3.8.92 \(\int \frac {(a+b x+c x^2)^3}{\sqrt {1-d x} \sqrt {1+d x}} \, dx\) [792]

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

[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

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Rubi [A]
time = 0.58, 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 = {913, 1829, 655, 222} \begin {gather*} -\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 {\text {ArcSin}(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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*(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])/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 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 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}{\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*}

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Mathematica [A]
time = 0.73, size = 264, normalized size = 0.81 \begin {gather*} \frac {-d^2 \sqrt {1-d^2 x^2} \left (80 b^3 d^2 \left (2+d^2 x^2\right )+90 b^2 d^2 x \left (4 a d^2+c \left (3+2 d^2 x^2\right )\right )+48 b \left (15 a^2 d^4+10 a c d^2 \left (2+d^2 x^2\right )+c^2 \left (8+4 d^2 x^2+3 d^4 x^4\right )\right )+5 c x \left (72 a^2 d^4+18 a c d^2 \left (3+2 d^2 x^2\right )+c^2 \left (15+10 d^2 x^2+8 d^4 x^4\right )\right )\right )+15 \sqrt {-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+16 a^3 d^6\right ) \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}\right )}{240 d^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(d^2*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*Sqrt[-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)*Log[-(Sqrt[-d^2]*x) + Sqrt[1 - d^2*x^2]])/(240*d^8)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.17, size = 602, normalized size = 1.86

method	result	size
risch	\(\frac {\left (40 c^{3} x^{5} d^{4}+144 b \,c^{2} x^{4} d^{4}+180 a \,c^{2} d^{4} x^{3}+180 b^{2} c \,d^{4} x^{3}+480 a b c \,d^{4} x^{2}+80 b^{3} d^{4} x^{2}+360 a^{2} c \,d^{4} x +360 a \,b^{2} d^{4} x +50 c^{3} d^{2} x^{3}+720 a^{2} b \,d^{4}+192 b \,c^{2} d^{2} x^{2}+270 a \,c^{2} d^{2} x +270 b^{2} c \,d^{2} x +960 a b c \,d^{2}+160 b^{3} d^{2}+75 c^{3} x +384 b \,c^{2}\right ) \left (d x -1\right ) \sqrt {d x +1}\, \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{240 d^{6} \sqrt {-\left (d x -1\right ) \left (d x +1\right )}\, \sqrt {-d x +1}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) a^{3}}{\sqrt {d^{2}}}+\frac {3 \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) a^{2} c}{2 d^{2} \sqrt {d^{2}}}+\frac {3 \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,b^{2}}{2 d^{2} \sqrt {d^{2}}}+\frac {9 \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,c^{2}}{8 d^{4} \sqrt {d^{2}}}+\frac {9 \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) b^{2} c}{8 d^{4} \sqrt {d^{2}}}+\frac {5 \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) c^{3}}{16 d^{6} \sqrt {d^{2}}}\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{\sqrt {-d x +1}\, \sqrt {d x +1}}\)	\(454\)
default	\(-\frac {\sqrt {-d x +1}\, \sqrt {d x +1}\, \left (40 \,\mathrm {csgn}\left (d \right ) c^{3} d^{5} x^{5} \sqrt {-d^{2} x^{2}+1}+144 \,\mathrm {csgn}\left (d \right ) b \,c^{2} d^{5} x^{4} \sqrt {-d^{2} x^{2}+1}+180 \,\mathrm {csgn}\left (d \right ) a \,c^{2} d^{5} x^{3} \sqrt {-d^{2} x^{2}+1}+180 \,\mathrm {csgn}\left (d \right ) b^{2} c \,d^{5} x^{3} \sqrt {-d^{2} x^{2}+1}+480 \,\mathrm {csgn}\left (d \right ) a b c \,d^{5} x^{2} \sqrt {-d^{2} x^{2}+1}+80 \,\mathrm {csgn}\left (d \right ) b^{3} d^{5} x^{2} \sqrt {-d^{2} x^{2}+1}+360 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{5} a^{2} c x +360 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{5} a \,b^{2} x +50 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{3} c^{3} x^{3}+720 \,\mathrm {csgn}\left (d \right ) d^{5} \sqrt {-d^{2} x^{2}+1}\, a^{2} b +192 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{3} b \,c^{2} x^{2}-240 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a^{3} d^{6}+270 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{3} a \,c^{2} x +270 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d^{3} b^{2} c x +960 \,\mathrm {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, a b c +160 \,\mathrm {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, b^{3}-360 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a^{2} c \,d^{4}-360 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,b^{2} d^{4}+75 \sqrt {-d^{2} x^{2}+1}\, \mathrm {csgn}\left (d \right ) d \,c^{3} x +384 \,\mathrm {csgn}\left (d \right ) d \sqrt {-d^{2} x^{2}+1}\, b \,c^{2}-270 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,c^{2} d^{2}-270 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) b^{2} c \,d^{2}-75 \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) c^{3}\right ) \mathrm {csgn}\left (d \right )}{240 d^{7} \sqrt {-d^{2} x^{2}+1}}\)	\(602\)

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,method=_RETURNVERBOSE)

[Out]

-1/240*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(40*csgn(d)*c^3*d^5*x^5*(-d^2*x^2+1)^(1/2)+144*csgn(d)*b*c^2*d^5*x^4*(-d^2
*x^2+1)^(1/2)+180*csgn(d)*a*c^2*d^5*x^3*(-d^2*x^2+1)^(1/2)+180*csgn(d)*b^2*c*d^5*x^3*(-d^2*x^2+1)^(1/2)+480*cs
gn(d)*a*b*c*d^5*x^2*(-d^2*x^2+1)^(1/2)+80*csgn(d)*b^3*d^5*x^2*(-d^2*x^2+1)^(1/2)+360*(-d^2*x^2+1)^(1/2)*csgn(d
)*d^5*a^2*c*x+360*(-d^2*x^2+1)^(1/2)*csgn(d)*d^5*a*b^2*x+50*(-d^2*x^2+1)^(1/2)*csgn(d)*d^3*c^3*x^3+720*csgn(d)
*d^5*(-d^2*x^2+1)^(1/2)*a^2*b+192*(-d^2*x^2+1)^(1/2)*csgn(d)*d^3*b*c^2*x^2-240*arctan(csgn(d)*d*x/(-d^2*x^2+1)
^(1/2))*a^3*d^6+270*(-d^2*x^2+1)^(1/2)*csgn(d)*d^3*a*c^2*x+270*(-d^2*x^2+1)^(1/2)*csgn(d)*d^3*b^2*c*x+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*(-d^2*x^2+1)^(1/2)*csgn(d)*d*c^3*x+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 = 0.50, size = 365, normalized size = 1.13 \begin {gather*} -\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}} \end {gather*}

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

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Fricas [A]
time = 1.72, size = 251, normalized size = 0.77 \begin {gather*} -\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 {gather*}

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)] 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)**(1/2)/(d*x+1)**(1/2),x)

[Out]

Timed out

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Giac [A]
time = 5.79, size = 353, normalized size = 1.09 \begin {gather*} -\frac {{\left (720 \, a^{2} b d^{5} - 360 \, a b^{2} d^{4} - 360 \, a^{2} c d^{4} + 240 \, b^{3} d^{3} + 1440 \, a b c d^{3} - 450 \, b^{2} c d^{2} - 450 \, a c^{2} d^{2} + 720 \, b c^{2} d - 165 \, c^{3} + {\left (360 \, a b^{2} d^{4} + 360 \, a^{2} c d^{4} - 160 \, b^{3} d^{3} - 960 \, a b c d^{3} + 810 \, b^{2} c d^{2} + 810 \, a c^{2} d^{2} - 960 \, b c^{2} d + 425 \, c^{3} + 2 \, {\left (40 \, b^{3} d^{3} + 240 \, a b c d^{3} - 270 \, b^{2} c d^{2} - 270 \, a c^{2} d^{2} + 528 \, b c^{2} d - 275 \, c^{3} + {\left (90 \, b^{2} c d^{2} + 90 \, a c^{2} d^{2} - 288 \, b c^{2} d + 225 \, c^{3} + 4 \, {\left (5 \, {\left (d x + 1\right )} c^{3} + 18 \, b c^{2} d - 25 \, c^{3}\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^{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 )}{240 \, d^{7}} \end {gather*}

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*((720*a^2*b*d^5 - 360*a*b^2*d^4 - 360*a^2*c*d^4 + 240*b^3*d^3 + 1440*a*b*c*d^3 - 450*b^2*c*d^2 - 450*a*
c^2*d^2 + 720*b*c^2*d - 165*c^3 + (360*a*b^2*d^4 + 360*a^2*c*d^4 - 160*b^3*d^3 - 960*a*b*c*d^3 + 810*b^2*c*d^2
+ 810*a*c^2*d^2 - 960*b*c^2*d + 425*c^3 + 2*(40*b^3*d^3 + 240*a*b*c*d^3 - 270*b^2*c*d^2 - 270*a*c^2*d^2 + 528
*b*c^2*d - 275*c^3 + (90*b^2*c*d^2 + 90*a*c^2*d^2 - 288*b*c^2*d + 225*c^3 + 4*(5*(d*x + 1)*c^3 + 18*b*c^2*d -
25*c^3)*(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^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)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))/d^7

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Mupad [B]
time = 31.33, size = 1768, normalized size = 5.46 \begin {gather*} \text {Too large to display} \end {gather*}

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)

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