{28} e^{3} + 24 \, C d^{27} f e^{2} - 15 \, C d^{26} f^{2} e + 8 \, C d^{25} f^{3}\right )}}{d^{30}}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} - \frac {30 \, {\left (8 \, A d^{4} e^{3} + 12 \, A d^{2} f^{2} e + 12 \, B d^{2} f e^{2} + 3 \, B f^{3} + 4 \, C d^{2} e^{3} + 9 \, C f^{2} e\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{4}}}{120 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*(C*x^2+B*x+A)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="giac")

[Out]

-1/120*(((2*(d*x + 1)*(3*(d*x + 1)*(4*(d*x + 1)*C*f^3/d^5 + (5*B*d^26*f^3 + 15*C*d^26*f^2*e - 16*C*d^25*f^3)/d
^30) + (20*A*d^27*f^3 + 60*B*d^27*f^2*e - 45*B*d^26*f^3 + 60*C*d^27*f*e^2 - 135*C*d^26*f^2*e + 88*C*d^25*f^3)/
d^30) + 5*(36*A*d^28*f^2*e - 16*A*d^27*f^3 + 36*B*d^28*f*e^2 - 48*B*d^27*f^2*e + 27*B*d^26*f^3 + 12*C*d^28*e^3
- 48*C*d^27*f*e^2 + 81*C*d^26*f^2*e - 32*C*d^25*f^3)/d^30)*(d*x + 1) + 15*(24*A*d^29*f*e^2 - 12*A*d^28*f^2*e
+ 8*A*d^27*f^3 + 8*B*d^29*e^3 - 12*B*d^28*f*e^2 + 24*B*d^27*f^2*e - 5*B*d^26*f^3 - 4*C*d^28*e^3 + 24*C*d^27*f*
e^2 - 15*C*d^26*f^2*e + 8*C*d^25*f^3)/d^30)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 30*(8*A*d^4*e^3 + 12*A*d^2*f^2*e +
12*B*d^2*f*e^2 + 3*B*f^3 + 4*C*d^2*e^3 + 9*C*f^2*e)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^4)/d

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maple [C]  time = 0.03, size = 643, normalized size = 1.89 \begin {gather*} -\frac {\sqrt {-d x +1}\, \sqrt {d x +1}\, \left (24 \sqrt {-d^{2} x^{2}+1}\, C \,d^{4} f^{3} x^{4} \mathrm {csgn}\relax (d )+30 \sqrt {-d^{2} x^{2}+1}\, B \,d^{4} f^{3} x^{3} \mathrm {csgn}\relax (d )+90 \sqrt {-d^{2} x^{2}+1}\, C \,d^{4} e \,f^{2} x^{3} \mathrm {csgn}\relax (d )+40 \sqrt {-d^{2} x^{2}+1}\, A \,d^{4} f^{3} x^{2} \mathrm {csgn}\relax (d )+120 \sqrt {-d^{2} x^{2}+1}\, B \,d^{4} e \,f^{2} x^{2} \mathrm {csgn}\relax (d )+120 \sqrt {-d^{2} x^{2}+1}\, C \,d^{4} e^{2} f \,x^{2} \mathrm {csgn}\relax (d )-120 A \,d^{5} e^{3} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+180 \sqrt {-d^{2} x^{2}+1}\, A \,d^{4} e \,f^{2} x \,\mathrm {csgn}\relax (d )+180 \sqrt {-d^{2} x^{2}+1}\, B \,d^{4} e^{2} f x \,\mathrm {csgn}\relax (d )+60 \sqrt {-d^{2} x^{2}+1}\, C \,d^{4} e^{3} x \,\mathrm {csgn}\relax (d )+360 \sqrt {-d^{2} x^{2}+1}\, A \,d^{4} e^{2} f \,\mathrm {csgn}\relax (d )+120 \sqrt {-d^{2} x^{2}+1}\, B \,d^{4} e^{3} \mathrm {csgn}\relax (d )+32 \sqrt {-d^{2} x^{2}+1}\, C \,d^{2} f^{3} x^{2} \mathrm {csgn}\relax (d )-180 A \,d^{3} e \,f^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-180 B \,d^{3} e^{2} f \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+45 \sqrt {-d^{2} x^{2}+1}\, B \,d^{2} f^{3} x \,\mathrm {csgn}\relax (d )-60 C \,d^{3} e^{3} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+135 \sqrt {-d^{2} x^{2}+1}\, C \,d^{2} e \,f^{2} x \,\mathrm {csgn}\relax (d )+80 \sqrt {-d^{2} x^{2}+1}\, A \,d^{2} f^{3} \mathrm {csgn}\relax (d )+240 \sqrt {-d^{2} x^{2}+1}\, B \,d^{2} e \,f^{2} \mathrm {csgn}\relax (d )+240 \sqrt {-d^{2} x^{2}+1}\, C \,d^{2} e^{2} f \,\mathrm {csgn}\relax (d )-45 B d \,f^{3} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-135 C d e \,f^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+64 \sqrt {-d^{2} x^{2}+1}\, C \,f^{3} \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{120 \sqrt {-d^{2} x^{2}+1}\, d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(24*(-d^2*x^2+1)^(1/2)*C*d^4*f^3*x^4*csgn(d)+30*(-d^2*x^2+1)^(1/2)*B*d^4*f
^3*x^3*csgn(d)+90*(-d^2*x^2+1)^(1/2)*C*d^4*e*f^2*x^3*csgn(d)+40*(-d^2*x^2+1)^(1/2)*A*d^4*f^3*x^2*csgn(d)+120*(
-d^2*x^2+1)^(1/2)*B*d^4*e*f^2*x^2*csgn(d)+120*(-d^2*x^2+1)^(1/2)*C*d^4*e^2*f*x^2*csgn(d)+180*(-d^2*x^2+1)^(1/2
)*A*d^4*e*f^2*x*csgn(d)+180*(-d^2*x^2+1)^(1/2)*B*d^4*e^2*f*x*csgn(d)+60*(-d^2*x^2+1)^(1/2)*C*d^4*e^3*x*csgn(d)
+360*(-d^2*x^2+1)^(1/2)*A*d^4*e^2*f*csgn(d)-120*A*d^5*e^3*arctan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))+120*(-d^2*x
^2+1)^(1/2)*B*d^4*e^3*csgn(d)+32*(-d^2*x^2+1)^(1/2)*C*d^2*f^3*x^2*csgn(d)+45*(-d^2*x^2+1)^(1/2)*B*d^2*f^3*x*cs
gn(d)+135*(-d^2*x^2+1)^(1/2)*C*d^2*e*f^2*x*csgn(d)+80*(-d^2*x^2+1)^(1/2)*A*d^2*f^3*csgn(d)-180*A*d^3*e*f^2*arc
tan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))+240*(-d^2*x^2+1)^(1/2)*B*d^2*e*f^2*csgn(d)-180*B*d^3*e^2*f*arctan(1/(-d^
2*x^2+1)^(1/2)*d*x*csgn(d))+240*(-d^2*x^2+1)^(1/2)*C*d^2*e^2*f*csgn(d)-60*C*d^3*e^3*arctan(1/(-d^2*x^2+1)^(1/2
)*d*x*csgn(d))-45*B*d*f^3*arctan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))+64*(-d^2*x^2+1)^(1/2)*C*f^3*csgn(d)-135*C*d
*e*f^2*arctan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d)))*csgn(d)/d^6/(-d^2*x^2+1)^(1/2)

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maxima [A]  time = 1.05, size = 355, normalized size = 1.04 \begin {gather*} -\frac {\sqrt {-d^{2} x^{2} + 1} C f^{3} x^{4}}{5 \, d^{2}} + \frac {A e^{3} \arcsin \left (d x\right )}{d} - \frac {\sqrt {-d^{2} x^{2} + 1} B e^{3}}{d^{2}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} A e^{2} f}{d^{2}} - \frac {4 \, \sqrt {-d^{2} x^{2} + 1} C f^{3} x^{2}}{15 \, d^{4}} - \frac {{\left (3 \, C e f^{2} + B f^{3}\right )} \sqrt {-d^{2} x^{2} + 1} x^{3}}{4 \, d^{2}} - \frac {{\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} \sqrt {-d^{2} x^{2} + 1} x^{2}}{3 \, d^{2}} - \frac {{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} \sqrt {-d^{2} x^{2} + 1} x}{2 \, d^{2}} + \frac {{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} \arcsin \left (d x\right )}{2 \, d^{3}} - \frac {8 \, \sqrt {-d^{2} x^{2} + 1} C f^{3}}{15 \, d^{6}} - \frac {3 \, {\left (3 \, C e f^{2} + B f^{3}\right )} \sqrt {-d^{2} x^{2} + 1} x}{8 \, d^{4}} - \frac {2 \, {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} \sqrt {-d^{2} x^{2} + 1}}{3 \, d^{4}} + \frac {3 \, {\left (3 \, C e f^{2} + B f^{3}\right )} \arcsin \left (d x\right )}{8 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 35.29, size = 2606, normalized size = 7.66

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))/((1 - d*x)^(1/2)*(d*x + 1)^(1/2)),x)

[Out]

- ((((2048*C*f^3)/3 + 640*C*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6 + (((2048*C*f^3)/3 + 6
40*C*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) - 1)^14 - (((4096*C*f^3)/3 - 832*C*d^2*e^2*f)*((1 -
d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8 - (((4096*C*f^3)/3 - 832*C*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^12)/((
d*x + 1)^(1/2) - 1)^12 + (((12288*C*f^3)/5 + 768*C*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^10)/((d*x + 1)^(1/2) - 1)^
10 + (((1 - d*x)^(1/2) - 1)^3*(2*C*d^3*e^3 - (87*C*d*e*f^2)/2))/((d*x + 1)^(1/2) - 1)^3 - (((1 - d*x)^(1/2) -
1)^17*(2*C*d^3*e^3 - (87*C*d*e*f^2)/2))/((d*x + 1)^(1/2) - 1)^17 + (((1 - d*x)^(1/2) - 1)^7*(88*C*d^3*e^3 - 42
*C*d*e*f^2))/((d*x + 1)^(1/2) - 1)^7 - (((1 - d*x)^(1/2) - 1)^13*(88*C*d^3*e^3 - 42*C*d*e*f^2))/((d*x + 1)^(1/
2) - 1)^13 + (((1 - d*x)^(1/2) - 1)^5*(40*C*d^3*e^3 + 426*C*d*e*f^2))/((d*x + 1)^(1/2) - 1)^5 - (((1 - d*x)^(1
/2) - 1)^15*(40*C*d^3*e^3 + 426*C*d*e*f^2))/((d*x + 1)^(1/2) - 1)^15 + (((1 - d*x)^(1/2) - 1)^9*(52*C*d^3*e^3
- 507*C*d*e*f^2))/((d*x + 1)^(1/2) - 1)^9 - (((1 - d*x)^(1/2) - 1)^11*(52*C*d^3*e^3 - 507*C*d*e*f^2))/((d*x +
1)^(1/2) - 1)^11 - (d*(4*C*d^2*e^3 + 9*C*e*f^2)*((1 - d*x)^(1/2) - 1))/(2*((d*x + 1)^(1/2) - 1)) + (d*(4*C*d^2
*e^3 + 9*C*e*f^2)*((1 - d*x)^(1/2) - 1)^19)/(2*((d*x + 1)^(1/2) - 1)^19) + (192*C*d^2*e^2*f*((1 - d*x)^(1/2) -
1)^4)/((d*x + 1)^(1/2) - 1)^4 + (192*C*d^2*e^2*f*((1 - d*x)^(1/2) - 1)^16)/((d*x + 1)^(1/2) - 1)^16)/(d^6 + (
10*d^6*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (45*d^6*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) -
1)^4 + (120*d^6*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6 + (210*d^6*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1
)^(1/2) - 1)^8 + (252*d^6*((1 - d*x)^(1/2) - 1)^10)/((d*x + 1)^(1/2) - 1)^10 + (210*d^6*((1 - d*x)^(1/2) - 1)^
12)/((d*x + 1)^(1/2) - 1)^12 + (120*d^6*((1 - d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) - 1)^14 + (45*d^6*((1 - d*x
)^(1/2) - 1)^16)/((d*x + 1)^(1/2) - 1)^16 + (10*d^6*((1 - d*x)^(1/2) - 1)^18)/((d*x + 1)^(1/2) - 1)^18 + (d^6*
((1 - d*x)^(1/2) - 1)^20)/((d*x + 1)^(1/2) - 1)^20) - (((64*A*f^3 + 96*A*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^4)/(
(d*x + 1)^(1/2) - 1)^4 + ((64*A*f^3 + 96*A*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8 - (((12
8*A*f^3)/3 - 144*A*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6 + (24*A*d^2*e^2*f*((1 - d*x)^(1
/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (24*A*d^2*e^2*f*((1 - d*x)^(1/2) - 1)^10)/((d*x + 1)^(1/2) - 1)^10 - (6*
A*d*e*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (30*A*d*e*f^2*((1 - d*x)^(1/2) - 1)^3)/((d*x + 1)^(1/
2) - 1)^3 + (36*A*d*e*f^2*((1 - d*x)^(1/2) - 1)^5)/((d*x + 1)^(1/2) - 1)^5 - (36*A*d*e*f^2*((1 - d*x)^(1/2) -
1)^7)/((d*x + 1)^(1/2) - 1)^7 - (30*A*d*e*f^2*((1 - d*x)^(1/2) - 1)^9)/((d*x + 1)^(1/2) - 1)^9 + (6*A*d*e*f^2*
((1 - d*x)^(1/2) - 1)^11)/((d*x + 1)^(1/2) - 1)^11)/(d^4 + (6*d^4*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) -
1)^2 + (15*d^4*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (20*d^4*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^
(1/2) - 1)^6 + (15*d^4*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8 + (6*d^4*((1 - d*x)^(1/2) - 1)^10)/((d
*x + 1)^(1/2) - 1)^10 + (d^4*((1 - d*x)^(1/2) - 1)^12)/((d*x + 1)^(1/2) - 1)^12) - ((((3*B*f^3)/2 + 6*B*d^2*e^
2*f)*((1 - d*x)^(1/2) - 1)^15)/((d*x + 1)^(1/2) - 1)^15 - (((23*B*f^3)/2 - 18*B*d^2*e^2*f)*((1 - d*x)^(1/2) -
1)^3)/((d*x + 1)^(1/2) - 1)^3 + (((23*B*f^3)/2 - 18*B*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^13)/((d*x + 1)^(1/2) -
1)^13 + (((333*B*f^3)/2 + 90*B*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^5)/((d*x + 1)^(1/2) - 1)^5 - (((333*B*f^3)/2 +
90*B*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^11)/((d*x + 1)^(1/2) - 1)^11 - (((671*B*f^3)/2 - 66*B*d^2*e^2*f)*((1 -
d*x)^(1/2) - 1)^7)/((d*x + 1)^(1/2) - 1)^7 + (((671*B*f^3)/2 - 66*B*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^9)/((d*x
+ 1)^(1/2) - 1)^9 + (((1 - d*x)^(1/2) - 1)^4*(48*B*d^3*e^3 + 192*B*d*e*f^2))/((d*x + 1)^(1/2) - 1)^4 + (((1 -
d*x)^(1/2) - 1)^12*(48*B*d^3*e^3 + 192*B*d*e*f^2))/((d*x + 1)^(1/2) - 1)^12 + (((1 - d*x)^(1/2) - 1)^8*(160*B*
d^3*e^3 + 128*B*d*e*f^2))/((d*x + 1)^(1/2) - 1)^8 + (((1 - d*x)^(1/2) - 1)^6*(120*B*d^3*e^3 + 256*B*d*e*f^2))/
((d*x + 1)^(1/2) - 1)^6 + (((1 - d*x)^(1/2) - 1)^10*(120*B*d^3*e^3 + 256*B*d*e*f^2))/((d*x + 1)^(1/2) - 1)^10
- (((3*B*f^3)/2 + 6*B*d^2*e^2*f)*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + (8*B*d^3*e^3*((1 - d*x)^(1/2)
- 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (8*B*d^3*e^3*((1 - d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) - 1)^14)/(d^5 + (8*d
^5*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (28*d^5*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4
+ (56*d^5*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6 + (70*d^5*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2
) - 1)^8 + (56*d^5*((1 - d*x)^(1/2) - 1)^10)/((d*x + 1)^(1/2) - 1)^10 + (28*d^5*((1 - d*x)^(1/2) - 1)^12)/((d*
x + 1)^(1/2) - 1)^12 + (8*d^5*((1 - d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) - 1)^14 + (d^5*((1 - d*x)^(1/2) - 1)^
16)/((d*x + 1)^(1/2) - 1)^16) - (3*B*f*atan((B*f*(f^2 + 4*d^2*e^2)*((1 - d*x)^(1/2) - 1))/((B*f^3 + 4*B*d^2*e^
2*f)*((d*x + 1)^(1/2) - 1)))*(f^2 + 4*d^2*e^2))/(2*d^5) - (2*A*e*atan((A*e*((1 - d*x)^(1/2) - 1)*(3*f^2 + 2*d^
2*e^2))/((2*A*d^2*e^3 + 3*A*e*f^2)*((d*x + 1)^(1/2) - 1)))*(3*f^2 + 2*d^2*e^2))/d^3 - (C*e*atan((C*e*((1 - d*x
)^(1/2) - 1)*(9*f^2 + 4*d^2*e^2))/((4*C*d^2*e^3 + 9*C*e*f^2)*((d*x + 1)^(1/2) - 1)))*(9*f^2 + 4*d^2*e^2))/(2*d
^5)

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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)**3*(C*x**2+B*x+A)/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

Timed out

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