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

Optimal. Leaf size=276 \[ \frac{x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac{6 a c}{d^2}+\frac{b^2}{d^2}+\frac{3 c^2}{d^4}\right )}{d^6 \sqrt{1-d^2 x^2}}-\frac{3 \sin ^{-1}(d x) \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )}{8 d^7}+\frac{c x \sqrt{1-d^2 x^2} \left (12 a c d^2+12 b^2 d^2+7 c^2\right )}{8 d^6}+\frac{b \sqrt{1-d^2 x^2} \left (6 a c d^2+b^2 d^2+5 c^2\right )}{d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4} \]

[Out]

(b*(3*a^2 + (3*c^2)/d^4 + b^2/d^2 + (6*a*c)/d^2)*d^4 + (c + a*d^2)*(c^2 + 3*b^2*d^2 + 2*a*c*d^2 + a^2*d^4)*x)/
(d^6*Sqrt[1 - d^2*x^2]) + (b*(5*c^2 + b^2*d^2 + 6*a*c*d^2)*Sqrt[1 - d^2*x^2])/d^6 + (c*(7*c^2 + 12*b^2*d^2 + 1
2*a*c*d^2)*x*Sqrt[1 - d^2*x^2])/(8*d^6) + (b*c^2*x^2*Sqrt[1 - d^2*x^2])/d^4 + (c^3*x^3*Sqrt[1 - d^2*x^2])/(4*d
^4) - (3*(5*c^3 + 12*b^2*c*d^2 + 12*a*c^2*d^2 + 8*a*b^2*d^4 + 8*a^2*c*d^4)*ArcSin[d*x])/(8*d^7)

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Rubi [A]  time = 0.597837, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {899, 1814, 1815, 641, 216} \[ \frac{x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac{6 a c}{d^2}+\frac{b^2}{d^2}+\frac{3 c^2}{d^4}\right )}{d^6 \sqrt{1-d^2 x^2}}-\frac{3 \sin ^{-1}(d x) \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )}{8 d^7}+\frac{c x \sqrt{1-d^2 x^2} \left (12 a c d^2+12 b^2 d^2+7 c^2\right )}{8 d^6}+\frac{b \sqrt{1-d^2 x^2} \left (6 a c d^2+b^2 d^2+5 c^2\right )}{d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(3*a^2 + (3*c^2)/d^4 + b^2/d^2 + (6*a*c)/d^2)*d^4 + (c + a*d^2)*(c^2 + 3*b^2*d^2 + 2*a*c*d^2 + a^2*d^4)*x)/
(d^6*Sqrt[1 - d^2*x^2]) + (b*(5*c^2 + b^2*d^2 + 6*a*c*d^2)*Sqrt[1 - d^2*x^2])/d^6 + (c*(7*c^2 + 12*b^2*d^2 + 1
2*a*c*d^2)*x*Sqrt[1 - d^2*x^2])/(8*d^6) + (b*c^2*x^2*Sqrt[1 - d^2*x^2])/d^4 + (c^3*x^3*Sqrt[1 - d^2*x^2])/(4*d
^4) - (3*(5*c^3 + 12*b^2*c*d^2 + 12*a*c^2*d^2 + 8*a*b^2*d^4 + 8*a^2*c*d^4)*ArcSin[d*x])/(8*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 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

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}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx &=\int \frac{\left (a+b x+c x^2\right )^3}{\left (1-d^2 x^2\right )^{3/2}} \, dx\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}-\int \frac{\frac{c^3+3 a c^2 d^2+3 a b^2 d^4+3 c d^2 \left (b^2+a^2 d^2\right )}{d^6}+\frac{b \left (b^2+3 c \left (2 a+\frac{c}{d^2}\right )\right ) x}{d^2}+\frac{c \left (3 b^2+c \left (3 a+\frac{c}{d^2}\right )\right ) x^2}{d^2}+\frac{3 b c^2 x^3}{d^2}+\frac{c^3 x^4}{d^2}}{\sqrt{1-d^2 x^2}} \, dx\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}+\frac{\int \frac{-\frac{4 \left (c^3+3 a c^2 d^2+3 a b^2 d^4+3 c d^2 \left (b^2+a^2 d^2\right )\right )}{d^4}-4 b \left (b^2+3 c \left (2 a+\frac{c}{d^2}\right )\right ) x-c \left (12 b^2+c \left (12 a+\frac{7 c}{d^2}\right )\right ) x^2-12 b c^2 x^3}{\sqrt{1-d^2 x^2}} \, dx}{4 d^2}\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}-\frac{\int \frac{12 \left (3 b^2 \left (c+a d^2\right )+c \left (3 a c+\frac{c^2}{d^2}+3 a^2 d^2\right )\right )+12 b \left (5 c^2+b^2 d^2+6 a c d^2\right ) x+3 c \left (7 c^2+12 b^2 d^2+12 a c d^2\right ) x^2}{\sqrt{1-d^2 x^2}} \, dx}{12 d^4}\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}+\frac{c \left (7 c^2+12 b^2 d^2+12 a c d^2\right ) x \sqrt{1-d^2 x^2}}{8 d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}+\frac{\int \frac{-9 \left (5 c^3+12 b^2 c d^2+12 a c^2 d^2+8 a b^2 d^4+8 a^2 c d^4\right )-24 b d^2 \left (5 c^2+b^2 d^2+6 a c d^2\right ) x}{\sqrt{1-d^2 x^2}} \, dx}{24 d^6}\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}+\frac{b \left (5 c^2+b^2 d^2+6 a c d^2\right ) \sqrt{1-d^2 x^2}}{d^6}+\frac{c \left (7 c^2+12 b^2 d^2+12 a c d^2\right ) x \sqrt{1-d^2 x^2}}{8 d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}-\frac{\left (3 \left (5 c^3+12 b^2 c d^2+12 a c^2 d^2+8 a b^2 d^4+8 a^2 c d^4\right )\right ) \int \frac{1}{\sqrt{1-d^2 x^2}} \, dx}{8 d^6}\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}+\frac{b \left (5 c^2+b^2 d^2+6 a c d^2\right ) \sqrt{1-d^2 x^2}}{d^6}+\frac{c \left (7 c^2+12 b^2 d^2+12 a c d^2\right ) x \sqrt{1-d^2 x^2}}{8 d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}-\frac{3 \left (5 c^3+12 b^2 c d^2+12 a c^2 d^2+8 a b^2 d^4+8 a^2 c d^4\right ) \sin ^{-1}(d x)}{8 d^7}\\ \end{align*}

Mathematica [A]  time = 0.24352, size = 239, normalized size = 0.87 \[ \frac{-3 \sqrt{1-d^2 x^2} \sin ^{-1}(d x) \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )-8 b \left (-3 a^2 d^5+6 a c d^3 \left (d^2 x^2-2\right )+c^2 d \left (d^4 x^4+4 d^2 x^2-8\right )\right )+d x \left (24 a^2 c d^4+8 a^3 d^6-12 a c^2 d^2 \left (d^2 x^2-3\right )+c^3 \left (-2 d^4 x^4-5 d^2 x^2+15\right )\right )-12 b^2 d^3 x \left (c \left (d^2 x^2-3\right )-2 a d^2\right )-8 b^3 d^3 \left (d^2 x^2-2\right )}{8 d^7 \sqrt{1-d^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b^3*d^3*(-2 + d^2*x^2) - 12*b^2*d^3*x*(-2*a*d^2 + c*(-3 + d^2*x^2)) + d*x*(24*a^2*c*d^4 + 8*a^3*d^6 - 12*a
*c^2*d^2*(-3 + d^2*x^2) + c^3*(15 - 5*d^2*x^2 - 2*d^4*x^4)) - 8*b*(-3*a^2*d^5 + 6*a*c*d^3*(-2 + d^2*x^2) + c^2
*d*(-8 + 4*d^2*x^2 + d^4*x^4)) - 3*(5*c^3 + 12*b^2*c*d^2 + 12*a*c^2*d^2 + 8*a*b^2*d^4 + 8*a^2*c*d^4)*Sqrt[1 -
d^2*x^2]*ArcSin[d*x])/(8*d^7*Sqrt[1 - d^2*x^2])

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Maple [C]  time = 0.225, size = 755, 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]

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

[Out]

1/8*(-d*x+1)^(1/2)*(15*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*c^3-15*csgn(d)*d*(-d^2*x^2+1)^(1/2)*x*c^3-64*csg
n(d)*d*(-d^2*x^2+1)^(1/2)*b*c^2+2*csgn(d)*x^5*c^3*d^5*(-d^2*x^2+1)^(1/2)+48*csgn(d)*x^2*a*b*c*d^5*(-d^2*x^2+1)
^(1/2)-8*csgn(d)*d^7*(-d^2*x^2+1)^(1/2)*x*a^3-24*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*x^2*a^2*c*d^6-24*arcta
n(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*x^2*a*b^2*d^6-36*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*x^2*a*c^2*d^4-36*arc
tan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*x^2*b^2*c*d^4-15*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*x^2*c^3*d^2+8*csgn
(d)*x^2*b^3*d^5*(-d^2*x^2+1)^(1/2)+5*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x^3*c^3-24*csgn(d)*d^5*(-d^2*x^2+1)^(1/2)*
a^2*b+36*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a*c^2*d^2+36*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*b^2*c*d^2-
16*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*b^3+24*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a^2*c*d^4+24*arctan(csgn(d)*d*
x/(-d^2*x^2+1)^(1/2))*a*b^2*d^4+8*csgn(d)*x^4*b*c^2*d^5*(-d^2*x^2+1)^(1/2)+12*csgn(d)*x^3*a*c^2*d^5*(-d^2*x^2+
1)^(1/2)+12*csgn(d)*x^3*b^2*c*d^5*(-d^2*x^2+1)^(1/2)-24*csgn(d)*d^5*(-d^2*x^2+1)^(1/2)*x*a^2*c-24*csgn(d)*d^5*
(-d^2*x^2+1)^(1/2)*x*a*b^2+32*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x^2*b*c^2-36*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x*a*c
^2-36*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x*b^2*c-96*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*a*b*c)*csgn(d)/(d*x-1)/(-d^2*x^
2+1)^(1/2)/d^7/(d*x+1)^(1/2)

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Maxima [A]  time = 1.67893, size = 549, normalized size = 1.99 \begin{align*} -\frac{c^{3} x^{5}}{4 \, \sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{b c^{2} x^{4}}{\sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{a^{3} x}{\sqrt{-d^{2} x^{2} + 1}} - \frac{5 \, c^{3} x^{3}}{8 \, \sqrt{-d^{2} x^{2} + 1} d^{4}} - \frac{3 \,{\left (b^{2} c + a c^{2}\right )} x^{3}}{2 \, \sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{3 \, a^{2} b}{\sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{4 \, b c^{2} x^{2}}{\sqrt{-d^{2} x^{2} + 1} d^{4}} - \frac{{\left (b^{3} + 6 \, a b c\right )} x^{2}}{\sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{3 \,{\left (a b^{2} + a^{2} c\right )} x}{\sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{3 \,{\left (a b^{2} + a^{2} c\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}} d^{2}} + \frac{15 \, c^{3} x}{8 \, \sqrt{-d^{2} x^{2} + 1} d^{6}} + \frac{9 \,{\left (b^{2} c + a c^{2}\right )} x}{2 \, \sqrt{-d^{2} x^{2} + 1} d^{4}} - \frac{15 \, c^{3} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \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 )}{2 \, \sqrt{d^{2}} d^{4}} + \frac{8 \, b c^{2}}{\sqrt{-d^{2} x^{2} + 1} d^{6}} + \frac{2 \,{\left (b^{3} + 6 \, a b c\right )}}{\sqrt{-d^{2} x^{2} + 1} 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)^(3/2)/(d*x+1)^(3/2),x, algorithm="maxima")

[Out]

-1/4*c^3*x^5/(sqrt(-d^2*x^2 + 1)*d^2) - b*c^2*x^4/(sqrt(-d^2*x^2 + 1)*d^2) + a^3*x/sqrt(-d^2*x^2 + 1) - 5/8*c^
3*x^3/(sqrt(-d^2*x^2 + 1)*d^4) - 3/2*(b^2*c + a*c^2)*x^3/(sqrt(-d^2*x^2 + 1)*d^2) + 3*a^2*b/(sqrt(-d^2*x^2 + 1
)*d^2) - 4*b*c^2*x^2/(sqrt(-d^2*x^2 + 1)*d^4) - (b^3 + 6*a*b*c)*x^2/(sqrt(-d^2*x^2 + 1)*d^2) + 3*(a*b^2 + a^2*
c)*x/(sqrt(-d^2*x^2 + 1)*d^2) - 3*(a*b^2 + a^2*c)*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^2) + 15/8*c^3*x/(sqrt(-
d^2*x^2 + 1)*d^6) + 9/2*(b^2*c + a*c^2)*x/(sqrt(-d^2*x^2 + 1)*d^4) - 15/8*c^3*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^
2)*d^6) - 9/2*(b^2*c + a*c^2)*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^4) + 8*b*c^2/(sqrt(-d^2*x^2 + 1)*d^6) + 2*(
b^3 + 6*a*b*c)/(sqrt(-d^2*x^2 + 1)*d^4)

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Fricas [A]  time = 1.71622, size = 802, normalized size = 2.91 \begin{align*} -\frac{24 \, a^{2} b d^{5} + 64 \, b c^{2} d + 16 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} - 8 \,{\left (3 \, a^{2} b d^{7} + 8 \, b c^{2} d^{3} + 2 \,{\left (b^{3} + 6 \, a b c\right )} d^{5}\right )} x^{2} -{\left (2 \, c^{3} d^{5} x^{5} + 8 \, b c^{2} d^{5} x^{4} - 24 \, a^{2} b d^{5} - 64 \, b c^{2} d - 16 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} +{\left (5 \, c^{3} d^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{5}\right )} x^{3} + 8 \,{\left (4 \, b c^{2} d^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{5}\right )} x^{2} -{\left (8 \, a^{3} d^{7} + 24 \,{\left (a b^{2} + a^{2} c\right )} d^{5} + 15 \, c^{3} d + 36 \,{\left (b^{2} c + a c^{2}\right )} d^{3}\right )} x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 6 \,{\left (8 \,{\left (a b^{2} + a^{2} c\right )} d^{4} + 5 \, c^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} -{\left (8 \,{\left (a b^{2} + a^{2} c\right )} d^{6} + 5 \, c^{3} d^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{8 \,{\left (d^{9} x^{2} - d^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(24*a^2*b*d^5 + 64*b*c^2*d + 16*(b^3 + 6*a*b*c)*d^3 - 8*(3*a^2*b*d^7 + 8*b*c^2*d^3 + 2*(b^3 + 6*a*b*c)*d^
5)*x^2 - (2*c^3*d^5*x^5 + 8*b*c^2*d^5*x^4 - 24*a^2*b*d^5 - 64*b*c^2*d - 16*(b^3 + 6*a*b*c)*d^3 + (5*c^3*d^3 +
12*(b^2*c + a*c^2)*d^5)*x^3 + 8*(4*b*c^2*d^3 + (b^3 + 6*a*b*c)*d^5)*x^2 - (8*a^3*d^7 + 24*(a*b^2 + a^2*c)*d^5
+ 15*c^3*d + 36*(b^2*c + a*c^2)*d^3)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 6*(8*(a*b^2 + a^2*c)*d^4 + 5*c^3 + 12*(
b^2*c + a*c^2)*d^2 - (8*(a*b^2 + a^2*c)*d^6 + 5*c^3*d^2 + 12*(b^2*c + a*c^2)*d^4)*x^2)*arctan((sqrt(d*x + 1)*s
qrt(-d*x + 1) - 1)/(d*x)))/(d^9*x^2 - 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)**(3/2)/(d*x+1)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.57657, size = 961, normalized size = 3.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/86016*(8*a*b^2*d^39 + 8*a^2*c*d^39 + 12*b^2*c*d^37 + 12*a*c^2*d^37 + 5*c^3*d^35)*arcsin(1/2*sqrt(2)*sqrt(d*
x + 1)) - 1/516096*(4*a^3*d^41 + 12*a^2*b*d^40 + 12*a*b^2*d^39 + 12*a^2*c*d^39 + 20*b^3*d^38 + 120*a*b*c*d^38
- 12*b^2*c*d^37 - 12*a*c^2*d^37 + 108*b*c^2*d^36 - 14*c^3*d^35 - (8*b^3*d^38 + 48*a*b*c*d^38 - 36*b^2*c*d^37 -
36*a*c^2*d^37 + 80*b*c^2*d^36 - 35*c^3*d^35 + (12*b^2*c*d^37 + 12*a*c^2*d^37 - 32*b*c^2*d^36 + 25*c^3*d^35 +
2*((d*x + 1)*c^3*d^35 + 4*b*c^2*d^36 - 5*c^3*d^35)*(d*x + 1))*(d*x + 1))*(d*x + 1))*sqrt(d*x + 1)*sqrt(-d*x +
1)/(d*x - 1) + 1/4*(a^3*d^6*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 3*a^2*b*d^5*(sqrt(2) - sqrt(-d*x + 1))/
sqrt(d*x + 1) + 3*a*b^2*d^4*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + 3*a^2*c*d^4*(sqrt(2) - sqrt(-d*x + 1))/
sqrt(d*x + 1) - b^3*d^3*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 6*a*b*c*d^3*(sqrt(2) - sqrt(-d*x + 1))/sqrt
(d*x + 1) + 3*b^2*c*d^2*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + 3*a*c^2*d^2*(sqrt(2) - sqrt(-d*x + 1))/sqrt
(d*x + 1) - 3*b*c^2*d*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + c^3*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1))
/d^7 - 1/4*(a^3*d^6 - 3*a^2*b*d^5 + 3*a*b^2*d^4 + 3*a^2*c*d^4 - b^3*d^3 - 6*a*b*c*d^3 + 3*b^2*c*d^2 + 3*a*c^2*
d^2 - 3*b*c^2*d + c^3)*sqrt(d*x + 1)/(d^7*(sqrt(2) - sqrt(-d*x + 1)))