3.8.97 \(\int \frac {(a+b x+c x^2)^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx\) [797]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.37, antiderivative size = 276, normalized size of antiderivative = 1.00, 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 = {913, 1828, 1829, 655, 222} \begin {gather*} -\frac {3 \text {ArcSin}(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 {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 {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} \end {gather*}

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 222

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 655

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 913

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 1828

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 1829

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}{(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 = 0.91, size = 261, normalized size = 0.95 \begin {gather*} -\frac {\frac {d^2 \left (8 b^3 d^2 \left (-2+d^2 x^2\right )+12 b^2 d^2 x \left (-2 a d^2+c \left (-3+d^2 x^2\right )\right )+8 b \left (-3 a^2 d^4+6 a c d^2 \left (-2+d^2 x^2\right )+c^2 \left (-8+4 d^2 x^2+d^4 x^4\right )\right )+x \left (-24 a^2 c d^4-8 a^3 d^6+12 a c^2 d^2 \left (-3+d^2 x^2\right )+c^3 \left (-15+5 d^2 x^2+2 d^4 x^4\right )\right )\right )}{\sqrt {1-d^2 x^2}}+3 \sqrt {-d^2} \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 ) \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}\right )}{8 d^8} \end {gather*}

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]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.18, size = 755, normalized size = 2.74

method	result	size
default	\(-\frac {\sqrt {-d x +1}\, \left (96 \,\mathrm {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, a b c +24 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{5} a^{2} c x +24 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{5} a \,b^{2} x +36 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{3} a \,c^{2} x +36 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{3} b^{2} c x -15 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) c^{3}+16 \,\mathrm {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, b^{3}-36 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) b^{2} c \,d^{2}-24 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a^{2} c \,d^{4}-24 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,b^{2} d^{4}-36 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,c^{2} d^{2}+15 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) c^{3} d^{2} x^{2}-8 \,\mathrm {csgn}\left (d \right ) b \,c^{2} d^{5} x^{4} \sqrt {-d^{2} x^{2}+1}-12 \,\mathrm {csgn}\left (d \right ) a \,c^{2} d^{5} x^{3} \sqrt {-d^{2} x^{2}+1}-12 \,\mathrm {csgn}\left (d \right ) b^{2} c \,d^{5} x^{3} \sqrt {-d^{2} x^{2}+1}-32 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{3} b \,c^{2} x^{2}-48 \,\mathrm {csgn}\left (d \right ) a b c \,d^{5} x^{2} \sqrt {-d^{2} x^{2}+1}-8 \,\mathrm {csgn}\left (d \right ) b^{3} d^{5} x^{2} \sqrt {-d^{2} x^{2}+1}-5 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{3} c^{3} x^{3}+8 \,\mathrm {csgn}\left (d \right ) d^{7} \sqrt {-d^{2} x^{2}+1}\, a^{3} x -2 \,\mathrm {csgn}\left (d \right ) c^{3} d^{5} x^{5} \sqrt {-d^{2} x^{2}+1}+64 \,\mathrm {csgn}\left (d \right ) d \sqrt {-d^{2} x^{2}+1}\, b \,c^{2}+24 \,\mathrm {csgn}\left (d \right ) d^{5} \sqrt {-d^{2} x^{2}+1}\, a^{2} b +36 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,c^{2} d^{4} x^{2}+36 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) b^{2} c \,d^{4} x^{2}+24 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a^{2} c \,d^{6} x^{2}+24 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,b^{2} d^{6} x^{2}+15 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d \,c^{3} x \right ) \mathrm {csgn}\left (d \right )}{8 \left (d x -1\right ) \sqrt {-d^{2} x^{2}+1}\, d^{7} \sqrt {d x +1}}\)	\(755\)
risch	\(\text {Expression too large to display}\)	\(1160\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 371, normalized size = 1.34 \begin {gather*} -\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 (d x\right )}{d^{3}} + \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 (d x\right )}{8 \, d^{7}} - \frac {9 \, {\left (b^{2} c + a c^{2}\right )} \arcsin \left (d x\right )}{2 \, d^{5}} + \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 {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 1.89, size = 376, normalized size = 1.36 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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)**3/(-d*x+1)**(3/2)/(d*x+1)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (260) = 520\).
time = 5.44, size = 736, normalized size = 2.67 \begin {gather*} \frac {\frac {{\left ({\left ({\left (d x + 1\right )} {\left (2 \, {\left (d x + 1\right )} {\left (\frac {{\left (d x + 1\right )} c^{3}}{d^{6}} + \frac {4 \, b c^{2} d^{31} - 5 \, c^{3} d^{30}}{d^{36}}\right )} + \frac {12 \, b^{2} c d^{32} + 12 \, a c^{2} d^{32} - 32 \, b c^{2} d^{31} + 25 \, c^{3} d^{30}}{d^{36}}\right )} + \frac {8 \, b^{3} d^{33} + 48 \, a b c d^{33} - 36 \, b^{2} c d^{32} - 36 \, a c^{2} d^{32} + 80 \, b c^{2} d^{31} - 35 \, c^{3} d^{30}}{d^{36}}\right )} {\left (d x + 1\right )} - \frac {2 \, {\left (2 \, a^{3} d^{36} + 6 \, a^{2} b d^{35} + 6 \, a b^{2} d^{34} + 6 \, a^{2} c d^{34} + 10 \, b^{3} d^{33} + 60 \, a b c d^{33} - 6 \, b^{2} c d^{32} - 6 \, a c^{2} d^{32} + 54 \, b c^{2} d^{31} - 7 \, c^{3} d^{30}\right )}}{d^{36}}\right )} \sqrt {d x + 1} \sqrt {-d x + 1}}{d x - 1} - \frac {6 \, {\left (8 \, a b^{2} d^{4} + 8 \, a^{2} c d^{4} + 12 \, b^{2} c d^{2} + 12 \, a c^{2} d^{2} + 5 \, c^{3}\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{6}} + \frac {2 \, {\left (\frac {a^{3} d^{6} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {3 \, a^{2} b d^{5} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {3 \, a b^{2} d^{4} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {3 \, a^{2} c d^{4} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {b^{3} d^{3} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {6 \, a b c d^{3} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {3 \, b^{2} c d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {3 \, a c^{2} d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {3 \, b c^{2} d {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {c^{3} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}}\right )}}{d^{6}} - \frac {2 \, {\left (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}\right )} \sqrt {d x + 1}}{d^{6} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}}{8 \, d} \end {gather*}

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]

1/8*((((d*x + 1)*(2*(d*x + 1)*((d*x + 1)*c^3/d^6 + (4*b*c^2*d^31 - 5*c^3*d^30)/d^36) + (12*b^2*c*d^32 + 12*a*c
^2*d^32 - 32*b*c^2*d^31 + 25*c^3*d^30)/d^36) + (8*b^3*d^33 + 48*a*b*c*d^33 - 36*b^2*c*d^32 - 36*a*c^2*d^32 + 8
0*b*c^2*d^31 - 35*c^3*d^30)/d^36)*(d*x + 1) - 2*(2*a^3*d^36 + 6*a^2*b*d^35 + 6*a*b^2*d^34 + 6*a^2*c*d^34 + 10*
b^3*d^33 + 60*a*b*c*d^33 - 6*b^2*c*d^32 - 6*a*c^2*d^32 + 54*b*c^2*d^31 - 7*c^3*d^30)/d^36)*sqrt(d*x + 1)*sqrt(
-d*x + 1)/(d*x - 1) - 6*(8*a*b^2*d^4 + 8*a^2*c*d^4 + 12*b^2*c*d^2 + 12*a*c^2*d^2 + 5*c^3)*arcsin(1/2*sqrt(2)*s
qrt(d*x + 1))/d^6 + 2*(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))/s
qrt(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))/s
qrt(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^6 - 2*(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^6*(sqrt(2) - sqrt(-d*x + 1))))/d

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^3}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________