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

Optimal. Leaf size=166 \[ \frac {\sin ^{-1}(d x) \left (8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right )}{8 d^5}-\frac {x \sqrt {1-d^2 x^2} \left (c \left (8 a+\frac {3 c}{d^2}\right )+4 b^2\right )}{8 d^2}-\frac {2 b \sqrt {1-d^2 x^2} \left (3 a d^2+2 c\right )}{3 d^4}-\frac {2 b c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {c^2 x^3 \sqrt {1-d^2 x^2}}{4 d^2} \]

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Rubi [A]  time = 0.32, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {899, 1815, 641, 216} \begin {gather*} \frac {\sin ^{-1}(d x) \left (8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right )}{8 d^5}-\frac {x \sqrt {1-d^2 x^2} \left (c \left (8 a+\frac {3 c}{d^2}\right )+4 b^2\right )}{8 d^2}-\frac {2 b \sqrt {1-d^2 x^2} \left (3 a d^2+2 c\right )}{3 d^4}-\frac {2 b c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {c^2 x^3 \sqrt {1-d^2 x^2}}{4 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(2*c + 3*a*d^2)*Sqrt[1 - d^2*x^2])/(3*d^4) - ((4*b^2 + c*(8*a + (3*c)/d^2))*x*Sqrt[1 - d^2*x^2])/(8*d^2)
- (2*b*c*x^2*Sqrt[1 - d^2*x^2])/(3*d^2) - (c^2*x^3*Sqrt[1 - d^2*x^2])/(4*d^2) + ((3*c^2 + 4*b^2*d^2 + 8*a*c*d
^2 + 8*a^2*d^4)*ArcSin[d*x])/(8*d^5)

Rule 216

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

Rule 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 )^2}{\sqrt {1-d x} \sqrt {1+d x}} \, dx &=\int \frac {\left (a+b x+c x^2\right )^2}{\sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {c^2 x^3 \sqrt {1-d^2 x^2}}{4 d^2}-\frac {\int \frac {-4 a^2 d^2-8 a b d^2 x-\left (3 c^2+4 b^2 d^2+8 a c d^2\right ) x^2-8 b c d^2 x^3}{\sqrt {1-d^2 x^2}} \, dx}{4 d^2}\\ &=-\frac {2 b c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {c^2 x^3 \sqrt {1-d^2 x^2}}{4 d^2}+\frac {\int \frac {12 a^2 d^4+8 b d^2 \left (2 c+3 a d^2\right ) x+3 d^2 \left (3 c^2+4 b^2 d^2+8 a c d^2\right ) x^2}{\sqrt {1-d^2 x^2}} \, dx}{12 d^4}\\ &=-\frac {\left (3 c^2+4 b^2 d^2+8 a c d^2\right ) x \sqrt {1-d^2 x^2}}{8 d^4}-\frac {2 b c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {c^2 x^3 \sqrt {1-d^2 x^2}}{4 d^2}-\frac {\int \frac {-3 d^2 \left (3 c^2+4 b^2 d^2+8 a c d^2+8 a^2 d^4\right )-16 b d^4 \left (2 c+3 a d^2\right ) x}{\sqrt {1-d^2 x^2}} \, dx}{24 d^6}\\ &=-\frac {2 b \left (2 c+3 a d^2\right ) \sqrt {1-d^2 x^2}}{3 d^4}-\frac {\left (3 c^2+4 b^2 d^2+8 a c d^2\right ) x \sqrt {1-d^2 x^2}}{8 d^4}-\frac {2 b c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {c^2 x^3 \sqrt {1-d^2 x^2}}{4 d^2}+\frac {\left (3 c^2+4 b^2 d^2+8 a c d^2+8 a^2 d^4\right ) \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{8 d^4}\\ &=-\frac {2 b \left (2 c+3 a d^2\right ) \sqrt {1-d^2 x^2}}{3 d^4}-\frac {\left (3 c^2+4 b^2 d^2+8 a c d^2\right ) x \sqrt {1-d^2 x^2}}{8 d^4}-\frac {2 b c x^2 \sqrt {1-d^2 x^2}}{3 d^2}-\frac {c^2 x^3 \sqrt {1-d^2 x^2}}{4 d^2}+\frac {\left (3 c^2+4 b^2 d^2+8 a c d^2+8 a^2 d^4\right ) \sin ^{-1}(d x)}{8 d^5}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 114, normalized size = 0.69 \begin {gather*} \frac {3 \sin ^{-1}(d x) \left (8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right )-d \sqrt {1-d^2 x^2} \left (16 b \left (3 a d^2+c d^2 x^2+2 c\right )+3 c x \left (8 a d^2+2 c d^2 x^2+3 c\right )+12 b^2 d^2 x\right )}{24 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(d*Sqrt[1 - d^2*x^2]*(12*b^2*d^2*x + 16*b*(2*c + 3*a*d^2 + c*d^2*x^2) + 3*c*x*(3*c + 8*a*d^2 + 2*c*d^2*x^2))
) + 3*(3*c^2 + 4*b^2*d^2 + 8*a*c*d^2 + 8*a^2*d^4)*ArcSin[d*x])/(24*d^5)

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IntegrateAlgebraic [B]  time = 0.29, size = 446, normalized size = 2.69 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {1-d x}}{\sqrt {d x+1}}\right ) \left (-8 a^2 d^4-8 a c d^2-4 b^2 d^2-3 c^2\right )}{4 d^5}+\frac {\sqrt {1-d x} \left (-\frac {144 a b d^3 (1-d x)}{d x+1}-\frac {144 a b d^3 (1-d x)^2}{(d x+1)^2}-\frac {48 a b d^3 (1-d x)^3}{(d x+1)^3}-48 a b d^3-\frac {24 a c d^2 (1-d x)}{d x+1}+\frac {24 a c d^2 (1-d x)^2}{(d x+1)^2}+\frac {24 a c d^2 (1-d x)^3}{(d x+1)^3}-24 a c d^2-\frac {12 b^2 d^2 (1-d x)}{d x+1}+\frac {12 b^2 d^2 (1-d x)^2}{(d x+1)^2}+\frac {12 b^2 d^2 (1-d x)^3}{(d x+1)^3}-12 b^2 d^2-\frac {80 b c d (1-d x)}{d x+1}-\frac {80 b c d (1-d x)^2}{(d x+1)^2}-\frac {48 b c d (1-d x)^3}{(d x+1)^3}-48 b c d+\frac {9 c^2 (1-d x)}{d x+1}-\frac {9 c^2 (1-d x)^2}{(d x+1)^2}+\frac {15 c^2 (1-d x)^3}{(d x+1)^3}-15 c^2\right )}{12 d^5 \sqrt {d x+1} \left (\frac {1-d x}{d x+1}+1\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - d*x]*(-15*c^2 - 48*b*c*d - 12*b^2*d^2 - 24*a*c*d^2 - 48*a*b*d^3 + (15*c^2*(1 - d*x)^3)/(1 + d*x)^3 -
(48*b*c*d*(1 - d*x)^3)/(1 + d*x)^3 + (12*b^2*d^2*(1 - d*x)^3)/(1 + d*x)^3 + (24*a*c*d^2*(1 - d*x)^3)/(1 + d*x
)^3 - (48*a*b*d^3*(1 - d*x)^3)/(1 + d*x)^3 - (9*c^2*(1 - d*x)^2)/(1 + d*x)^2 - (80*b*c*d*(1 - d*x)^2)/(1 + d*x
)^2 + (12*b^2*d^2*(1 - d*x)^2)/(1 + d*x)^2 + (24*a*c*d^2*(1 - d*x)^2)/(1 + d*x)^2 - (144*a*b*d^3*(1 - d*x)^2)/
(1 + d*x)^2 + (9*c^2*(1 - d*x))/(1 + d*x) - (80*b*c*d*(1 - d*x))/(1 + d*x) - (12*b^2*d^2*(1 - d*x))/(1 + d*x)
- (24*a*c*d^2*(1 - d*x))/(1 + d*x) - (144*a*b*d^3*(1 - d*x))/(1 + d*x)))/(12*d^5*Sqrt[1 + d*x]*(1 + (1 - d*x)/
(1 + d*x))^4) + ((-3*c^2 - 4*b^2*d^2 - 8*a*c*d^2 - 8*a^2*d^4)*ArcTan[Sqrt[1 - d*x]/Sqrt[1 + d*x]])/(4*d^5)

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fricas [A]  time = 0.42, size = 134, normalized size = 0.81 \begin {gather*} -\frac {{\left (6 \, c^{2} d^{3} x^{3} + 16 \, b c d^{3} x^{2} + 48 \, a b d^{3} + 32 \, b c d + 3 \, {\left (4 \, {\left (b^{2} + 2 \, a c\right )} d^{3} + 3 \, c^{2} d\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 6 \, {\left (8 \, a^{2} d^{4} + 4 \, {\left (b^{2} + 2 \, a c\right )} d^{2} + 3 \, c^{2}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{24 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((6*c^2*d^3*x^3 + 16*b*c*d^3*x^2 + 48*a*b*d^3 + 32*b*c*d + 3*(4*(b^2 + 2*a*c)*d^3 + 3*c^2*d)*x)*sqrt(d*x
+ 1)*sqrt(-d*x + 1) + 6*(8*a^2*d^4 + 4*(b^2 + 2*a*c)*d^2 + 3*c^2)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(
d*x)))/d^5

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giac [A]  time = 0.42, size = 196, normalized size = 1.18 \begin {gather*} -\frac {{\left ({\left (d x + 1\right )} {\left (2 \, {\left (d x + 1\right )} {\left (\frac {3 \, {\left (d x + 1\right )} c^{2}}{d^{4}} + \frac {8 \, b c d^{17} - 9 \, c^{2} d^{16}}{d^{20}}\right )} + \frac {12 \, b^{2} d^{18} + 24 \, a c d^{18} - 32 \, b c d^{17} + 27 \, c^{2} d^{16}}{d^{20}}\right )} + \frac {3 \, {\left (16 \, a b d^{19} - 4 \, b^{2} d^{18} - 8 \, a c d^{18} + 16 \, b c d^{17} - 5 \, c^{2} d^{16}\right )}}{d^{20}}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} - \frac {6 \, {\left (8 \, a^{2} d^{4} + 4 \, b^{2} d^{2} + 8 \, a c d^{2} + 3 \, c^{2}\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{4}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(((d*x + 1)*(2*(d*x + 1)*(3*(d*x + 1)*c^2/d^4 + (8*b*c*d^17 - 9*c^2*d^16)/d^20) + (12*b^2*d^18 + 24*a*c*
d^18 - 32*b*c*d^17 + 27*c^2*d^16)/d^20) + 3*(16*a*b*d^19 - 4*b^2*d^18 - 8*a*c*d^18 + 16*b*c*d^17 - 5*c^2*d^16)
/d^20)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 6*(8*a^2*d^4 + 4*b^2*d^2 + 8*a*c*d^2 + 3*c^2)*arcsin(1/2*sqrt(2)*sqrt(d*
x + 1))/d^4)/d

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maple [C]  time = 0.03, size = 291, normalized size = 1.75 \begin {gather*} -\frac {\sqrt {-d x +1}\, \sqrt {d x +1}\, \left (6 \sqrt {-d^{2} x^{2}+1}\, c^{2} d^{3} x^{3} \mathrm {csgn}\relax (d )+16 \sqrt {-d^{2} x^{2}+1}\, b c \,d^{3} x^{2} \mathrm {csgn}\relax (d )-24 a^{2} d^{4} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+24 \sqrt {-d^{2} x^{2}+1}\, a c \,d^{3} x \,\mathrm {csgn}\relax (d )+12 \sqrt {-d^{2} x^{2}+1}\, b^{2} d^{3} x \,\mathrm {csgn}\relax (d )+48 \sqrt {-d^{2} x^{2}+1}\, a b \,d^{3} \mathrm {csgn}\relax (d )-24 a c \,d^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )-12 b^{2} d^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )+9 \sqrt {-d^{2} x^{2}+1}\, c^{2} d x \,\mathrm {csgn}\relax (d )+32 \sqrt {-d^{2} x^{2}+1}\, b c d \,\mathrm {csgn}\relax (d )-9 c^{2} \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}}\right )\right ) \mathrm {csgn}\relax (d )}{24 \sqrt {-d^{2} x^{2}+1}\, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(6*csgn(d)*x^3*c^2*d^3*(-d^2*x^2+1)^(1/2)+16*csgn(d)*x^2*b*c*d^3*(-d^2*x^2+
1)^(1/2)+24*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x*a*c+12*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x*b^2+48*(-d^2*x^2+1)^(1/2)
*csgn(d)*d^3*a*b-24*arctan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))*a^2*d^4+9*csgn(d)*d*(-d^2*x^2+1)^(1/2)*x*c^2+32*(
-d^2*x^2+1)^(1/2)*csgn(d)*d*b*c-24*arctan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))*a*c*d^2-12*arctan(1/(-d^2*x^2+1)^(
1/2)*d*x*csgn(d))*b^2*d^2-9*arctan(1/(-d^2*x^2+1)^(1/2)*d*x*csgn(d))*c^2)*csgn(d)/d^5/(-d^2*x^2+1)^(1/2)

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maxima [A]  time = 0.97, size = 171, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {-d^{2} x^{2} + 1} c^{2} x^{3}}{4 \, d^{2}} - \frac {2 \, \sqrt {-d^{2} x^{2} + 1} b c x^{2}}{3 \, d^{2}} + \frac {a^{2} \arcsin \left (d x\right )}{d} - \frac {2 \, \sqrt {-d^{2} x^{2} + 1} a b}{d^{2}} - \frac {\sqrt {-d^{2} x^{2} + 1} {\left (b^{2} + 2 \, a c\right )} x}{2 \, d^{2}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} c^{2} x}{8 \, d^{4}} + \frac {{\left (b^{2} + 2 \, a c\right )} \arcsin \left (d x\right )}{2 \, d^{3}} - \frac {4 \, \sqrt {-d^{2} x^{2} + 1} b c}{3 \, d^{4}} + \frac {3 \, c^{2} \arcsin \left (d x\right )}{8 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(-d^2*x^2 + 1)*c^2*x^3/d^2 - 2/3*sqrt(-d^2*x^2 + 1)*b*c*x^2/d^2 + a^2*arcsin(d*x)/d - 2*sqrt(-d^2*x^2
+ 1)*a*b/d^2 - 1/2*sqrt(-d^2*x^2 + 1)*(b^2 + 2*a*c)*x/d^2 - 3/8*sqrt(-d^2*x^2 + 1)*c^2*x/d^4 + 1/2*(b^2 + 2*a
*c)*arcsin(d*x)/d^3 - 4/3*sqrt(-d^2*x^2 + 1)*b*c/d^4 + 3/8*c^2*arcsin(d*x)/d^5

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mupad [B]  time = 13.85, size = 897, normalized size = 5.40 \begin {gather*} -\frac {\frac {{\left (\sqrt {1-d\,x}-1\right )}^{15}\,\left (2\,b^2\,d^2+\frac {3\,c^2}{2}+4\,a\,c\,d^2\right )}{{\left (\sqrt {d\,x+1}-1\right )}^{15}}+\frac {{\left (\sqrt {1-d\,x}-1\right )}^3\,\left (6\,b^2\,d^2-\frac {23\,c^2}{2}+12\,a\,c\,d^2\right )}{{\left (\sqrt {d\,x+1}-1\right )}^3}-\frac {{\left (\sqrt {1-d\,x}-1\right )}^{13}\,\left (6\,b^2\,d^2-\frac {23\,c^2}{2}+12\,a\,c\,d^2\right )}{{\left (\sqrt {d\,x+1}-1\right )}^{13}}+\frac {{\left (\sqrt {1-d\,x}-1\right )}^5\,\left (30\,b^2\,d^2+\frac {333\,c^2}{2}+60\,a\,c\,d^2\right )}{{\left (\sqrt {d\,x+1}-1\right )}^5}-\frac {{\left (\sqrt {1-d\,x}-1\right )}^{11}\,\left (30\,b^2\,d^2+\frac {333\,c^2}{2}+60\,a\,c\,d^2\right )}{{\left (\sqrt {d\,x+1}-1\right )}^{11}}+\frac {{\left (\sqrt {1-d\,x}-1\right )}^7\,\left (22\,b^2\,d^2-\frac {671\,c^2}{2}+44\,a\,c\,d^2\right )}{{\left (\sqrt {d\,x+1}-1\right )}^7}-\frac {{\left (\sqrt {1-d\,x}-1\right )}^9\,\left (22\,b^2\,d^2-\frac {671\,c^2}{2}+44\,a\,c\,d^2\right )}{{\left (\sqrt {d\,x+1}-1\right )}^9}+\frac {{\left (\sqrt {1-d\,x}-1\right )}^4\,\left (96\,a\,b\,d^3+128\,b\,c\,d\right )}{{\left (\sqrt {d\,x+1}-1\right )}^4}+\frac {{\left (\sqrt {1-d\,x}-1\right )}^{12}\,\left (96\,a\,b\,d^3+128\,b\,c\,d\right )}{{\left (\sqrt {d\,x+1}-1\right )}^{12}}+\frac {{\left (\sqrt {1-d\,x}-1\right )}^8\,\left (320\,a\,b\,d^3+\frac {256\,b\,c\,d}{3}\right )}{{\left (\sqrt {d\,x+1}-1\right )}^8}+\frac {{\left (\sqrt {1-d\,x}-1\right )}^6\,\left (240\,a\,b\,d^3+\frac {512\,b\,c\,d}{3}\right )}{{\left (\sqrt {d\,x+1}-1\right )}^6}+\frac {{\left (\sqrt {1-d\,x}-1\right )}^{10}\,\left (240\,a\,b\,d^3+\frac {512\,b\,c\,d}{3}\right )}{{\left (\sqrt {d\,x+1}-1\right )}^{10}}-\frac {\left (\sqrt {1-d\,x}-1\right )\,\left (2\,b^2\,d^2+\frac {3\,c^2}{2}+4\,a\,c\,d^2\right )}{\sqrt {d\,x+1}-1}+\frac {16\,a\,b\,d^3\,{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+\frac {16\,a\,b\,d^3\,{\left (\sqrt {1-d\,x}-1\right )}^{14}}{{\left (\sqrt {d\,x+1}-1\right )}^{14}}}{d^5+\frac {8\,d^5\,{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+\frac {28\,d^5\,{\left (\sqrt {1-d\,x}-1\right )}^4}{{\left (\sqrt {d\,x+1}-1\right )}^4}+\frac {56\,d^5\,{\left (\sqrt {1-d\,x}-1\right )}^6}{{\left (\sqrt {d\,x+1}-1\right )}^6}+\frac {70\,d^5\,{\left (\sqrt {1-d\,x}-1\right )}^8}{{\left (\sqrt {d\,x+1}-1\right )}^8}+\frac {56\,d^5\,{\left (\sqrt {1-d\,x}-1\right )}^{10}}{{\left (\sqrt {d\,x+1}-1\right )}^{10}}+\frac {28\,d^5\,{\left (\sqrt {1-d\,x}-1\right )}^{12}}{{\left (\sqrt {d\,x+1}-1\right )}^{12}}+\frac {8\,d^5\,{\left (\sqrt {1-d\,x}-1\right )}^{14}}{{\left (\sqrt {d\,x+1}-1\right )}^{14}}+\frac {d^5\,{\left (\sqrt {1-d\,x}-1\right )}^{16}}{{\left (\sqrt {d\,x+1}-1\right )}^{16}}}-\frac {\mathrm {atan}\left (\frac {\sqrt {1-d\,x}-1}{\sqrt {d\,x+1}-1}\right )\,\left (8\,a^2\,d^4+8\,a\,c\,d^2+4\,b^2\,d^2+3\,c^2\right )}{2\,d^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((((1 - d*x)^(1/2) - 1)^15*((3*c^2)/2 + 2*b^2*d^2 + 4*a*c*d^2))/((d*x + 1)^(1/2) - 1)^15 + (((1 - d*x)^(1/2)
- 1)^3*(6*b^2*d^2 - (23*c^2)/2 + 12*a*c*d^2))/((d*x + 1)^(1/2) - 1)^3 - (((1 - d*x)^(1/2) - 1)^13*(6*b^2*d^2
- (23*c^2)/2 + 12*a*c*d^2))/((d*x + 1)^(1/2) - 1)^13 + (((1 - d*x)^(1/2) - 1)^5*((333*c^2)/2 + 30*b^2*d^2 + 60
*a*c*d^2))/((d*x + 1)^(1/2) - 1)^5 - (((1 - d*x)^(1/2) - 1)^11*((333*c^2)/2 + 30*b^2*d^2 + 60*a*c*d^2))/((d*x
+ 1)^(1/2) - 1)^11 + (((1 - d*x)^(1/2) - 1)^7*(22*b^2*d^2 - (671*c^2)/2 + 44*a*c*d^2))/((d*x + 1)^(1/2) - 1)^7
- (((1 - d*x)^(1/2) - 1)^9*(22*b^2*d^2 - (671*c^2)/2 + 44*a*c*d^2))/((d*x + 1)^(1/2) - 1)^9 + (((1 - d*x)^(1/
2) - 1)^4*(128*b*c*d + 96*a*b*d^3))/((d*x + 1)^(1/2) - 1)^4 + (((1 - d*x)^(1/2) - 1)^12*(128*b*c*d + 96*a*b*d^
3))/((d*x + 1)^(1/2) - 1)^12 + (((1 - d*x)^(1/2) - 1)^8*((256*b*c*d)/3 + 320*a*b*d^3))/((d*x + 1)^(1/2) - 1)^8
+ (((1 - d*x)^(1/2) - 1)^6*((512*b*c*d)/3 + 240*a*b*d^3))/((d*x + 1)^(1/2) - 1)^6 + (((1 - d*x)^(1/2) - 1)^10
*((512*b*c*d)/3 + 240*a*b*d^3))/((d*x + 1)^(1/2) - 1)^10 - (((1 - d*x)^(1/2) - 1)*((3*c^2)/2 + 2*b^2*d^2 + 4*a
*c*d^2))/((d*x + 1)^(1/2) - 1) + (16*a*b*d^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (16*a*b*d^3*((
1 - d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) - 1)^14)/(d^5 + (8*d^5*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)
^2 + (28*d^5*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (56*d^5*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1
/2) - 1)^6 + (70*d^5*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8 + (56*d^5*((1 - d*x)^(1/2) - 1)^10)/((d*
x + 1)^(1/2) - 1)^10 + (28*d^5*((1 - d*x)^(1/2) - 1)^12)/((d*x + 1)^(1/2) - 1)^12 + (8*d^5*((1 - d*x)^(1/2) -
1)^14)/((d*x + 1)^(1/2) - 1)^14 + (d^5*((1 - d*x)^(1/2) - 1)^16)/((d*x + 1)^(1/2) - 1)^16) - (atan(((1 - d*x)^
(1/2) - 1)/((d*x + 1)^(1/2) - 1))*(3*c^2 + 8*a^2*d^4 + 4*b^2*d^2 + 8*a*c*d^2))/(2*d^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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