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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.93, antiderivative size = 324, normalized size of antiderivative = 1.00, 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 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 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 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]

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 = 0.26, size = 229, normalized size = 0.71 \[ \frac {15 \sin ^{-1}(d x) \left (16 a^3 d^6+24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right )-d \sqrt {1-d^2 x^2} \left (48 b \left (15 a^2 d^4+10 a c d^2 \left (d^2 x^2+2\right )+c^2 \left (3 d^4 x^4+4 d^2 x^2+8\right )\right )+5 c x \left (72 a^2 d^4+18 a c d^2 \left (2 d^2 x^2+3\right )+c^2 \left (8 d^4 x^4+10 d^2 x^2+15\right )\right )+90 b^2 d^2 x \left (4 a d^2+c \left (2 d^2 x^2+3\right )\right )+80 b^3 d^2 \left (d^2 x^2+2\right )\right )}{240 d^7} \]

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)

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 251, normalized size = 0.77 \[ -\frac {{\left (40 \, c^{3} d^{5} x^{5} + 144 \, b c^{2} d^{5} x^{4} + 720 \, a^{2} b d^{5} + 384 \, b c^{2} d + 160 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} + 10 \, {\left (5 \, c^{3} d^{3} + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{5}\right )} x^{3} + 16 \, {\left (12 \, b c^{2} d^{3} + 5 \, {\left (b^{3} + 6 \, a b c\right )} d^{5}\right )} x^{2} + 15 \, {\left (24 \, {\left (a b^{2} + a^{2} c\right )} d^{5} + 5 \, c^{3} d + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{3}\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 30 \, {\left (16 \, a^{3} d^{6} + 24 \, {\left (a b^{2} + a^{2} c\right )} d^{4} + 5 \, c^{3} + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{240 \, d^{7}} \]

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

________________________________________________________________________________________

giac [A]  time = 0.62, size = 412, normalized size = 1.27 \[ -\frac {{\left ({\left (2 \, {\left ({\left (d x + 1\right )} {\left (4 \, {\left (d x + 1\right )} {\left (\frac {5 \, {\left (d x + 1\right )} c^{3}}{d^{6}} + \frac {18 \, b c^{2} d^{37} - 25 \, c^{3} d^{36}}{d^{42}}\right )} + \frac {9 \, {\left (10 \, b^{2} c d^{38} + 10 \, a c^{2} d^{38} - 32 \, b c^{2} d^{37} + 25 \, c^{3} d^{36}\right )}}{d^{42}}\right )} + \frac {40 \, b^{3} d^{39} + 240 \, a b c d^{39} - 270 \, b^{2} c d^{38} - 270 \, a c^{2} d^{38} + 528 \, b c^{2} d^{37} - 275 \, c^{3} d^{36}}{d^{42}}\right )} {\left (d x + 1\right )} + \frac {5 \, {\left (72 \, a b^{2} d^{40} + 72 \, a^{2} c d^{40} - 32 \, b^{3} d^{39} - 192 \, a b c d^{39} + 162 \, b^{2} c d^{38} + 162 \, a c^{2} d^{38} - 192 \, b c^{2} d^{37} + 85 \, c^{3} d^{36}\right )}}{d^{42}}\right )} {\left (d x + 1\right )} + \frac {15 \, {\left (48 \, a^{2} b d^{41} - 24 \, a b^{2} d^{40} - 24 \, a^{2} c d^{40} + 16 \, b^{3} d^{39} + 96 \, a b c d^{39} - 30 \, b^{2} c d^{38} - 30 \, a c^{2} d^{38} + 48 \, b c^{2} d^{37} - 11 \, c^{3} d^{36}\right )}}{d^{42}}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} - \frac {30 \, {\left (16 \, a^{3} d^{6} + 24 \, a b^{2} d^{4} + 24 \, a^{2} c d^{4} + 18 \, b^{2} c d^{2} + 18 \, a c^{2} d^{2} + 5 \, c^{3}\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{6}}}{240 \, d} \]

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]

-1/240*(((2*((d*x + 1)*(4*(d*x + 1)*(5*(d*x + 1)*c^3/d^6 + (18*b*c^2*d^37 - 25*c^3*d^36)/d^42) + 9*(10*b^2*c*d
^38 + 10*a*c^2*d^38 - 32*b*c^2*d^37 + 25*c^3*d^36)/d^42) + (40*b^3*d^39 + 240*a*b*c*d^39 - 270*b^2*c*d^38 - 27
0*a*c^2*d^38 + 528*b*c^2*d^37 - 275*c^3*d^36)/d^42)*(d*x + 1) + 5*(72*a*b^2*d^40 + 72*a^2*c*d^40 - 32*b^3*d^39
- 192*a*b*c*d^39 + 162*b^2*c*d^38 + 162*a*c^2*d^38 - 192*b*c^2*d^37 + 85*c^3*d^36)/d^42)*(d*x + 1) + 15*(48*a
^2*b*d^41 - 24*a*b^2*d^40 - 24*a^2*c*d^40 + 16*b^3*d^39 + 96*a*b*c*d^39 - 30*b^2*c*d^38 - 30*a*c^2*d^38 + 48*b
*c^2*d^37 - 11*c^3*d^36)/d^42)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 30*(16*a^3*d^6 + 24*a*b^2*d^4 + 24*a^2*c*d^4 + 1
8*b^2*c*d^2 + 18*a*c^2*d^2 + 5*c^3)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^6)/d

________________________________________________________________________________________

maple [C]  time = 0.06, size = 602, normalized size = 1.86 \[ -\frac {\sqrt {-d x +1}\, \sqrt {d x +1}\, \left (40 \sqrt {-d^{2} x^{2}+1}\, c^{3} d^{5} x^{5} \mathrm {csgn}\relax (d )+144 \sqrt {-d^{2} x^{2}+1}\, b \,c^{2} d^{5} x^{4} \mathrm {csgn}\relax (d )+180 \sqrt {-d^{2} x^{2}+1}\, a \,c^{2} d^{5} x^{3} \mathrm {csgn}\relax (d )+180 \sqrt {-d^{2} x^{2}+1}\, b^{2} c \,d^{5} x^{3} \mathrm {csgn}\relax (d )+480 \sqrt {-d^{2} x^{2}+1}\, a b c \,d^{5} x^{2} \mathrm {csgn}\relax (d )+80 \sqrt {-d^{2} x^{2}+1}\, b^{3} d^{5} x^{2} \mathrm {csgn}\relax (d )-240 a^{3} d^{6} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+360 \sqrt {-d^{2} x^{2}+1}\, a^{2} c \,d^{5} x \,\mathrm {csgn}\relax (d )+360 \sqrt {-d^{2} x^{2}+1}\, a \,b^{2} d^{5} x \,\mathrm {csgn}\relax (d )+50 \sqrt {-d^{2} x^{2}+1}\, c^{3} d^{3} x^{3} \mathrm {csgn}\relax (d )+720 \sqrt {-d^{2} x^{2}+1}\, a^{2} b \,d^{5} \mathrm {csgn}\relax (d )+192 \sqrt {-d^{2} x^{2}+1}\, b \,c^{2} d^{3} x^{2} \mathrm {csgn}\relax (d )-360 a^{2} c \,d^{4} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-360 a \,b^{2} d^{4} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+270 \sqrt {-d^{2} x^{2}+1}\, a \,c^{2} d^{3} x \,\mathrm {csgn}\relax (d )+270 \sqrt {-d^{2} x^{2}+1}\, b^{2} c \,d^{3} x \,\mathrm {csgn}\relax (d )+960 \sqrt {-d^{2} x^{2}+1}\, a b c \,d^{3} \mathrm {csgn}\relax (d )+160 \sqrt {-d^{2} x^{2}+1}\, b^{3} d^{3} \mathrm {csgn}\relax (d )-270 a \,c^{2} d^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-270 b^{2} c \,d^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+75 \sqrt {-d^{2} x^{2}+1}\, c^{3} d x \,\mathrm {csgn}\relax (d )+384 \sqrt {-d^{2} x^{2}+1}\, b \,c^{2} d \,\mathrm {csgn}\relax (d )-75 c^{3} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )\right ) \mathrm {csgn}\relax (d )}{240 \sqrt {-d^{2} x^{2}+1}\, d^{7}} \]

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*(-d^2*x^2+1)^(1/2)*csgn(d)*d^5*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*(-d^
2*x^2+1)^(1/2)*csgn(d)*d^3*a*b*c+160*(-d^2*x^2+1)^(1/2)*csgn(d)*d^3*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*
(-d^2*x^2+1)^(1/2)*csgn(d)*d*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)

________________________________________________________________________________________

maxima [A]  time = 0.98, size = 365, normalized size = 1.13 \[ -\frac {\sqrt {-d^{2} x^{2} + 1} c^{3} x^{5}}{6 \, d^{2}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} b c^{2} x^{4}}{5 \, d^{2}} + \frac {a^{3} \arcsin \left (d x\right )}{d} - \frac {5 \, \sqrt {-d^{2} x^{2} + 1} c^{3} x^{3}}{24 \, d^{4}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} {\left (b^{2} c + a c^{2}\right )} x^{3}}{4 \, d^{2}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} a^{2} b}{d^{2}} - \frac {4 \, \sqrt {-d^{2} x^{2} + 1} b c^{2} x^{2}}{5 \, d^{4}} - \frac {\sqrt {-d^{2} x^{2} + 1} {\left (b^{3} + 6 \, a b c\right )} x^{2}}{3 \, d^{2}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} {\left (a b^{2} + a^{2} c\right )} x}{2 \, d^{2}} + \frac {3 \, {\left (a b^{2} + a^{2} c\right )} \arcsin \left (d x\right )}{2 \, d^{3}} - \frac {5 \, \sqrt {-d^{2} x^{2} + 1} c^{3} x}{16 \, d^{6}} - \frac {9 \, \sqrt {-d^{2} x^{2} + 1} {\left (b^{2} c + a c^{2}\right )} x}{8 \, d^{4}} - \frac {8 \, \sqrt {-d^{2} x^{2} + 1} b c^{2}}{5 \, d^{6}} - \frac {2 \, \sqrt {-d^{2} x^{2} + 1} {\left (b^{3} + 6 \, a b c\right )}}{3 \, d^{4}} + \frac {5 \, c^{3} \arcsin \left (d x\right )}{16 \, d^{7}} + \frac {9 \, {\left (b^{2} c + a c^{2}\right )} \arcsin \left (d x\right )}{8 \, d^{5}} \]

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*x)/d - 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*sqrt(-d^2*x^2 + 1)*(a
*b^2 + a^2*c)*x/d^2 + 3/2*(a*b^2 + a^2*c)*arcsin(d*x)/d^3 - 5/16*sqrt(-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*x)/d^7 + 9/8*(b^2*c + a*c^2)*arcsin(d*x)/d^5

________________________________________________________________________________________

mupad [B]  time = 31.33, size = 1768, normalized size = 5.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((((1 - d*x)^(1/2) - 1)^23*((5*c^3)/4 + 6*a*b^2*d^4 + (9*a*c^2*d^2)/2 + 6*a^2*c*d^4 + (9*b^2*c*d^2)/2))/((d*
x + 1)^(1/2) - 1)^23 - (((1 - d*x)^(1/2) - 1)*((5*c^3)/4 + 6*a*b^2*d^4 + (9*a*c^2*d^2)/2 + 6*a^2*c*d^4 + (9*b^
2*c*d^2)/2))/((d*x + 1)^(1/2) - 1) - (((1 - d*x)^(1/2) - 1)^3*((175*c^3)/12 + 6*a*b^2*d^4 + (105*a*c^2*d^2)/2
+ 6*a^2*c*d^4 + (105*b^2*c*d^2)/2))/((d*x + 1)^(1/2) - 1)^3 + (((1 - d*x)^(1/2) - 1)^21*((175*c^3)/12 + 6*a*b^
2*d^4 + (105*a*c^2*d^2)/2 + 6*a^2*c*d^4 + (105*b^2*c*d^2)/2))/((d*x + 1)^(1/2) - 1)^21 + (((1 - d*x)^(1/2) - 1
)^5*(126*a*b^2*d^4 - (311*c^3)/4 + (669*a*c^2*d^2)/2 + 126*a^2*c*d^4 + (669*b^2*c*d^2)/2))/((d*x + 1)^(1/2) -
1)^5 - (((1 - d*x)^(1/2) - 1)^19*(126*a*b^2*d^4 - (311*c^3)/4 + (669*a*c^2*d^2)/2 + 126*a^2*c*d^4 + (669*b^2*c
*d^2)/2))/((d*x + 1)^(1/2) - 1)^19 + (((1 - d*x)^(1/2) - 1)^7*((8361*c^3)/4 + 510*a*b^2*d^4 + (1533*a*c^2*d^2)
/2 + 510*a^2*c*d^4 + (1533*b^2*c*d^2)/2))/((d*x + 1)^(1/2) - 1)^7 - (((1 - d*x)^(1/2) - 1)^17*((8361*c^3)/4 +
510*a*b^2*d^4 + (1533*a*c^2*d^2)/2 + 510*a^2*c*d^4 + (1533*b^2*c*d^2)/2))/((d*x + 1)^(1/2) - 1)^17 + (((1 - d*
x)^(1/2) - 1)^11*((25295*c^3)/2 + 420*a*b^2*d^4 - 549*a*c^2*d^2 + 420*a^2*c*d^4 - 549*b^2*c*d^2))/((d*x + 1)^(
1/2) - 1)^11 - (((1 - d*x)^(1/2) - 1)^13*((25295*c^3)/2 + 420*a*b^2*d^4 - 549*a*c^2*d^2 + 420*a^2*c*d^4 - 549*
b^2*c*d^2))/((d*x + 1)^(1/2) - 1)^13 - (((1 - d*x)^(1/2) - 1)^9*((42259*c^3)/6 - 804*a*b^2*d^4 + 165*a*c^2*d^2
- 804*a^2*c*d^4 + 165*b^2*c*d^2))/((d*x + 1)^(1/2) - 1)^9 + (((1 - d*x)^(1/2) - 1)^15*((42259*c^3)/6 - 804*a*
b^2*d^4 + 165*a*c^2*d^2 - 804*a^2*c*d^4 + 165*b^2*c*d^2))/((d*x + 1)^(1/2) - 1)^15 + (((1 - d*x)^(1/2) - 1)^6*
((1024*b^3*d^3)/3 + 1080*a^2*b*d^5 + 2048*b*c^2*d + 2048*a*b*c*d^3))/((d*x + 1)^(1/2) - 1)^6 + (((1 - d*x)^(1/
2) - 1)^18*((1024*b^3*d^3)/3 + 1080*a^2*b*d^5 + 2048*b*c^2*d + 2048*a*b*c*d^3))/((d*x + 1)^(1/2) - 1)^18 + (((
1 - d*x)^(1/2) - 1)^10*(1024*b^3*d^3 + 5040*a^2*b*d^5 + (6144*b*c^2*d)/5 + 6144*a*b*c*d^3))/((d*x + 1)^(1/2) -
1)^10 + (((1 - d*x)^(1/2) - 1)^14*(1024*b^3*d^3 + 5040*a^2*b*d^5 + (6144*b*c^2*d)/5 + 6144*a*b*c*d^3))/((d*x
+ 1)^(1/2) - 1)^14 + (((1 - d*x)^(1/2) - 1)^12*((3200*b^3*d^3)/3 + 6048*a^2*b*d^5 + (32768*b*c^2*d)/5 + 6400*a
*b*c*d^3))/((d*x + 1)^(1/2) - 1)^12 + (((1 - d*x)^(1/2) - 1)^4*(64*b^3*d^3 + 240*a^2*b*d^5 + 384*a*b*c*d^3))/(
(d*x + 1)^(1/2) - 1)^4 + (((1 - d*x)^(1/2) - 1)^20*(64*b^3*d^3 + 240*a^2*b*d^5 + 384*a*b*c*d^3))/((d*x + 1)^(1
/2) - 1)^20 + (((1 - d*x)^(1/2) - 1)^8*(768*b^3*d^3 + 2880*a^2*b*d^5 + 4608*a*b*c*d^3))/((d*x + 1)^(1/2) - 1)^
8 + (((1 - d*x)^(1/2) - 1)^16*(768*b^3*d^3 + 2880*a^2*b*d^5 + 4608*a*b*c*d^3))/((d*x + 1)^(1/2) - 1)^16 + (24*
a^2*b*d^5*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (24*a^2*b*d^5*((1 - d*x)^(1/2) - 1)^22)/((d*x + 1
)^(1/2) - 1)^22)/(d^7 + (12*d^7*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (66*d^7*((1 - d*x)^(1/2) -
1)^4)/((d*x + 1)^(1/2) - 1)^4 + (220*d^7*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6 + (495*d^7*((1 - d*x
)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8 + (792*d^7*((1 - d*x)^(1/2) - 1)^10)/((d*x + 1)^(1/2) - 1)^10 + (924*d
^7*((1 - d*x)^(1/2) - 1)^12)/((d*x + 1)^(1/2) - 1)^12 + (792*d^7*((1 - d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) -
1)^14 + (495*d^7*((1 - d*x)^(1/2) - 1)^16)/((d*x + 1)^(1/2) - 1)^16 + (220*d^7*((1 - d*x)^(1/2) - 1)^18)/((d*x
+ 1)^(1/2) - 1)^18 + (66*d^7*((1 - d*x)^(1/2) - 1)^20)/((d*x + 1)^(1/2) - 1)^20 + (12*d^7*((1 - d*x)^(1/2) -
1)^22)/((d*x + 1)^(1/2) - 1)^22 + (d^7*((1 - d*x)^(1/2) - 1)^24)/((d*x + 1)^(1/2) - 1)^24) - (atan(((1 - d*x)^
(1/2) - 1)/((d*x + 1)^(1/2) - 1))*(5*c^3 + 16*a^3*d^6 + 24*a*b^2*d^4 + 18*a*c^2*d^2 + 24*a^2*c*d^4 + 18*b^2*c*
d^2))/(4*d^7)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________