3.792 \(\int \frac{(a+b x+c x^2)^3}{\sqrt{1-d x} \sqrt{1+d x}} \, dx\)

Optimal. Leaf size=324 \[ -\frac{x \sqrt{1-d^2 x^2} \left (24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^6}-\frac{b \sqrt{1-d^2 x^2} \left (45 a^2 d^4+60 a c d^2+10 b^2 d^2+24 c^2\right )}{15 d^6}+\frac{\sin ^{-1}(d x) \left (24 a^2 c d^4+16 a^3 d^6+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^7}-\frac{c x^3 \sqrt{1-d^2 x^2} \left (18 a c d^2+18 b^2 d^2+5 c^2\right )}{24 d^4}-\frac{b x^2 \sqrt{1-d^2 x^2} \left (30 a c d^2+5 b^2 d^2+12 c^2\right )}{15 d^4}-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2} \]

[Out]

-(b*(24*c^2 + 10*b^2*d^2 + 60*a*c*d^2 + 45*a^2*d^4)*Sqrt[1 - d^2*x^2])/(15*d^6) - ((5*c^3 + 18*b^2*c*d^2 + 18*
a*c^2*d^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4)*x*Sqrt[1 - d^2*x^2])/(16*d^6) - (b*(12*c^2 + 5*b^2*d^2 + 30*a*c*d^2)*
x^2*Sqrt[1 - d^2*x^2])/(15*d^4) - (c*(5*c^2 + 18*b^2*d^2 + 18*a*c*d^2)*x^3*Sqrt[1 - d^2*x^2])/(24*d^4) - (3*b*
c^2*x^4*Sqrt[1 - d^2*x^2])/(5*d^2) - (c^3*x^5*Sqrt[1 - d^2*x^2])/(6*d^2) + ((5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d
^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4 + 16*a^3*d^6)*ArcSin[d*x])/(16*d^7)

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Rubi [A]  time = 0.933511, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {899, 1815, 641, 216} \[ -\frac{x \sqrt{1-d^2 x^2} \left (24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^6}-\frac{b \sqrt{1-d^2 x^2} \left (45 a^2 d^4+60 a c d^2+10 b^2 d^2+24 c^2\right )}{15 d^6}+\frac{\sin ^{-1}(d x) \left (24 a^2 c d^4+16 a^3 d^6+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )}{16 d^7}-\frac{c x^3 \sqrt{1-d^2 x^2} \left (18 a c d^2+18 b^2 d^2+5 c^2\right )}{24 d^4}-\frac{b x^2 \sqrt{1-d^2 x^2} \left (30 a c d^2+5 b^2 d^2+12 c^2\right )}{15 d^4}-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]

[Out]

-(b*(24*c^2 + 10*b^2*d^2 + 60*a*c*d^2 + 45*a^2*d^4)*Sqrt[1 - d^2*x^2])/(15*d^6) - ((5*c^3 + 18*b^2*c*d^2 + 18*
a*c^2*d^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4)*x*Sqrt[1 - d^2*x^2])/(16*d^6) - (b*(12*c^2 + 5*b^2*d^2 + 30*a*c*d^2)*
x^2*Sqrt[1 - d^2*x^2])/(15*d^4) - (c*(5*c^2 + 18*b^2*d^2 + 18*a*c*d^2)*x^3*Sqrt[1 - d^2*x^2])/(24*d^4) - (3*b*
c^2*x^4*Sqrt[1 - d^2*x^2])/(5*d^2) - (c^3*x^5*Sqrt[1 - d^2*x^2])/(6*d^2) + ((5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d
^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4 + 16*a^3*d^6)*ArcSin[d*x])/(16*d^7)

Rule 899

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

Rule 1815

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[(e*x^(q - 1)*(a + b*x^2)^(p + 1))/(b*(q + 2*p + 1)), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x]
&& PolyQ[Pq, x] &&  !LeQ[p, -1]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

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 \frac{\left (a+b x+c x^2\right )^3}{\sqrt{1-d x} \sqrt{1+d x}} \, dx &=\int \frac{\left (a+b x+c x^2\right )^3}{\sqrt{1-d^2 x^2}} \, dx\\ &=-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2}-\frac{\int \frac{-6 a^3 d^2-18 a^2 b d^2 x-18 a \left (b^2+a c\right ) d^2 x^2-6 b \left (b^2+6 a c\right ) d^2 x^3-c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^4-18 b c^2 d^2 x^5}{\sqrt{1-d^2 x^2}} \, dx}{6 d^2}\\ &=-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2}+\frac{\int \frac{30 a^3 d^4+90 a^2 b d^4 x+90 a \left (b^2+a c\right ) d^4 x^2+6 b d^2 \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^3+5 c d^2 \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^4}{\sqrt{1-d^2 x^2}} \, dx}{30 d^4}\\ &=-\frac{c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^3 \sqrt{1-d^2 x^2}}{24 d^4}-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2}-\frac{\int \frac{-120 a^3 d^6-360 a^2 b d^6 x-15 d^2 \left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4\right ) x^2-24 b d^4 \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^3}{\sqrt{1-d^2 x^2}} \, dx}{120 d^6}\\ &=-\frac{b \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^2 \sqrt{1-d^2 x^2}}{15 d^4}-\frac{c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^3 \sqrt{1-d^2 x^2}}{24 d^4}-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2}+\frac{\int \frac{360 a^3 d^8+24 b d^4 \left (24 c^2+10 b^2 d^2+60 a c d^2+45 a^2 d^4\right ) x+45 d^4 \left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4\right ) x^2}{\sqrt{1-d^2 x^2}} \, dx}{360 d^8}\\ &=-\frac{\left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4\right ) x \sqrt{1-d^2 x^2}}{16 d^6}-\frac{b \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^2 \sqrt{1-d^2 x^2}}{15 d^4}-\frac{c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^3 \sqrt{1-d^2 x^2}}{24 d^4}-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2}-\frac{\int \frac{-45 d^4 \left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4+16 a^3 d^6\right )-48 b d^6 \left (24 c^2+10 b^2 d^2+60 a c d^2+45 a^2 d^4\right ) x}{\sqrt{1-d^2 x^2}} \, dx}{720 d^{10}}\\ &=-\frac{b \left (24 c^2+10 b^2 d^2+60 a c d^2+45 a^2 d^4\right ) \sqrt{1-d^2 x^2}}{15 d^6}-\frac{\left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4\right ) x \sqrt{1-d^2 x^2}}{16 d^6}-\frac{b \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^2 \sqrt{1-d^2 x^2}}{15 d^4}-\frac{c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^3 \sqrt{1-d^2 x^2}}{24 d^4}-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2}+\frac{\left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4+16 a^3 d^6\right ) \int \frac{1}{\sqrt{1-d^2 x^2}} \, dx}{16 d^6}\\ &=-\frac{b \left (24 c^2+10 b^2 d^2+60 a c d^2+45 a^2 d^4\right ) \sqrt{1-d^2 x^2}}{15 d^6}-\frac{\left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4\right ) x \sqrt{1-d^2 x^2}}{16 d^6}-\frac{b \left (12 c^2+5 b^2 d^2+30 a c d^2\right ) x^2 \sqrt{1-d^2 x^2}}{15 d^4}-\frac{c \left (5 c^2+18 b^2 d^2+18 a c d^2\right ) x^3 \sqrt{1-d^2 x^2}}{24 d^4}-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2}+\frac{\left (5 c^3+18 b^2 c d^2+18 a c^2 d^2+24 a b^2 d^4+24 a^2 c d^4+16 a^3 d^6\right ) \sin ^{-1}(d x)}{16 d^7}\\ \end{align*}

Mathematica [A]  time = 0.262266, size = 229, normalized size = 0.71 \[ \frac{15 \sin ^{-1}(d x) \left (24 a^2 c d^4+16 a^3 d^6+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )-d \sqrt{1-d^2 x^2} \left (48 b \left (15 a^2 d^4+10 a c d^2 \left (d^2 x^2+2\right )+c^2 \left (3 d^4 x^4+4 d^2 x^2+8\right )\right )+5 c x \left (72 a^2 d^4+18 a c d^2 \left (2 d^2 x^2+3\right )+c^2 \left (8 d^4 x^4+10 d^2 x^2+15\right )\right )+90 b^2 d^2 x \left (4 a d^2+c \left (2 d^2 x^2+3\right )\right )+80 b^3 d^2 \left (d^2 x^2+2\right )\right )}{240 d^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]

[Out]

(-(d*Sqrt[1 - d^2*x^2]*(80*b^3*d^2*(2 + d^2*x^2) + 90*b^2*d^2*x*(4*a*d^2 + c*(3 + 2*d^2*x^2)) + 48*b*(15*a^2*d
^4 + 10*a*c*d^2*(2 + d^2*x^2) + c^2*(8 + 4*d^2*x^2 + 3*d^4*x^4)) + 5*c*x*(72*a^2*d^4 + 18*a*c*d^2*(3 + 2*d^2*x
^2) + c^2*(15 + 10*d^2*x^2 + 8*d^4*x^4)))) + 15*(5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d^2 + 24*a*b^2*d^4 + 24*a^2*c
*d^4 + 16*a^3*d^6)*ArcSin[d*x])/(240*d^7)

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Maple [C]  time = 0.183, size = 602, 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((c*x^2+b*x+a)^3/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x)

[Out]

-1/240*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(40*csgn(d)*x^5*c^3*d^5*(-d^2*x^2+1)^(1/2)+144*csgn(d)*x^4*b*c^2*d^5*(-d^2
*x^2+1)^(1/2)+180*csgn(d)*x^3*a*c^2*d^5*(-d^2*x^2+1)^(1/2)+180*csgn(d)*x^3*b^2*c*d^5*(-d^2*x^2+1)^(1/2)+480*cs
gn(d)*x^2*a*b*c*d^5*(-d^2*x^2+1)^(1/2)+80*csgn(d)*x^2*b^3*d^5*(-d^2*x^2+1)^(1/2)+50*csgn(d)*d^3*(-d^2*x^2+1)^(
1/2)*x^3*c^3+360*csgn(d)*d^5*(-d^2*x^2+1)^(1/2)*x*a^2*c+360*csgn(d)*d^5*(-d^2*x^2+1)^(1/2)*x*a*b^2+192*csgn(d)
*d^3*(-d^2*x^2+1)^(1/2)*x^2*b*c^2+720*csgn(d)*d^5*(-d^2*x^2+1)^(1/2)*a^2*b-240*arctan(csgn(d)*d*x/(-d^2*x^2+1)
^(1/2))*a^3*d^6+270*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x*a*c^2+270*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x*b^2*c+960*csgn
(d)*d^3*(-d^2*x^2+1)^(1/2)*a*b*c+160*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*b^3-360*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1
/2))*a^2*c*d^4-360*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a*b^2*d^4+75*csgn(d)*d*(-d^2*x^2+1)^(1/2)*x*c^3+384*
csgn(d)*d*(-d^2*x^2+1)^(1/2)*b*c^2-270*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a*c^2*d^2-270*arctan(csgn(d)*d*x
/(-d^2*x^2+1)^(1/2))*b^2*c*d^2-75*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*c^3)*csgn(d)/d^7/(-d^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.80824, size = 554, normalized size = 1.71 \begin{align*} -\frac{\sqrt{-d^{2} x^{2} + 1} c^{3} x^{5}}{6 \, d^{2}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1} b c^{2} x^{4}}{5 \, d^{2}} + \frac{a^{3} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}}} - \frac{5 \, \sqrt{-d^{2} x^{2} + 1} c^{3} x^{3}}{24 \, d^{4}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1}{\left (b^{2} c + a c^{2}\right )} x^{3}}{4 \, d^{2}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1} a^{2} b}{d^{2}} - \frac{4 \, \sqrt{-d^{2} x^{2} + 1} b c^{2} x^{2}}{5 \, d^{4}} - \frac{\sqrt{-d^{2} x^{2} + 1}{\left (b^{3} + 6 \, a b c\right )} x^{2}}{3 \, d^{2}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1}{\left (a b^{2} + a^{2} c\right )} x}{2 \, d^{2}} + \frac{3 \,{\left (a b^{2} + a^{2} c\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{2}} - \frac{5 \, \sqrt{-d^{2} x^{2} + 1} c^{3} x}{16 \, d^{6}} - \frac{9 \, \sqrt{-d^{2} x^{2} + 1}{\left (b^{2} c + a c^{2}\right )} x}{8 \, d^{4}} - \frac{8 \, \sqrt{-d^{2} x^{2} + 1} b c^{2}}{5 \, d^{6}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1}{\left (b^{3} + 6 \, a b c\right )}}{3 \, d^{4}} + \frac{5 \, c^{3} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{16 \, \sqrt{d^{2}} d^{6}} + \frac{9 \,{\left (b^{2} c + a c^{2}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="maxima")

[Out]

-1/6*sqrt(-d^2*x^2 + 1)*c^3*x^5/d^2 - 3/5*sqrt(-d^2*x^2 + 1)*b*c^2*x^4/d^2 + a^3*arcsin(d^2*x/sqrt(d^2))/sqrt(
d^2) - 5/24*sqrt(-d^2*x^2 + 1)*c^3*x^3/d^4 - 3/4*sqrt(-d^2*x^2 + 1)*(b^2*c + a*c^2)*x^3/d^2 - 3*sqrt(-d^2*x^2
+ 1)*a^2*b/d^2 - 4/5*sqrt(-d^2*x^2 + 1)*b*c^2*x^2/d^4 - 1/3*sqrt(-d^2*x^2 + 1)*(b^3 + 6*a*b*c)*x^2/d^2 - 3/2*s
qrt(-d^2*x^2 + 1)*(a*b^2 + a^2*c)*x/d^2 + 3/2*(a*b^2 + a^2*c)*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^2) - 5/16*s
qrt(-d^2*x^2 + 1)*c^3*x/d^6 - 9/8*sqrt(-d^2*x^2 + 1)*(b^2*c + a*c^2)*x/d^4 - 8/5*sqrt(-d^2*x^2 + 1)*b*c^2/d^6
- 2/3*sqrt(-d^2*x^2 + 1)*(b^3 + 6*a*b*c)/d^4 + 5/16*c^3*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^6) + 9/8*(b^2*c +
a*c^2)*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^4)

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Fricas [A]  time = 1.39823, size = 570, normalized size = 1.76 \begin{align*} -\frac{{\left (40 \, c^{3} d^{5} x^{5} + 144 \, b c^{2} d^{5} x^{4} + 720 \, a^{2} b d^{5} + 384 \, b c^{2} d + 160 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} + 10 \,{\left (5 \, c^{3} d^{3} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{5}\right )} x^{3} + 16 \,{\left (12 \, b c^{2} d^{3} + 5 \,{\left (b^{3} + 6 \, a b c\right )} d^{5}\right )} x^{2} + 15 \,{\left (24 \,{\left (a b^{2} + a^{2} c\right )} d^{5} + 5 \, c^{3} d + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{3}\right )} x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 30 \,{\left (16 \, a^{3} d^{6} + 24 \,{\left (a b^{2} + a^{2} c\right )} d^{4} + 5 \, c^{3} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{240 \, d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="fricas")

[Out]

-1/240*((40*c^3*d^5*x^5 + 144*b*c^2*d^5*x^4 + 720*a^2*b*d^5 + 384*b*c^2*d + 160*(b^3 + 6*a*b*c)*d^3 + 10*(5*c^
3*d^3 + 18*(b^2*c + a*c^2)*d^5)*x^3 + 16*(12*b*c^2*d^3 + 5*(b^3 + 6*a*b*c)*d^5)*x^2 + 15*(24*(a*b^2 + a^2*c)*d
^5 + 5*c^3*d + 18*(b^2*c + a*c^2)*d^3)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 30*(16*a^3*d^6 + 24*(a*b^2 + a^2*c)*d
^4 + 5*c^3 + 18*(b^2*c + a*c^2)*d^2)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/d^7

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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((c*x**2+b*x+a)**3/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.58294, size = 518, normalized size = 1.6 \begin{align*} -\frac{{\left (720 \, a^{2} b d^{41} - 360 \, a b^{2} d^{40} - 360 \, a^{2} c d^{40} + 240 \, b^{3} d^{39} + 1440 \, a b c d^{39} - 450 \, b^{2} c d^{38} - 450 \, a c^{2} d^{38} + 720 \, b c^{2} d^{37} - 165 \, c^{3} d^{36} +{\left (360 \, a b^{2} d^{40} + 360 \, a^{2} c d^{40} - 160 \, b^{3} d^{39} - 960 \, a b c d^{39} + 810 \, b^{2} c d^{38} + 810 \, a c^{2} d^{38} - 960 \, b c^{2} d^{37} + 425 \, c^{3} d^{36} + 2 \,{\left (40 \, b^{3} d^{39} + 240 \, a b c d^{39} - 270 \, b^{2} c d^{38} - 270 \, a c^{2} d^{38} + 528 \, b c^{2} d^{37} - 275 \, c^{3} d^{36} +{\left (90 \, b^{2} c d^{38} + 90 \, a c^{2} d^{38} - 288 \, b c^{2} d^{37} + 225 \, c^{3} d^{36} + 4 \,{\left (5 \,{\left (d x + 1\right )} c^{3} d^{36} + 18 \, b c^{2} d^{37} - 25 \, c^{3} d^{36}\right )}{\left (d x + 1\right )}\right )}{\left (d x + 1\right )}\right )}{\left (d x + 1\right )}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 30 \,{\left (16 \, a^{3} d^{42} + 24 \, a b^{2} d^{40} + 24 \, a^{2} c d^{40} + 18 \, b^{2} c d^{38} + 18 \, a c^{2} d^{38} + 5 \, c^{3} d^{36}\right )} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{21626880 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="giac")

[Out]

-1/21626880*((720*a^2*b*d^41 - 360*a*b^2*d^40 - 360*a^2*c*d^40 + 240*b^3*d^39 + 1440*a*b*c*d^39 - 450*b^2*c*d^
38 - 450*a*c^2*d^38 + 720*b*c^2*d^37 - 165*c^3*d^36 + (360*a*b^2*d^40 + 360*a^2*c*d^40 - 160*b^3*d^39 - 960*a*
b*c*d^39 + 810*b^2*c*d^38 + 810*a*c^2*d^38 - 960*b*c^2*d^37 + 425*c^3*d^36 + 2*(40*b^3*d^39 + 240*a*b*c*d^39 -
270*b^2*c*d^38 - 270*a*c^2*d^38 + 528*b*c^2*d^37 - 275*c^3*d^36 + (90*b^2*c*d^38 + 90*a*c^2*d^38 - 288*b*c^2*
d^37 + 225*c^3*d^36 + 4*(5*(d*x + 1)*c^3*d^36 + 18*b*c^2*d^37 - 25*c^3*d^36)*(d*x + 1))*(d*x + 1))*(d*x + 1))*
(d*x + 1))*sqrt(d*x + 1)*sqrt(-d*x + 1) - 30*(16*a^3*d^42 + 24*a*b^2*d^40 + 24*a^2*c*d^40 + 18*b^2*c*d^38 + 18
*a*c^2*d^38 + 5*c^3*d^36)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))/d