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

Optimal. Leaf size=415 \[ -\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^2 \left (7 d^2 f (2 A f+B e)-C \left (3 d^2 e^2-8 f^2\right )\right )}{70 d^4 f}+\frac{\left (1-d^2 x^2\right )^{3/2} \left (3 d^2 f x \left (-98 A d^2 e f^2-14 B d^2 e^2 f-35 B f^3+6 C d^2 e^3-41 C e f^2\right )+8 \left (C \left (-30 d^2 e^2 f^2+3 d^4 e^4-8 f^4\right )-7 d^2 f \left (2 A f \left (6 d^2 e^2+f^2\right )+B \left (d^2 e^3+6 e f^2\right )\right )\right )\right )}{840 d^6 f}+\frac{x \sqrt{1-d^2 x^2} \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{16 d^4}+\frac{\sin ^{-1}(d x) \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{16 d^5}+\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^3 (3 C e-7 B f)}{42 d^2 f}-\frac{C \left (1-d^2 x^2\right )^{3/2} (e+f x)^4}{7 d^2 f} \]

[Out]

((2*C*d^2*e^3 + 8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2 + B*f^3)*x*Sqrt[1 - d^2*x^2])/(16*d^4)
- ((7*d^2*f*(B*e + 2*A*f) - C*(3*d^2*e^2 - 8*f^2))*(e + f*x)^2*(1 - d^2*x^2)^(3/2))/(70*d^4*f) + ((3*C*e - 7*
B*f)*(e + f*x)^3*(1 - d^2*x^2)^(3/2))/(42*d^2*f) - (C*(e + f*x)^4*(1 - d^2*x^2)^(3/2))/(7*d^2*f) + ((8*(C*(3*d
^4*e^4 - 30*d^2*e^2*f^2 - 8*f^4) - 7*d^2*f*(2*A*f*(6*d^2*e^2 + f^2) + B*(d^2*e^3 + 6*e*f^2))) + 3*d^2*f*(6*C*d
^2*e^3 - 14*B*d^2*e^2*f - 41*C*e*f^2 - 98*A*d^2*e*f^2 - 35*B*f^3)*x)*(1 - d^2*x^2)^(3/2))/(840*d^6*f) + ((2*C*
d^2*e^3 + 8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2 + B*f^3)*ArcSin[d*x])/(16*d^5)

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Rubi [A]  time = 0.672762, antiderivative size = 415, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {1609, 1654, 833, 780, 195, 216} \[ -\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^2 \left (7 d^2 f (2 A f+B e)-C \left (3 d^2 e^2-8 f^2\right )\right )}{70 d^4 f}+\frac{\left (1-d^2 x^2\right )^{3/2} \left (3 d^2 f x \left (-98 A d^2 e f^2-14 B d^2 e^2 f-35 B f^3+6 C d^2 e^3-41 C e f^2\right )+8 \left (C \left (-30 d^2 e^2 f^2+3 d^4 e^4-8 f^4\right )-7 d^2 f \left (2 A f \left (6 d^2 e^2+f^2\right )+B \left (d^2 e^3+6 e f^2\right )\right )\right )\right )}{840 d^6 f}+\frac{x \sqrt{1-d^2 x^2} \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{16 d^4}+\frac{\sin ^{-1}(d x) \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{16 d^5}+\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^3 (3 C e-7 B f)}{42 d^2 f}-\frac{C \left (1-d^2 x^2\right )^{3/2} (e+f x)^4}{7 d^2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*C*d^2*e^3 + 8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2 + B*f^3)*x*Sqrt[1 - d^2*x^2])/(16*d^4)
- ((7*d^2*f*(B*e + 2*A*f) - C*(3*d^2*e^2 - 8*f^2))*(e + f*x)^2*(1 - d^2*x^2)^(3/2))/(70*d^4*f) + ((3*C*e - 7*
B*f)*(e + f*x)^3*(1 - d^2*x^2)^(3/2))/(42*d^2*f) - (C*(e + f*x)^4*(1 - d^2*x^2)^(3/2))/(7*d^2*f) + ((8*(C*(3*d
^4*e^4 - 30*d^2*e^2*f^2 - 8*f^4) - 7*d^2*f*(2*A*f*(6*d^2*e^2 + f^2) + B*(d^2*e^3 + 6*e*f^2))) + 3*d^2*f*(6*C*d
^2*e^3 - 14*B*d^2*e^2*f - 41*C*e*f^2 - 98*A*d^2*e*f^2 - 35*B*f^3)*x)*(1 - d^2*x^2)^(3/2))/(840*d^6*f) + ((2*C*
d^2*e^3 + 8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2 + B*f^3)*ArcSin[d*x])/(16*d^5)

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]))

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 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 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]

Rubi steps

\begin{align*} \int \sqrt{1-d x} \sqrt{1+d x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx &=\int (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 \left (1-d^2 x^2\right )^{3/2}}{7 d^2 f}-\frac{\int (e+f x)^3 \left (-\left (4 C+7 A d^2\right ) f^2+d^2 f (3 C e-7 B f) x\right ) \sqrt{1-d^2 x^2} \, dx}{7 d^2 f^2}\\ &=\frac{(3 C e-7 B f) (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{42 d^2 f}-\frac{C (e+f x)^4 \left (1-d^2 x^2\right )^{3/2}}{7 d^2 f}+\frac{\int (e+f x)^2 \left (3 d^2 f^2 \left (5 C e+14 A d^2 e+7 B f\right )+3 d^2 f \left (2 \left (4 C+7 A d^2\right ) f^2-d^2 e (3 C e-7 B f)\right ) x\right ) \sqrt{1-d^2 x^2} \, dx}{42 d^4 f^2}\\ &=-\frac{\left (7 d^2 f (B e+2 A f)-C \left (3 d^2 e^2-8 f^2\right )\right ) (e+f x)^2 \left (1-d^2 x^2\right )^{3/2}}{70 d^4 f}+\frac{(3 C e-7 B f) (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{42 d^2 f}-\frac{C (e+f x)^4 \left (1-d^2 x^2\right )^{3/2}}{7 d^2 f}-\frac{\int (e+f x) \left (-3 d^2 f^2 \left (19 C d^2 e^2+70 A d^4 e^2+49 B d^2 e f+16 C f^2+28 A d^2 f^2\right )+3 d^4 f \left (6 C d^2 e^3-14 B d^2 e^2 f-41 C e f^2-98 A d^2 e f^2-35 B f^3\right ) x\right ) \sqrt{1-d^2 x^2} \, dx}{210 d^6 f^2}\\ &=-\frac{\left (7 d^2 f (B e+2 A f)-C \left (3 d^2 e^2-8 f^2\right )\right ) (e+f x)^2 \left (1-d^2 x^2\right )^{3/2}}{70 d^4 f}+\frac{(3 C e-7 B f) (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{42 d^2 f}-\frac{C (e+f x)^4 \left (1-d^2 x^2\right )^{3/2}}{7 d^2 f}+\frac{\left (8 \left (C \left (3 d^4 e^4-30 d^2 e^2 f^2-8 f^4\right )-7 d^2 f \left (2 A f \left (6 d^2 e^2+f^2\right )+B \left (d^2 e^3+6 e f^2\right )\right )\right )+3 d^2 f \left (6 C d^2 e^3-14 B d^2 e^2 f-41 C e f^2-98 A d^2 e f^2-35 B f^3\right ) x\right ) \left (1-d^2 x^2\right )^{3/2}}{840 d^6 f}+\frac{\left (2 C d^2 e^3+8 A d^4 e^3+6 B d^2 e^2 f+3 C e f^2+6 A d^2 e f^2+B f^3\right ) \int \sqrt{1-d^2 x^2} \, dx}{8 d^4}\\ &=\frac{\left (2 C d^2 e^3+8 A d^4 e^3+6 B d^2 e^2 f+3 C e f^2+6 A d^2 e f^2+B f^3\right ) x \sqrt{1-d^2 x^2}}{16 d^4}-\frac{\left (7 d^2 f (B e+2 A f)-C \left (3 d^2 e^2-8 f^2\right )\right ) (e+f x)^2 \left (1-d^2 x^2\right )^{3/2}}{70 d^4 f}+\frac{(3 C e-7 B f) (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{42 d^2 f}-\frac{C (e+f x)^4 \left (1-d^2 x^2\right )^{3/2}}{7 d^2 f}+\frac{\left (8 \left (C \left (3 d^4 e^4-30 d^2 e^2 f^2-8 f^4\right )-7 d^2 f \left (2 A f \left (6 d^2 e^2+f^2\right )+B \left (d^2 e^3+6 e f^2\right )\right )\right )+3 d^2 f \left (6 C d^2 e^3-14 B d^2 e^2 f-41 C e f^2-98 A d^2 e f^2-35 B f^3\right ) x\right ) \left (1-d^2 x^2\right )^{3/2}}{840 d^6 f}+\frac{\left (2 C d^2 e^3+8 A d^4 e^3+6 B d^2 e^2 f+3 C e f^2+6 A d^2 e f^2+B f^3\right ) \int \frac{1}{\sqrt{1-d^2 x^2}} \, dx}{16 d^4}\\ &=\frac{\left (2 C d^2 e^3+8 A d^4 e^3+6 B d^2 e^2 f+3 C e f^2+6 A d^2 e f^2+B f^3\right ) x \sqrt{1-d^2 x^2}}{16 d^4}-\frac{\left (7 d^2 f (B e+2 A f)-C \left (3 d^2 e^2-8 f^2\right )\right ) (e+f x)^2 \left (1-d^2 x^2\right )^{3/2}}{70 d^4 f}+\frac{(3 C e-7 B f) (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{42 d^2 f}-\frac{C (e+f x)^4 \left (1-d^2 x^2\right )^{3/2}}{7 d^2 f}+\frac{\left (8 \left (C \left (3 d^4 e^4-30 d^2 e^2 f^2-8 f^4\right )-7 d^2 f \left (2 A f \left (6 d^2 e^2+f^2\right )+B \left (d^2 e^3+6 e f^2\right )\right )\right )+3 d^2 f \left (6 C d^2 e^3-14 B d^2 e^2 f-41 C e f^2-98 A d^2 e f^2-35 B f^3\right ) x\right ) \left (1-d^2 x^2\right )^{3/2}}{840 d^6 f}+\frac{\left (2 C d^2 e^3+8 A d^4 e^3+6 B d^2 e^2 f+3 C e f^2+6 A d^2 e f^2+B f^3\right ) \sin ^{-1}(d x)}{16 d^5}\\ \end{align*}

Mathematica [A]  time = 0.493917, size = 355, normalized size = 0.86 \[ \frac{\sqrt{1-d^2 x^2} \left (14 A d^2 \left (6 d^4 x \left (20 e^2 f x+10 e^3+15 e f^2 x^2+4 f^3 x^3\right )-d^2 f \left (120 e^2+45 e f x+8 f^2 x^2\right )-16 f^3\right )+7 B \left (4 d^6 x^2 \left (45 e^2 f x+20 e^3+36 e f^2 x^2+10 f^3 x^3\right )-2 d^4 \left (45 e^2 f x+40 e^3+24 e f^2 x^2+5 f^3 x^3\right )-3 d^2 f^2 (32 e+5 f x)\right )-C \left (-12 d^6 x^3 \left (84 e^2 f x+35 e^3+70 e f^2 x^2+20 f^3 x^3\right )+6 d^4 x \left (56 e^2 f x+35 e^3+35 e f^2 x^2+8 f^3 x^3\right )+d^2 f \left (672 e^2+315 e f x+64 f^2 x^2\right )+128 f^3\right )\right )+105 d \sin ^{-1}(d x) \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{1680 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - d^2*x^2]*(14*A*d^2*(-16*f^3 - d^2*f*(120*e^2 + 45*e*f*x + 8*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)) + 7*B*(-3*d^2*f^2*(32*e + 5*f*x) - 2*d^4*(40*e^3 + 45*e^2*f*x + 24*e*f^2*x^2 + 5*
f^3*x^3) + 4*d^6*x^2*(20*e^3 + 45*e^2*f*x + 36*e*f^2*x^2 + 10*f^3*x^3)) - C*(128*f^3 + d^2*f*(672*e^2 + 315*e*
f*x + 64*f^2*x^2) + 6*d^4*x*(35*e^3 + 56*e^2*f*x + 35*e*f^2*x^2 + 8*f^3*x^3) - 12*d^6*x^3*(35*e^3 + 84*e^2*f*x
+ 70*e*f^2*x^2 + 20*f^3*x^3))) + 105*d*(2*C*d^2*e^3 + 8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2
+ B*f^3)*ArcSin[d*x])/(1680*d^6)

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Maple [C]  time = 0.029, size = 959, normalized size = 2.3 \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)*(-d*x+1)^(1/2)*(d*x+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.99422, size = 644, normalized size = 1.55 \begin{align*} -\frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f^{3} x^{4}}{7 \, d^{2}} + \frac{1}{2} \, \sqrt{-d^{2} x^{2} + 1} A e^{3} x + \frac{A e^{3} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} B e^{3}}{3 \, d^{2}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} A e^{2} f}{d^{2}} - \frac{4 \,{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f^{3} x^{2}}{35 \, d^{4}} - \frac{{\left (3 \, C e f^{2} + B f^{3}\right )}{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{3}}{6 \, d^{2}} - \frac{{\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )}{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{2}}{5 \, d^{2}} - \frac{{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )}{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{4 \, 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}{8 \, d^{2}} - \frac{8 \,{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f^{3}}{105 \, d^{6}} - \frac{{\left (3 \, C e f^{2} + B f^{3}\right )}{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{8 \, d^{4}} + \frac{{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{2}} - \frac{2 \,{\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )}{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{15 \, d^{4}} + \frac{{\left (3 \, C e f^{2} + B f^{3}\right )} \sqrt{-d^{2} x^{2} + 1} x}{16 \, d^{4}} + \frac{{\left (3 \, C e f^{2} + B f^{3}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{16 \, \sqrt{d^{2}} d^{4}} \end{align*}

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]

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

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Fricas [A]  time = 1.10033, size = 888, normalized size = 2.14 \begin{align*} \frac{{\left (240 \, C d^{6} f^{3} x^{6} - 560 \, B d^{4} e^{3} - 672 \, B d^{2} e f^{2} + 280 \,{\left (3 \, C d^{6} e f^{2} + B d^{6} f^{3}\right )} x^{5} + 48 \,{\left (21 \, C d^{6} e^{2} f + 21 \, B d^{6} e f^{2} +{\left (7 \, A d^{6} - C d^{4}\right )} f^{3}\right )} x^{4} - 336 \,{\left (5 \, A d^{4} + 2 \, C d^{2}\right )} e^{2} f - 32 \,{\left (7 \, A d^{2} + 4 \, C\right )} f^{3} + 70 \,{\left (6 \, C d^{6} e^{3} + 18 \, B d^{6} e^{2} f - B d^{4} f^{3} + 3 \,{\left (6 \, A d^{6} - C d^{4}\right )} e f^{2}\right )} x^{3} + 16 \,{\left (35 \, B d^{6} e^{3} - 21 \, B d^{4} e f^{2} + 21 \,{\left (5 \, A d^{6} - C d^{4}\right )} e^{2} f -{\left (7 \, A d^{4} + 4 \, C d^{2}\right )} f^{3}\right )} x^{2} - 105 \,{\left (6 \, B d^{4} e^{2} f + B d^{2} f^{3} - 2 \,{\left (4 \, A d^{6} - C d^{4}\right )} e^{3} + 3 \,{\left (2 \, A d^{4} + C d^{2}\right )} e f^{2}\right )} x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 210 \,{\left (6 \, B d^{3} e^{2} f + B d f^{3} + 2 \,{\left (4 \, A d^{5} + C d^{3}\right )} e^{3} + 3 \,{\left (2 \, A d^{3} + C d\right )} e f^{2}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{1680 \, d^{6}} \end{align*}

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]

1/1680*((240*C*d^6*f^3*x^6 - 560*B*d^4*e^3 - 672*B*d^2*e*f^2 + 280*(3*C*d^6*e*f^2 + B*d^6*f^3)*x^5 + 48*(21*C*
d^6*e^2*f + 21*B*d^6*e*f^2 + (7*A*d^6 - C*d^4)*f^3)*x^4 - 336*(5*A*d^4 + 2*C*d^2)*e^2*f - 32*(7*A*d^2 + 4*C)*f
^3 + 70*(6*C*d^6*e^3 + 18*B*d^6*e^2*f - B*d^4*f^3 + 3*(6*A*d^6 - C*d^4)*e*f^2)*x^3 + 16*(35*B*d^6*e^3 - 21*B*d
^4*e*f^2 + 21*(5*A*d^6 - C*d^4)*e^2*f - (7*A*d^4 + 4*C*d^2)*f^3)*x^2 - 105*(6*B*d^4*e^2*f + B*d^2*f^3 - 2*(4*A
*d^6 - C*d^4)*e^3 + 3*(2*A*d^4 + C*d^2)*e*f^2)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 210*(6*B*d^3*e^2*f + B*d*f^3
+ 2*(4*A*d^5 + C*d^3)*e^3 + 3*(2*A*d^3 + C*d)*e*f^2)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/d^6

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

[Out]

Timed out

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Giac [B]  time = 3.14709, size = 1122, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

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]

1/1680*(112*((d*x + 1)*(3*(d*x + 1)*((d*x + 1)/d^3 - 4/d^3) + 17/d^3) - 10/d^3)*(d*x + 1)^(3/2)*sqrt(-d*x + 1)
*A*f^3 + 16*((3*((d*x + 1)*(5*(d*x + 1)*((d*x + 1)/d^5 - 6/d^5) + 74/d^5) - 96/d^5)*(d*x + 1) + 203/d^5)*(d*x
+ 1) - 70/d^5)*(d*x + 1)^(3/2)*sqrt(-d*x + 1)*C*f^3 + 336*((d*x + 1)*(3*(d*x + 1)*((d*x + 1)/d^3 - 4/d^3) + 17
/d^3) - 10/d^3)*(d*x + 1)^(3/2)*sqrt(-d*x + 1)*B*f^2*e + 336*((d*x + 1)*(3*(d*x + 1)*((d*x + 1)/d^3 - 4/d^3) +
17/d^3) - 10/d^3)*(d*x + 1)^(3/2)*sqrt(-d*x + 1)*C*f*e^2 + 35*(((2*((d*x + 1)*(4*(d*x + 1)*((d*x + 1)/d^4 - 5
/d^4) + 39/d^4) - 37/d^4)*(d*x + 1) + 31/d^4)*(d*x + 1) - 3/d^4)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 6*arcsin(1/2*s
qrt(2)*sqrt(d*x + 1))/d^4)*B*f^3 + 1680*(d*x + 1)^(3/2)*(d*x - 1)*sqrt(-d*x + 1)*A*f*e^2/d + 630*(((d*x + 1)*(
2*(d*x + 1)*((d*x + 1)/d^2 - 3/d^2) + 5/d^2) - 1/d^2)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*arcsin(1/2*sqrt(2)*sqrt
(d*x + 1))/d^2)*A*f^2*e + 105*(((2*((d*x + 1)*(4*(d*x + 1)*((d*x + 1)/d^4 - 5/d^4) + 39/d^4) - 37/d^4)*(d*x +
1) + 31/d^4)*(d*x + 1) - 3/d^4)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 6*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^4)*C*f^2*
e + 560*(d*x + 1)^(3/2)*(d*x - 1)*sqrt(-d*x + 1)*B*e^3/d + 630*(((d*x + 1)*(2*(d*x + 1)*((d*x + 1)/d^2 - 3/d^2
) + 5/d^2) - 1/d^2)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^2)*B*f*e^2 + 840*(sqr
t(d*x + 1)*sqrt(-d*x + 1)*d*x + 2*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*A*e^3 + 210*(((d*x + 1)*(2*(d*x + 1)*((d*
x + 1)/d^2 - 3/d^2) + 5/d^2) - 1/d^2)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^2)*
C*e^3)/d