3.1.1 \(\int \sqrt {1-d x} \sqrt {1+d x} (e+f x)^3 (A+B x+C x^2) \, dx\)
Optimal. Leaf size=415 \[ -\frac {\left (1-d^2 x^2\right )^{3/2} (e+f x)^2 \left (7 d^2 f (2 A f+B e)-C \left (3 d^2 e^2-8 f^2\right )\right )}{70 d^4 f}+\frac {x \sqrt {1-d^2 x^2} \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{16 d^4}+\frac {\left (1-d^2 x^2\right )^{3/2} \left (3 d^2 f x \left (-98 A d^2 e f^2-14 B d^2 e^2 f-35 B f^3+6 C d^2 e^3-41 C e f^2\right )+8 \left (C \left (3 d^4 e^4-30 d^2 e^2 f^2-8 f^4\right )-7 d^2 f \left (2 A f \left (6 d^2 e^2+f^2\right )+B \left (d^2 e^3+6 e f^2\right )\right )\right )\right )}{840 d^6 f}+\frac {\sin ^{-1}(d x) \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{16 d^5}+\frac {\left (1-d^2 x^2\right )^{3/2} (e+f x)^3 (3 C e-7 B f)}{42 d^2 f}-\frac {C \left (1-d^2 x^2\right )^{3/2} (e+f x)^4}{7 d^2 f} \]
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Rubi [A] time = 0.67, antiderivative size = 415, normalized size of antiderivative = 1.00,
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} \begin {gather*} -\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} \end {gather*}
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 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]
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 \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 (\left (4 C+7 A d^2\right ) f^2\right )+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*}
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Mathematica [A] time = 0.54, size = 355, normalized size = 0.86 \begin {gather*} \frac {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 )+\sqrt {1-d^2 x^2} \left (14 A d^2 \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 (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 (20 e^3+45 e^2 f x+36 e f^2 x^2+10 f^3 x^3\right )-2 d^4 \left (40 e^3+45 e^2 f x+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 (35 e^3+84 e^2 f x+70 e f^2 x^2+20 f^3 x^3\right )+6 d^4 x \left (35 e^3+56 e^2 f x+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 )}{1680 d^6} \end {gather*}
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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IntegrateAlgebraic [B] time = 1.06, size = 1590, normalized size = 3.83 \begin {gather*} \frac {\left (-8 A e^3 d^4-2 C e^3 d^2-6 A e f^2 d^2-6 B e^2 f d^2-B f^3-3 C e f^2\right ) \tan ^{-1}\left (\frac {\sqrt {1-d x}}{\sqrt {d x+1}}\right )}{8 d^5}-\frac {\sqrt {1-d x} \left (\frac {840 A d^5 e^3 (1-d x)^6}{(d x+1)^6}+\frac {210 C d^3 e^3 (1-d x)^6}{(d x+1)^6}+\frac {105 B d f^3 (1-d x)^6}{(d x+1)^6}+\frac {630 A d^3 e f^2 (1-d x)^6}{(d x+1)^6}+\frac {315 C d e f^2 (1-d x)^6}{(d x+1)^6}+\frac {630 B d^3 e^2 f (1-d x)^6}{(d x+1)^6}+\frac {3360 A d^5 e^3 (1-d x)^5}{(d x+1)^5}+\frac {2240 B d^4 e^3 (1-d x)^5}{(d x+1)^5}-\frac {840 C d^3 e^3 (1-d x)^5}{(d x+1)^5}+\frac {2240 A d^2 f^3 (1-d x)^5}{(d x+1)^5}+\frac {2240 C f^3 (1-d x)^5}{(d x+1)^5}-\frac {1540 B d f^3 (1-d x)^5}{(d x+1)^5}-\frac {2520 A d^3 e f^2 (1-d x)^5}{(d x+1)^5}+\frac {6720 B d^2 e f^2 (1-d x)^5}{(d x+1)^5}-\frac {4620 C d e f^2 (1-d x)^5}{(d x+1)^5}+\frac {6720 A d^4 e^2 f (1-d x)^5}{(d x+1)^5}-\frac {2520 B d^3 e^2 f (1-d x)^5}{(d x+1)^5}+\frac {6720 C d^2 e^2 f (1-d x)^5}{(d x+1)^5}+\frac {4200 A d^5 e^3 (1-d x)^4}{(d x+1)^4}+\frac {8960 B d^4 e^3 (1-d x)^4}{(d x+1)^4}-\frac {2310 C d^3 e^3 (1-d x)^4}{(d x+1)^4}+\frac {3584 A d^2 f^3 (1-d x)^4}{(d x+1)^4}-\frac {1792 C f^3 (1-d x)^4}{(d x+1)^4}+\frac {1085 B d f^3 (1-d x)^4}{(d x+1)^4}-\frac {6930 A d^3 e f^2 (1-d x)^4}{(d x+1)^4}+\frac {10752 B d^2 e f^2 (1-d x)^4}{(d x+1)^4}+\frac {3255 C d e f^2 (1-d x)^4}{(d x+1)^4}+\frac {26880 A d^4 e^2 f (1-d x)^4}{(d x+1)^4}-\frac {6930 B d^3 e^2 f (1-d x)^4}{(d x+1)^4}+\frac {10752 C d^2 e^2 f (1-d x)^4}{(d x+1)^4}+\frac {13440 B d^4 e^3 (1-d x)^3}{(d x+1)^3}+\frac {2688 A d^2 f^3 (1-d x)^3}{(d x+1)^3}+\frac {7296 C f^3 (1-d x)^3}{(d x+1)^3}+\frac {8064 B d^2 e f^2 (1-d x)^3}{(d x+1)^3}+\frac {40320 A d^4 e^2 f (1-d x)^3}{(d x+1)^3}+\frac {8064 C d^2 e^2 f (1-d x)^3}{(d x+1)^3}-\frac {4200 A d^5 e^3 (1-d x)^2}{(d x+1)^2}+\frac {8960 B d^4 e^3 (1-d x)^2}{(d x+1)^2}+\frac {2310 C d^3 e^3 (1-d x)^2}{(d x+1)^2}+\frac {3584 A d^2 f^3 (1-d x)^2}{(d x+1)^2}-\frac {1792 C f^3 (1-d x)^2}{(d x+1)^2}-\frac {1085 B d f^3 (1-d x)^2}{(d x+1)^2}+\frac {6930 A d^3 e f^2 (1-d x)^2}{(d x+1)^2}+\frac {10752 B d^2 e f^2 (1-d x)^2}{(d x+1)^2}-\frac {3255 C d e f^2 (1-d x)^2}{(d x+1)^2}+\frac {26880 A d^4 e^2 f (1-d x)^2}{(d x+1)^2}+\frac {6930 B d^3 e^2 f (1-d x)^2}{(d x+1)^2}+\frac {10752 C d^2 e^2 f (1-d x)^2}{(d x+1)^2}-\frac {3360 A d^5 e^3 (1-d x)}{d x+1}+\frac {2240 B d^4 e^3 (1-d x)}{d x+1}+\frac {840 C d^3 e^3 (1-d x)}{d x+1}+\frac {2240 A d^2 f^3 (1-d x)}{d x+1}+\frac {2240 C f^3 (1-d x)}{d x+1}+\frac {1540 B d f^3 (1-d x)}{d x+1}+\frac {2520 A d^3 e f^2 (1-d x)}{d x+1}+\frac {6720 B d^2 e f^2 (1-d x)}{d x+1}+\frac {4620 C d e f^2 (1-d x)}{d x+1}+\frac {6720 A d^4 e^2 f (1-d x)}{d x+1}+\frac {2520 B d^3 e^2 f (1-d x)}{d x+1}+\frac {6720 C d^2 e^2 f (1-d x)}{d x+1}-840 A d^5 e^3-210 C d^3 e^3-105 B d f^3-630 A d^3 e f^2-315 C d e f^2-630 B d^3 e^2 f\right )}{840 d^6 \sqrt {d x+1} \left (\frac {1-d x}{d x+1}+1\right )^7} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^3*(A + B*x + C*x^2),x]
[Out]
-1/840*(Sqrt[1 - d*x]*(-210*C*d^3*e^3 - 840*A*d^5*e^3 - 630*B*d^3*e^2*f - 315*C*d*e*f^2 - 630*A*d^3*e*f^2 - 10
5*B*d*f^3 + (210*C*d^3*e^3*(1 - d*x)^6)/(1 + d*x)^6 + (840*A*d^5*e^3*(1 - d*x)^6)/(1 + d*x)^6 + (630*B*d^3*e^2
*f*(1 - d*x)^6)/(1 + d*x)^6 + (315*C*d*e*f^2*(1 - d*x)^6)/(1 + d*x)^6 + (630*A*d^3*e*f^2*(1 - d*x)^6)/(1 + d*x
)^6 + (105*B*d*f^3*(1 - d*x)^6)/(1 + d*x)^6 - (840*C*d^3*e^3*(1 - d*x)^5)/(1 + d*x)^5 + (2240*B*d^4*e^3*(1 - d
*x)^5)/(1 + d*x)^5 + (3360*A*d^5*e^3*(1 - d*x)^5)/(1 + d*x)^5 + (6720*C*d^2*e^2*f*(1 - d*x)^5)/(1 + d*x)^5 - (
2520*B*d^3*e^2*f*(1 - d*x)^5)/(1 + d*x)^5 + (6720*A*d^4*e^2*f*(1 - d*x)^5)/(1 + d*x)^5 - (4620*C*d*e*f^2*(1 -
d*x)^5)/(1 + d*x)^5 + (6720*B*d^2*e*f^2*(1 - d*x)^5)/(1 + d*x)^5 - (2520*A*d^3*e*f^2*(1 - d*x)^5)/(1 + d*x)^5
+ (2240*C*f^3*(1 - d*x)^5)/(1 + d*x)^5 - (1540*B*d*f^3*(1 - d*x)^5)/(1 + d*x)^5 + (2240*A*d^2*f^3*(1 - d*x)^5)
/(1 + d*x)^5 - (2310*C*d^3*e^3*(1 - d*x)^4)/(1 + d*x)^4 + (8960*B*d^4*e^3*(1 - d*x)^4)/(1 + d*x)^4 + (4200*A*d
^5*e^3*(1 - d*x)^4)/(1 + d*x)^4 + (10752*C*d^2*e^2*f*(1 - d*x)^4)/(1 + d*x)^4 - (6930*B*d^3*e^2*f*(1 - d*x)^4)
/(1 + d*x)^4 + (26880*A*d^4*e^2*f*(1 - d*x)^4)/(1 + d*x)^4 + (3255*C*d*e*f^2*(1 - d*x)^4)/(1 + d*x)^4 + (10752
*B*d^2*e*f^2*(1 - d*x)^4)/(1 + d*x)^4 - (6930*A*d^3*e*f^2*(1 - d*x)^4)/(1 + d*x)^4 - (1792*C*f^3*(1 - d*x)^4)/
(1 + d*x)^4 + (1085*B*d*f^3*(1 - d*x)^4)/(1 + d*x)^4 + (3584*A*d^2*f^3*(1 - d*x)^4)/(1 + d*x)^4 + (13440*B*d^4
*e^3*(1 - d*x)^3)/(1 + d*x)^3 + (8064*C*d^2*e^2*f*(1 - d*x)^3)/(1 + d*x)^3 + (40320*A*d^4*e^2*f*(1 - d*x)^3)/(
1 + d*x)^3 + (8064*B*d^2*e*f^2*(1 - d*x)^3)/(1 + d*x)^3 + (7296*C*f^3*(1 - d*x)^3)/(1 + d*x)^3 + (2688*A*d^2*f
^3*(1 - d*x)^3)/(1 + d*x)^3 + (2310*C*d^3*e^3*(1 - d*x)^2)/(1 + d*x)^2 + (8960*B*d^4*e^3*(1 - d*x)^2)/(1 + d*x
)^2 - (4200*A*d^5*e^3*(1 - d*x)^2)/(1 + d*x)^2 + (10752*C*d^2*e^2*f*(1 - d*x)^2)/(1 + d*x)^2 + (6930*B*d^3*e^2
*f*(1 - d*x)^2)/(1 + d*x)^2 + (26880*A*d^4*e^2*f*(1 - d*x)^2)/(1 + d*x)^2 - (3255*C*d*e*f^2*(1 - d*x)^2)/(1 +
d*x)^2 + (10752*B*d^2*e*f^2*(1 - d*x)^2)/(1 + d*x)^2 + (6930*A*d^3*e*f^2*(1 - d*x)^2)/(1 + d*x)^2 - (1792*C*f^
3*(1 - d*x)^2)/(1 + d*x)^2 - (1085*B*d*f^3*(1 - d*x)^2)/(1 + d*x)^2 + (3584*A*d^2*f^3*(1 - d*x)^2)/(1 + d*x)^2
+ (840*C*d^3*e^3*(1 - d*x))/(1 + d*x) + (2240*B*d^4*e^3*(1 - d*x))/(1 + d*x) - (3360*A*d^5*e^3*(1 - d*x))/(1
+ d*x) + (6720*C*d^2*e^2*f*(1 - d*x))/(1 + d*x) + (2520*B*d^3*e^2*f*(1 - d*x))/(1 + d*x) + (6720*A*d^4*e^2*f*(
1 - d*x))/(1 + d*x) + (4620*C*d*e*f^2*(1 - d*x))/(1 + d*x) + (6720*B*d^2*e*f^2*(1 - d*x))/(1 + d*x) + (2520*A*
d^3*e*f^2*(1 - d*x))/(1 + d*x) + (2240*C*f^3*(1 - d*x))/(1 + d*x) + (1540*B*d*f^3*(1 - d*x))/(1 + d*x) + (2240
*A*d^2*f^3*(1 - d*x))/(1 + d*x)))/(d^6*Sqrt[1 + d*x]*(1 + (1 - d*x)/(1 + d*x))^7) + ((-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)*ArcTan[Sqrt[1 - d*x]/Sqrt[1 + d*x]])/(8*d^5)
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fricas [A] time = 0.72, size = 406, normalized size = 0.98 \begin {gather*} \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 {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]
1/1680*((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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giac [B] time = 3.11, size = 1948, normalized size = 4.69
result too large to display
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*(14*(((2*(d*x + 1)*(3*(d*x + 1)*(4*(d*x + 1)/d^4 - 21/d^4) + 133/d^4) - 295/d^4)*(d*x + 1) + 195/d^4)*s
qrt(d*x + 1)*sqrt(-d*x + 1) + 90*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^4)*A*d*f^3 + 7*(((2*((d*x + 1)*(4*(d*x +
1)*(5*(d*x + 1)/d^5 - 31/d^5) + 321/d^5) - 451/d^5)*(d*x + 1) + 745/d^5)*(d*x + 1) - 405/d^5)*sqrt(d*x + 1)*sq
rt(-d*x + 1) - 150*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^5)*B*d*f^3 + (((2*((4*(d*x + 1)*(5*(d*x + 1)*(6*(d*x +
1)/d^6 - 43/d^6) + 661/d^6) - 4551/d^6)*(d*x + 1) + 4781/d^6)*(d*x + 1) - 6335/d^6)*(d*x + 1) + 2835/d^6)*sqrt
(d*x + 1)*sqrt(-d*x + 1) + 1050*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^6)*C*d*f^3 + 210*(((d*x + 1)*(2*(d*x + 1)*
(3*(d*x + 1)/d^3 - 13/d^3) + 43/d^3) - 39/d^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 18*arcsin(1/2*sqrt(2)*sqrt(d*x +
1))/d^3)*A*d*f^2*e + 42*(((2*(d*x + 1)*(3*(d*x + 1)*(4*(d*x + 1)/d^4 - 21/d^4) + 133/d^4) - 295/d^4)*(d*x + 1
) + 195/d^4)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 90*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^4)*B*d*f^2*e + 21*(((2*((d*
x + 1)*(4*(d*x + 1)*(5*(d*x + 1)/d^5 - 31/d^5) + 321/d^5) - 451/d^5)*(d*x + 1) + 745/d^5)*(d*x + 1) - 405/d^5)
*sqrt(d*x + 1)*sqrt(-d*x + 1) - 150*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^5)*C*d*f^2*e + 70*(((d*x + 1)*(2*(d*x
+ 1)*(3*(d*x + 1)/d^3 - 13/d^3) + 43/d^3) - 39/d^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 18*arcsin(1/2*sqrt(2)*sqrt(
d*x + 1))/d^3)*A*f^3 + 14*(((2*(d*x + 1)*(3*(d*x + 1)*(4*(d*x + 1)/d^4 - 21/d^4) + 133/d^4) - 295/d^4)*(d*x +
1) + 195/d^4)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 90*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^4)*B*f^3 + 7*(((2*((d*x +
1)*(4*(d*x + 1)*(5*(d*x + 1)/d^5 - 31/d^5) + 321/d^5) - 451/d^5)*(d*x + 1) + 745/d^5)*(d*x + 1) - 405/d^5)*sqr
t(d*x + 1)*sqrt(-d*x + 1) - 150*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^5)*C*f^3 + 840*(sqrt(d*x + 1)*sqrt(-d*x +
1)*((d*x + 1)*(2*(d*x + 1)/d^2 - 7/d^2) + 9/d^2) + 6*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^2)*A*d*f*e^2 + 210*((
(d*x + 1)*(2*(d*x + 1)*(3*(d*x + 1)/d^3 - 13/d^3) + 43/d^3) - 39/d^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 18*arcsin
(1/2*sqrt(2)*sqrt(d*x + 1))/d^3)*B*d*f*e^2 + 42*(((2*(d*x + 1)*(3*(d*x + 1)*(4*(d*x + 1)/d^4 - 21/d^4) + 133/d
^4) - 295/d^4)*(d*x + 1) + 195/d^4)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 90*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^4)*C
*d*f*e^2 + 840*(sqrt(d*x + 1)*sqrt(-d*x + 1)*((d*x + 1)*(2*(d*x + 1)/d^2 - 7/d^2) + 9/d^2) + 6*arcsin(1/2*sqrt
(2)*sqrt(d*x + 1))/d^2)*A*f^2*e + 210*(((d*x + 1)*(2*(d*x + 1)*(3*(d*x + 1)/d^3 - 13/d^3) + 43/d^3) - 39/d^3)*
sqrt(d*x + 1)*sqrt(-d*x + 1) - 18*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^3)*B*f^2*e + 42*(((2*(d*x + 1)*(3*(d*x +
1)*(4*(d*x + 1)/d^4 - 21/d^4) + 133/d^4) - 295/d^4)*(d*x + 1) + 195/d^4)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 90*ar
csin(1/2*sqrt(2)*sqrt(d*x + 1))/d^4)*C*f^2*e + 280*(sqrt(d*x + 1)*sqrt(-d*x + 1)*((d*x + 1)*(2*(d*x + 1)/d^2 -
7/d^2) + 9/d^2) + 6*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^2)*B*d*e^3 + 70*(((d*x + 1)*(2*(d*x + 1)*(3*(d*x + 1)
/d^3 - 13/d^3) + 43/d^3) - 39/d^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 18*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^3)*C*
d*e^3 + 840*(sqrt(d*x + 1)*sqrt(-d*x + 1)*((d*x + 1)*(2*(d*x + 1)/d^2 - 7/d^2) + 9/d^2) + 6*arcsin(1/2*sqrt(2)
*sqrt(d*x + 1))/d^2)*B*f*e^2 + 210*(((d*x + 1)*(2*(d*x + 1)*(3*(d*x + 1)/d^3 - 13/d^3) + 43/d^3) - 39/d^3)*sqr
t(d*x + 1)*sqrt(-d*x + 1) - 18*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^3)*C*f*e^2 + 840*(sqrt(d*x + 1)*(d*x - 2)*s
qrt(-d*x + 1) - 2*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*A*e^3 + 1680*(sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*arcsin(1/2
*sqrt(2)*sqrt(d*x + 1)))*A*e^3 + 280*(sqrt(d*x + 1)*sqrt(-d*x + 1)*((d*x + 1)*(2*(d*x + 1)/d^2 - 7/d^2) + 9/d^
2) + 6*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^2)*C*e^3 + 2520*(sqrt(d*x + 1)*(d*x - 2)*sqrt(-d*x + 1) - 2*arcsin(
1/2*sqrt(2)*sqrt(d*x + 1)))*A*f*e^2/d + 840*(sqrt(d*x + 1)*(d*x - 2)*sqrt(-d*x + 1) - 2*arcsin(1/2*sqrt(2)*sqr
t(d*x + 1)))*B*e^3/d)/d
________________________________________________________________________________________
maple [C] time = 0.04, size = 959, normalized size = 2.31
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x+e)^3*(C*x^2+B*x+A)*(-d*x+1)^(1/2)*(d*x+1)^(1/2),x)
[Out]
1/1680*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(-128*C*csgn(d)*(-d^2*x^2+1)^(1/2)*f^3+840*A*arctan(1/(-d^2*x^2+1)^(1/2)*d
*x*csgn(d))*d^5*e^3+210*C*arctan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))*d^3*e^3+105*B*arctan(1/(-d^2*x^2+1)^(1/2)*d
*x*csgn(d))*d*f^3-560*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+630*A*arct
an(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))*d^3*e*f^2+630*B*arctan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))*d^3*e^2*f+315*C*
arctan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))*d*e*f^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+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^3-672*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+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)-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)+1008*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)+12
60*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 = 1.00, size = 444, normalized size = 1.07 \begin {gather*} -\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 (d x\right )}{2 \, d} - \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 (d x\right )}{8 \, d^{3}} - \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 (d x\right )}{16 \, 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]
-1/7*(-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*x)/d - 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)*sq
rt(-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*x)/d^3 - 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*x)/d^5
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mupad [B] time = 47.79, size = 3993, normalized size = 9.62
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e + f*x)^3*(1 - d*x)^(1/2)*(d*x + 1)^(1/2)*(A + B*x + C*x^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)^22)/((d*x + 1)^(1/2) - 1)^22 - (((20480*C*f^3)/3 - 448*C*d^2*e^2*f)*((1
- d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8 - (((20480*C*f^3)/3 - 448*C*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^20)/
((d*x + 1)^(1/2) - 1)^20 + (((458752*C*f^3)/15 + (27136*C*d^2*e^2*f)/5)*((1 - d*x)^(1/2) - 1)^10)/((d*x + 1)^(
1/2) - 1)^10 + (((458752*C*f^3)/15 + (27136*C*d^2*e^2*f)/5)*((1 - d*x)^(1/2) - 1)^18)/((d*x + 1)^(1/2) - 1)^18
- (((1011712*C*f^3)/15 - (13184*C*d^2*e^2*f)/5)*((1 - d*x)^(1/2) - 1)^12)/((d*x + 1)^(1/2) - 1)^12 - (((10117
12*C*f^3)/15 - (13184*C*d^2*e^2*f)/5)*((1 - d*x)^(1/2) - 1)^16)/((d*x + 1)^(1/2) - 1)^16 + (((9293824*C*f^3)/1
05 - (15104*C*d^2*e^2*f)/5)*((1 - d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) - 1)^14 + (((1 - d*x)^(1/2) - 1)^3*((29
*C*d^3*e^3)/2 - (41*C*d*e*f^2)/4))/((d*x + 1)^(1/2) - 1)^3 - (((1 - d*x)^(1/2) - 1)^25*((29*C*d^3*e^3)/2 - (41
*C*d*e*f^2)/4))/((d*x + 1)^(1/2) - 1)^25 - (((1 - d*x)^(1/2) - 1)^5*(39*C*d^3*e^3 - (1099*C*d*e*f^2)/2))/((d*x
+ 1)^(1/2) - 1)^5 + (((1 - d*x)^(1/2) - 1)^23*(39*C*d^3*e^3 - (1099*C*d*e*f^2)/2))/((d*x + 1)^(1/2) - 1)^23 -
(((1 - d*x)^(1/2) - 1)^7*(209*C*d^3*e^3 + (8755*C*d*e*f^2)/2))/((d*x + 1)^(1/2) - 1)^7 + (((1 - d*x)^(1/2) -
1)^21*(209*C*d^3*e^3 + (8755*C*d*e*f^2)/2))/((d*x + 1)^(1/2) - 1)^21 + (((1 - d*x)^(1/2) - 1)^11*((1767*C*d^3*
e^3)/2 - (8267*C*d*e*f^2)/4))/((d*x + 1)^(1/2) - 1)^11 - (((1 - d*x)^(1/2) - 1)^17*((1767*C*d^3*e^3)/2 - (8267
*C*d*e*f^2)/4))/((d*x + 1)^(1/2) - 1)^17 + (((1 - d*x)^(1/2) - 1)^13*(646*C*d^3*e^3 - 17527*C*d*e*f^2))/((d*x
+ 1)^(1/2) - 1)^13 - (((1 - d*x)^(1/2) - 1)^15*(646*C*d^3*e^3 - 17527*C*d*e*f^2))/((d*x + 1)^(1/2) - 1)^15 + (
((1 - d*x)^(1/2) - 1)^9*((165*C*d^3*e^3)/2 + (42095*C*d*e*f^2)/4))/((d*x + 1)^(1/2) - 1)^9 - (((1 - d*x)^(1/2)
- 1)^19*((165*C*d^3*e^3)/2 + (42095*C*d*e*f^2)/4))/((d*x + 1)^(1/2) - 1)^19 - (d*(2*C*d^2*e^3 + 3*C*e*f^2)*((
1 - d*x)^(1/2) - 1))/(4*((d*x + 1)^(1/2) - 1)) + (d*(2*C*d^2*e^3 + 3*C*e*f^2)*((1 - d*x)^(1/2) - 1)^27)/(4*((d
*x + 1)^(1/2) - 1)^27) + (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)^24)/((d*x + 1)^(1/2) - 1)^24)/(d^6 + (14*d^6*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) -
1)^2 + (91*d^6*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (364*d^6*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1
)^(1/2) - 1)^6 + (1001*d^6*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8 + (2002*d^6*((1 - d*x)^(1/2) - 1)^
10)/((d*x + 1)^(1/2) - 1)^10 + (3003*d^6*((1 - d*x)^(1/2) - 1)^12)/((d*x + 1)^(1/2) - 1)^12 + (3432*d^6*((1 -
d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) - 1)^14 + (3003*d^6*((1 - d*x)^(1/2) - 1)^16)/((d*x + 1)^(1/2) - 1)^16 +
(2002*d^6*((1 - d*x)^(1/2) - 1)^18)/((d*x + 1)^(1/2) - 1)^18 + (1001*d^6*((1 - d*x)^(1/2) - 1)^20)/((d*x + 1)^
(1/2) - 1)^20 + (364*d^6*((1 - d*x)^(1/2) - 1)^22)/((d*x + 1)^(1/2) - 1)^22 + (91*d^6*((1 - d*x)^(1/2) - 1)^24
)/((d*x + 1)^(1/2) - 1)^24 + (14*d^6*((1 - d*x)^(1/2) - 1)^26)/((d*x + 1)^(1/2) - 1)^26 + (d^6*((1 - d*x)^(1/2
) - 1)^28)/((d*x + 1)^(1/2) - 1)^28) - ((((4928*A*f^3)/3 + 512*A*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1
)^(1/2) - 1)^8 - (((1408*A*f^3)/3 - 32*A*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) - 1)^14 - (((14
08*A*f^3)/3 - 32*A*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6 + (((4928*A*f^3)/3 + 512*A*d^2*
e^2*f)*((1 - d*x)^(1/2) - 1)^12)/((d*x + 1)^(1/2) - 1)^12 - (((11008*A*f^3)/5 - 912*A*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)*(2*A*d^3*e^3 - (3*A*d*e*f^2)/2))/((d*x + 1)^(1/
2) - 1) - (((1 - d*x)^(1/2) - 1)^19*(2*A*d^3*e^3 - (3*A*d*e*f^2)/2))/((d*x + 1)^(1/2) - 1)^19 - (((1 - d*x)^(1
/2) - 1)^3*(2*A*d^3*e^3 - (99*A*d*e*f^2)/2))/((d*x + 1)^(1/2) - 1)^3 + (((1 - d*x)^(1/2) - 1)^17*(2*A*d^3*e^3
- (99*A*d*e*f^2)/2))/((d*x + 1)^(1/2) - 1)^17 - (((1 - d*x)^(1/2) - 1)^5*(40*A*d^3*e^3 + 306*A*d*e*f^2))/((d*x
+ 1)^(1/2) - 1)^5 + (((1 - d*x)^(1/2) - 1)^15*(40*A*d^3*e^3 + 306*A*d*e*f^2))/((d*x + 1)^(1/2) - 1)^15 - (((1
- d*x)^(1/2) - 1)^7*(88*A*d^3*e^3 - 306*A*d*e*f^2))/((d*x + 1)^(1/2) - 1)^7 + (((1 - d*x)^(1/2) - 1)^13*(88*A
*d^3*e^3 - 306*A*d*e*f^2))/((d*x + 1)^(1/2) - 1)^13 - (((1 - d*x)^(1/2) - 1)^9*(52*A*d^3*e^3 - 663*A*d*e*f^2))
/((d*x + 1)^(1/2) - 1)^9 + (((1 - d*x)^(1/2) - 1)^11*(52*A*d^3*e^3 - 663*A*d*e*f^2))/((d*x + 1)^(1/2) - 1)^11
+ (64*A*f^3*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (64*A*f^3*((1 - d*x)^(1/2) - 1)^16)/((d*x + 1)^
(1/2) - 1)^16 + (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)^18)/((d*x + 1)^(1/2) - 1)^18)/(d^4 + (10*d^4*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (45
*d^4*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (120*d^4*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1
)^6 + (210*d^4*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8 + (252*d^4*((1 - d*x)^(1/2) - 1)^10)/((d*x + 1
)^(1/2) - 1)^10 + (210*d^4*((1 - d*x)^(1/2) - 1)^12)/((d*x + 1)^(1/2) - 1)^12 + (120*d^4*((1 - d*x)^(1/2) - 1)
^14)/((d*x + 1)^(1/2) - 1)^14 + (45*d^4*((1 - d*x)^(1/2) - 1)^16)/((d*x + 1)^(1/2) - 1)^16 + (10*d^4*((1 - d*x
)^(1/2) - 1)^18)/((d*x + 1)^(1/2) - 1)^18 + (d^4*((1 - d*x)^(1/2) - 1)^20)/((d*x + 1)^(1/2) - 1)^20) - ((((B*f
^3)/4 + (3*B*d^2*e^2*f)/2)*((1 - d*x)^(1/2) - 1)^23)/((d*x + 1)^(1/2) - 1)^23 - (((35*B*f^3)/12 - (93*B*d^2*e^
2*f)/2)*((1 - d*x)^(1/2) - 1)^3)/((d*x + 1)^(1/2) - 1)^3 + (((35*B*f^3)/12 - (93*B*d^2*e^2*f)/2)*((1 - d*x)^(1
/2) - 1)^21)/((d*x + 1)^(1/2) - 1)^21 + (((757*B*f^3)/4 - (417*B*d^2*e^2*f)/2)*((1 - d*x)^(1/2) - 1)^5)/((d*x
+ 1)^(1/2) - 1)^5 - (((757*B*f^3)/4 - (417*B*d^2*e^2*f)/2)*((1 - d*x)^(1/2) - 1)^19)/((d*x + 1)^(1/2) - 1)^19
- (((7339*B*f^3)/4 + (513*B*d^2*e^2*f)/2)*((1 - d*x)^(1/2) - 1)^7)/((d*x + 1)^(1/2) - 1)^7 + (((7339*B*f^3)/4
+ (513*B*d^2*e^2*f)/2)*((1 - d*x)^(1/2) - 1)^17)/((d*x + 1)^(1/2) - 1)^17 - (((25661*B*f^3)/2 - 969*B*d^2*e^2*
f)*((1 - d*x)^(1/2) - 1)^11)/((d*x + 1)^(1/2) - 1)^11 + (((25661*B*f^3)/2 - 969*B*d^2*e^2*f)*((1 - d*x)^(1/2)
- 1)^13)/((d*x + 1)^(1/2) - 1)^13 + (((41929*B*f^3)/6 + 969*B*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^9)/((d*x + 1)^(
1/2) - 1)^9 - (((41929*B*f^3)/6 + 969*B*d^2*e^2*f)*((1 - d*x)^(1/2) - 1)^15)/((d*x + 1)^(1/2) - 1)^15 + (((1 -
d*x)^(1/2) - 1)^4*(16*B*d^3*e^3 + 192*B*d*e*f^2))/((d*x + 1)^(1/2) - 1)^4 + (((1 - d*x)^(1/2) - 1)^20*(16*B*d
^3*e^3 + 192*B*d*e*f^2))/((d*x + 1)^(1/2) - 1)^20 + (((1 - d*x)^(1/2) - 1)^6*((56*B*d^3*e^3)/3 - 1024*B*d*e*f^
2))/((d*x + 1)^(1/2) - 1)^6 + (((1 - d*x)^(1/2) - 1)^18*((56*B*d^3*e^3)/3 - 1024*B*d*e*f^2))/((d*x + 1)^(1/2)
- 1)^18 + (((1 - d*x)^(1/2) - 1)^8*(192*B*d^3*e^3 + 2304*B*d*e*f^2))/((d*x + 1)^(1/2) - 1)^8 + (((1 - d*x)^(1/
2) - 1)^16*(192*B*d^3*e^3 + 2304*B*d*e*f^2))/((d*x + 1)^(1/2) - 1)^16 + (((1 - d*x)^(1/2) - 1)^10*(656*B*d^3*e
^3 + (9216*B*d*e*f^2)/5))/((d*x + 1)^(1/2) - 1)^10 + (((1 - d*x)^(1/2) - 1)^14*(656*B*d^3*e^3 + (9216*B*d*e*f^
2)/5))/((d*x + 1)^(1/2) - 1)^14 + (((1 - d*x)^(1/2) - 1)^12*((2848*B*d^3*e^3)/3 - (16768*B*d*e*f^2)/5))/((d*x
+ 1)^(1/2) - 1)^12 - (((B*f^3)/4 + (3*B*d^2*e^2*f)/2)*((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)^22)/((d*x + 1)^(1/2)
- 1)^22)/(d^5 + (12*d^5*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (66*d^5*((1 - d*x)^(1/2) - 1)^4)/(
(d*x + 1)^(1/2) - 1)^4 + (220*d^5*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6 + (495*d^5*((1 - d*x)^(1/2)
- 1)^8)/((d*x + 1)^(1/2) - 1)^8 + (792*d^5*((1 - d*x)^(1/2) - 1)^10)/((d*x + 1)^(1/2) - 1)^10 + (924*d^5*((1
- d*x)^(1/2) - 1)^12)/((d*x + 1)^(1/2) - 1)^12 + (792*d^5*((1 - d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) - 1)^14 +
(495*d^5*((1 - d*x)^(1/2) - 1)^16)/((d*x + 1)^(1/2) - 1)^16 + (220*d^5*((1 - d*x)^(1/2) - 1)^18)/((d*x + 1)^(
1/2) - 1)^18 + (66*d^5*((1 - d*x)^(1/2) - 1)^20)/((d*x + 1)^(1/2) - 1)^20 + (12*d^5*((1 - d*x)^(1/2) - 1)^22)/
((d*x + 1)^(1/2) - 1)^22 + (d^5*((1 - d*x)^(1/2) - 1)^24)/((d*x + 1)^(1/2) - 1)^24) - (B*f*atan((B*f*(f^2 + 6*
d^2*e^2)*((1 - d*x)^(1/2) - 1))/((B*f^3 + 6*B*d^2*e^2*f)*((d*x + 1)^(1/2) - 1)))*(f^2 + 6*d^2*e^2))/(4*d^5) -
(A*e*atan((A*e*((1 - d*x)^(1/2) - 1)*(3*f^2 + 4*d^2*e^2))/((4*A*d^2*e^3 + 3*A*e*f^2)*((d*x + 1)^(1/2) - 1)))*(
3*f^2 + 4*d^2*e^2))/(2*d^3) - (C*e*atan((C*e*((1 - d*x)^(1/2) - 1)*(3*f^2 + 2*d^2*e^2))/((2*C*d^2*e^3 + 3*C*e*
f^2)*((d*x + 1)^(1/2) - 1)))*(3*f^2 + 2*d^2*e^2))/(4*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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