3.109 \(\int \frac {(d+e x+f x^2)^3}{\sqrt {a+b x+c x^2}} \, dx\)

Optimal. Leaf size=717 \[ \frac {x \sqrt {a+b x+c x^2} \left (24 c^2 f \left (50 a^2 f^2+322 a b e f+175 b^2 \left (d f+e^2\right )\right )-252 b^2 c f^2 (14 a f+15 b e)-160 c^3 \left (27 a f \left (d f+e^2\right )+10 b \left (6 d e f+e^3\right )\right )+1155 b^4 f^3+5760 c^4 d \left (d f+e^2\right )\right )}{3840 c^5}+\frac {\sqrt {a+b x+c x^2} \left (96 c^3 \left (128 a^2 e f^2+275 a b f \left (d f+e^2\right )+50 b^2 \left (6 d e f+e^3\right )\right )-504 b c^2 f \left (22 a^2 f^2+70 a b e f+25 b^2 \left (d f+e^2\right )\right )+420 b^3 c f^2 (34 a f+27 b e)-640 c^4 \left (8 a e \left (6 d f+e^2\right )+27 b d \left (d f+e^2\right )\right )-3465 b^5 f^3+23040 c^5 d^2 e\right )}{7680 c^6}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (384 c^4 \left (3 a^2 f \left (d f+e^2\right )+2 a b e \left (6 d f+e^2\right )+3 b^2 d \left (d f+e^2\right )\right )+840 b^2 c^2 f \left (2 a^2 f^2+4 a b e f+b^2 \left (d f+e^2\right )\right )-320 c^3 \left (a^3 f^3+9 a^2 b e f^2+9 a b^2 f \left (d f+e^2\right )+b^3 \left (6 d e f+e^3\right )\right )-252 b^4 c f^2 (5 a f+3 b e)-1536 c^5 d \left (a \left (d f+e^2\right )+b d e\right )+231 b^6 f^3+1024 c^6 d^3\right )}{1024 c^{13/2}}-\frac {x^2 \sqrt {a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{960 c^4}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{480 c^3}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c} \]

[Out]

1/1024*(1024*c^6*d^3+231*b^6*f^3-252*b^4*c*f^2*(5*a*f+3*b*e)-1536*c^5*d*(b*d*e+a*(d*f+e^2))+840*b^2*c^2*f*(4*a
*b*e*f+2*a^2*f^2+b^2*(d*f+e^2))+384*c^4*(3*b^2*d*(d*f+e^2)+3*a^2*f*(d*f+e^2)+2*a*b*e*(6*d*f+e^2))-320*c^3*(9*a
^2*b*e*f^2+a^3*f^3+9*a*b^2*f*(d*f+e^2)+b^3*(6*d*e*f+e^3)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/
c^(13/2)+1/7680*(23040*c^5*d^2*e-3465*b^5*f^3+420*b^3*c*f^2*(34*a*f+27*b*e)-504*b*c^2*f*(70*a*b*e*f+22*a^2*f^2
+25*b^2*(d*f+e^2))-640*c^4*(27*b*d*(d*f+e^2)+8*a*e*(6*d*f+e^2))+96*c^3*(128*a^2*e*f^2+275*a*b*f*(d*f+e^2)+50*b
^2*(6*d*e*f+e^3)))*(c*x^2+b*x+a)^(1/2)/c^6+1/3840*(1155*b^4*f^3-252*b^2*c*f^2*(14*a*f+15*b*e)+5760*c^4*d*(d*f+
e^2)+24*c^2*f*(322*a*b*e*f+50*a^2*f^2+175*b^2*(d*f+e^2))-160*c^3*(27*a*f*(d*f+e^2)+10*b*(6*d*e*f+e^3)))*x*(c*x
^2+b*x+a)^(1/2)/c^5-1/960*(231*b^3*f^3-36*b*c*f^2*(13*a*f+21*b*e)-320*c^3*(6*d*e*f+e^3)+24*c^2*f*(32*a*e*f+35*
b*(d*f+e^2)))*x^2*(c*x^2+b*x+a)^(1/2)/c^4+1/480*f*(99*b^2*f^2-4*c*f*(25*a*f+81*b*e)+360*c^2*(d*f+e^2))*x^3*(c*
x^2+b*x+a)^(1/2)/c^3+1/60*f^2*(-11*b*f+36*c*e)*x^4*(c*x^2+b*x+a)^(1/2)/c^2+1/6*f^3*x^5*(c*x^2+b*x+a)^(1/2)/c

________________________________________________________________________________________

Rubi [A]  time = 2.71, antiderivative size = 717, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1661, 640, 621, 206} \[ \frac {\sqrt {a+b x+c x^2} \left (-504 b c^2 f \left (22 a^2 f^2+70 a b e f+25 b^2 \left (d f+e^2\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (d f+e^2\right )+50 b^2 \left (6 d e f+e^3\right )\right )+420 b^3 c f^2 (34 a f+27 b e)-640 c^4 \left (8 a e \left (6 d f+e^2\right )+27 b d \left (d f+e^2\right )\right )-3465 b^5 f^3+23040 c^5 d^2 e\right )}{7680 c^6}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (840 b^2 c^2 f \left (2 a^2 f^2+4 a b e f+b^2 \left (d f+e^2\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (d f+e^2\right )+b^3 \left (6 d e f+e^3\right )\right )+384 c^4 \left (3 a^2 f \left (d f+e^2\right )+2 a b e \left (6 d f+e^2\right )+3 b^2 d \left (d f+e^2\right )\right )-252 b^4 c f^2 (5 a f+3 b e)-1536 c^5 d \left (a \left (d f+e^2\right )+b d e\right )+231 b^6 f^3+1024 c^6 d^3\right )}{1024 c^{13/2}}+\frac {x \sqrt {a+b x+c x^2} \left (24 c^2 f \left (50 a^2 f^2+322 a b e f+175 b^2 \left (d f+e^2\right )\right )-252 b^2 c f^2 (14 a f+15 b e)-160 c^3 \left (27 a f \left (d f+e^2\right )+10 b \left (6 d e f+e^3\right )\right )+1155 b^4 f^3+5760 c^4 d \left (d f+e^2\right )\right )}{3840 c^5}+\frac {f x^3 \sqrt {a+b x+c x^2} \left (-4 c f (25 a f+81 b e)+99 b^2 f^2+360 c^2 \left (d f+e^2\right )\right )}{480 c^3}-\frac {x^2 \sqrt {a+b x+c x^2} \left (24 c^2 f \left (32 a e f+35 b \left (d f+e^2\right )\right )-36 b c f^2 (13 a f+21 b e)+231 b^3 f^3-320 c^3 \left (6 d e f+e^3\right )\right )}{960 c^4}+\frac {f^2 x^4 \sqrt {a+b x+c x^2} (36 c e-11 b f)}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((23040*c^5*d^2*e - 3465*b^5*f^3 + 420*b^3*c*f^2*(27*b*e + 34*a*f) - 504*b*c^2*f*(70*a*b*e*f + 22*a^2*f^2 + 25
*b^2*(e^2 + d*f)) - 640*c^4*(27*b*d*(e^2 + d*f) + 8*a*e*(e^2 + 6*d*f)) + 96*c^3*(128*a^2*e*f^2 + 275*a*b*f*(e^
2 + d*f) + 50*b^2*(e^3 + 6*d*e*f)))*Sqrt[a + b*x + c*x^2])/(7680*c^6) + ((1155*b^4*f^3 - 252*b^2*c*f^2*(15*b*e
+ 14*a*f) + 5760*c^4*d*(e^2 + d*f) + 24*c^2*f*(322*a*b*e*f + 50*a^2*f^2 + 175*b^2*(e^2 + d*f)) - 160*c^3*(27*
a*f*(e^2 + d*f) + 10*b*(e^3 + 6*d*e*f)))*x*Sqrt[a + b*x + c*x^2])/(3840*c^5) - ((231*b^3*f^3 - 36*b*c*f^2*(21*
b*e + 13*a*f) - 320*c^3*(e^3 + 6*d*e*f) + 24*c^2*f*(32*a*e*f + 35*b*(e^2 + d*f)))*x^2*Sqrt[a + b*x + c*x^2])/(
960*c^4) + (f*(99*b^2*f^2 - 4*c*f*(81*b*e + 25*a*f) + 360*c^2*(e^2 + d*f))*x^3*Sqrt[a + b*x + c*x^2])/(480*c^3
) + (f^2*(36*c*e - 11*b*f)*x^4*Sqrt[a + b*x + c*x^2])/(60*c^2) + (f^3*x^5*Sqrt[a + b*x + c*x^2])/(6*c) + ((102
4*c^6*d^3 + 231*b^6*f^3 - 252*b^4*c*f^2*(3*b*e + 5*a*f) - 1536*c^5*d*(b*d*e + a*(e^2 + d*f)) + 840*b^2*c^2*f*(
4*a*b*e*f + 2*a^2*f^2 + b^2*(e^2 + d*f)) + 384*c^4*(3*b^2*d*(e^2 + d*f) + 3*a^2*f*(e^2 + d*f) + 2*a*b*e*(e^2 +
6*d*f)) - 320*c^3*(9*a^2*b*e*f^2 + a^3*f^3 + 9*a*b^2*f*(e^2 + d*f) + b^3*(e^3 + 6*d*e*f)))*ArcTanh[(b + 2*c*x
)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(13/2))

Rule 206

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

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (d+e x+f x^2\right )^3}{\sqrt {a+b x+c x^2}} \, dx &=\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {6 c d^3+18 c d^2 e x+18 c d \left (e^2+d f\right ) x^2+6 c e \left (e^2+6 d f\right ) x^3-f \left (5 a f^2-18 c \left (e^2+d f\right )\right ) x^4+\frac {1}{2} f^2 (36 c e-11 b f) x^5}{\sqrt {a+b x+c x^2}} \, dx}{6 c}\\ &=\frac {f^2 (36 c e-11 b f) x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {30 c^2 d^3+90 c^2 d^2 e x+90 c^2 d \left (e^2+d f\right ) x^2-2 \left (36 a c e f^2-11 a b f^3-15 c^2 \left (e^3+6 d e f\right )\right ) x^3+\frac {1}{4} f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^4}{\sqrt {a+b x+c x^2}} \, dx}{30 c^2}\\ &=\frac {f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}+\frac {f^2 (36 c e-11 b f) x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {120 c^3 d^3+360 c^3 d^2 e x-\frac {3}{4} \left (99 a b^2 f^3-4 a c f^2 (81 b e+25 a f)-480 c^3 d \left (e^2+d f\right )+360 a c^2 f \left (e^2+d f\right )\right ) x^2-\frac {3}{8} \left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^3}{\sqrt {a+b x+c x^2}} \, dx}{120 c^3}\\ &=-\frac {\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt {a+b x+c x^2}}{960 c^4}+\frac {f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}+\frac {f^2 (36 c e-11 b f) x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {360 c^4 d^3+\frac {3}{4} \left (1440 c^4 d^2 e+231 a b^3 f^3-36 a b c f^2 (21 b e+13 a f)-320 a c^3 e \left (e^2+6 d f\right )+24 a c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x+\frac {3}{16} \left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x^2}{\sqrt {a+b x+c x^2}} \, dx}{360 c^4}\\ &=\frac {\left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x \sqrt {a+b x+c x^2}}{3840 c^5}-\frac {\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt {a+b x+c x^2}}{960 c^4}+\frac {f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}+\frac {f^2 (36 c e-11 b f) x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {\frac {3}{16} \left (3840 c^5 d^3-1155 a b^4 f^3+252 a b^2 c f^2 (15 b e+14 a f)-5760 a c^4 d \left (e^2+d f\right )-24 a c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )+160 a c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right )+\frac {3}{32} \left (23040 c^5 d^2 e-3465 b^5 f^3+420 b^3 c f^2 (27 b e+34 a f)-504 b c^2 f \left (70 a b e f+22 a^2 f^2+25 b^2 \left (e^2+d f\right )\right )-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (e^2+d f\right )+50 b^2 \left (e^3+6 d e f\right )\right )\right ) x}{\sqrt {a+b x+c x^2}} \, dx}{720 c^5}\\ &=\frac {\left (23040 c^5 d^2 e-3465 b^5 f^3+420 b^3 c f^2 (27 b e+34 a f)-504 b c^2 f \left (70 a b e f+22 a^2 f^2+25 b^2 \left (e^2+d f\right )\right )-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (e^2+d f\right )+50 b^2 \left (e^3+6 d e f\right )\right )\right ) \sqrt {a+b x+c x^2}}{7680 c^6}+\frac {\left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x \sqrt {a+b x+c x^2}}{3840 c^5}-\frac {\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt {a+b x+c x^2}}{960 c^4}+\frac {f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}+\frac {f^2 (36 c e-11 b f) x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\left (1024 c^6 d^3+231 b^6 f^3-252 b^4 c f^2 (3 b e+5 a f)-1536 c^5 d \left (b d e+a \left (e^2+d f\right )\right )+840 b^2 c^2 f \left (4 a b e f+2 a^2 f^2+b^2 \left (e^2+d f\right )\right )+384 c^4 \left (3 b^2 d \left (e^2+d f\right )+3 a^2 f \left (e^2+d f\right )+2 a b e \left (e^2+6 d f\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^6}\\ &=\frac {\left (23040 c^5 d^2 e-3465 b^5 f^3+420 b^3 c f^2 (27 b e+34 a f)-504 b c^2 f \left (70 a b e f+22 a^2 f^2+25 b^2 \left (e^2+d f\right )\right )-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (e^2+d f\right )+50 b^2 \left (e^3+6 d e f\right )\right )\right ) \sqrt {a+b x+c x^2}}{7680 c^6}+\frac {\left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x \sqrt {a+b x+c x^2}}{3840 c^5}-\frac {\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt {a+b x+c x^2}}{960 c^4}+\frac {f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}+\frac {f^2 (36 c e-11 b f) x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\left (1024 c^6 d^3+231 b^6 f^3-252 b^4 c f^2 (3 b e+5 a f)-1536 c^5 d \left (b d e+a \left (e^2+d f\right )\right )+840 b^2 c^2 f \left (4 a b e f+2 a^2 f^2+b^2 \left (e^2+d f\right )\right )+384 c^4 \left (3 b^2 d \left (e^2+d f\right )+3 a^2 f \left (e^2+d f\right )+2 a b e \left (e^2+6 d f\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{512 c^6}\\ &=\frac {\left (23040 c^5 d^2 e-3465 b^5 f^3+420 b^3 c f^2 (27 b e+34 a f)-504 b c^2 f \left (70 a b e f+22 a^2 f^2+25 b^2 \left (e^2+d f\right )\right )-640 c^4 \left (27 b d \left (e^2+d f\right )+8 a e \left (e^2+6 d f\right )\right )+96 c^3 \left (128 a^2 e f^2+275 a b f \left (e^2+d f\right )+50 b^2 \left (e^3+6 d e f\right )\right )\right ) \sqrt {a+b x+c x^2}}{7680 c^6}+\frac {\left (1155 b^4 f^3-252 b^2 c f^2 (15 b e+14 a f)+5760 c^4 d \left (e^2+d f\right )+24 c^2 f \left (322 a b e f+50 a^2 f^2+175 b^2 \left (e^2+d f\right )\right )-160 c^3 \left (27 a f \left (e^2+d f\right )+10 b \left (e^3+6 d e f\right )\right )\right ) x \sqrt {a+b x+c x^2}}{3840 c^5}-\frac {\left (231 b^3 f^3-36 b c f^2 (21 b e+13 a f)-320 c^3 \left (e^3+6 d e f\right )+24 c^2 f \left (32 a e f+35 b \left (e^2+d f\right )\right )\right ) x^2 \sqrt {a+b x+c x^2}}{960 c^4}+\frac {f \left (99 b^2 f^2-4 c f (81 b e+25 a f)+360 c^2 \left (e^2+d f\right )\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}+\frac {f^2 (36 c e-11 b f) x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {f^3 x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\left (1024 c^6 d^3+231 b^6 f^3-252 b^4 c f^2 (3 b e+5 a f)-1536 c^5 d \left (b d e+a \left (e^2+d f\right )\right )+840 b^2 c^2 f \left (4 a b e f+2 a^2 f^2+b^2 \left (e^2+d f\right )\right )+384 c^4 \left (3 b^2 d \left (e^2+d f\right )+3 a^2 f \left (e^2+d f\right )+2 a b e \left (e^2+6 d f\right )\right )-320 c^3 \left (9 a^2 b e f^2+a^3 f^3+9 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{13/2}}\\ \end {align*}

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Mathematica [A]  time = 1.35, size = 615, normalized size = 0.86 \[ \frac {\sqrt {a+x (b+c x)} \left (48 c^3 \left (2 a^2 f^2 (128 e+25 f x)+2 a b f \left (f \left (275 d+39 f x^2\right )+275 e^2+161 e f x\right )+b^2 \left (6 e f \left (100 d+21 f x^2\right )+f^2 x \left (175 d+33 f x^2\right )+100 e^3+175 e^2 f x\right )\right )-168 b c^2 f \left (66 a^2 f^2+42 a b f (5 e+f x)+b^2 \left (75 d f+75 e^2+45 e f x+11 f^2 x^2\right )\right )+210 b^3 c f^2 (68 a f+54 b e+11 b f x)-64 c^4 \left (a \left (96 e f \left (5 d+f x^2\right )+5 f^2 x \left (27 d+5 f x^2\right )+80 e^3+135 e^2 f x\right )+b \left (270 d^2 f+15 d \left (18 e^2+20 e f x+7 f^2 x^2\right )+x \left (50 e^3+105 e^2 f x+81 e f^2 x^2+22 f^3 x^3\right )\right )\right )-3465 b^5 f^3+128 c^5 \left (90 d^2 (2 e+f x)+15 d x \left (6 e^2+8 e f x+3 f^2 x^2\right )+x^2 \left (20 e^3+45 e^2 f x+36 e f^2 x^2+10 f^3 x^3\right )\right )\right )}{7680 c^6}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \left (384 c^4 \left (3 a^2 f \left (d f+e^2\right )+2 a b e \left (6 d f+e^2\right )+3 b^2 d \left (d f+e^2\right )\right )+840 b^2 c^2 f \left (2 a^2 f^2+4 a b e f+b^2 \left (d f+e^2\right )\right )-320 c^3 \left (a^3 f^3+9 a^2 b e f^2+9 a b^2 f \left (d f+e^2\right )+b^3 \left (6 d e f+e^3\right )\right )-252 b^4 c f^2 (5 a f+3 b e)-1536 c^5 d \left (a \left (d f+e^2\right )+b d e\right )+231 b^6 f^3+1024 c^6 d^3\right )}{1024 c^{13/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + x*(b + c*x)]*(-3465*b^5*f^3 + 210*b^3*c*f^2*(54*b*e + 68*a*f + 11*b*f*x) - 168*b*c^2*f*(66*a^2*f^2 +
42*a*b*f*(5*e + f*x) + b^2*(75*e^2 + 75*d*f + 45*e*f*x + 11*f^2*x^2)) + 128*c^5*(90*d^2*(2*e + f*x) + 15*d*x*
(6*e^2 + 8*e*f*x + 3*f^2*x^2) + x^2*(20*e^3 + 45*e^2*f*x + 36*e*f^2*x^2 + 10*f^3*x^3)) + 48*c^3*(2*a^2*f^2*(12
8*e + 25*f*x) + b^2*(100*e^3 + 175*e^2*f*x + 6*e*f*(100*d + 21*f*x^2) + f^2*x*(175*d + 33*f*x^2)) + 2*a*b*f*(2
75*e^2 + 161*e*f*x + f*(275*d + 39*f*x^2))) - 64*c^4*(a*(80*e^3 + 135*e^2*f*x + 96*e*f*(5*d + f*x^2) + 5*f^2*x
*(27*d + 5*f*x^2)) + b*(270*d^2*f + 15*d*(18*e^2 + 20*e*f*x + 7*f^2*x^2) + x*(50*e^3 + 105*e^2*f*x + 81*e*f^2*
x^2 + 22*f^3*x^3)))))/(7680*c^6) + ((1024*c^6*d^3 + 231*b^6*f^3 - 252*b^4*c*f^2*(3*b*e + 5*a*f) - 1536*c^5*d*(
b*d*e + a*(e^2 + d*f)) + 840*b^2*c^2*f*(4*a*b*e*f + 2*a^2*f^2 + b^2*(e^2 + d*f)) + 384*c^4*(3*b^2*d*(e^2 + d*f
) + 3*a^2*f*(e^2 + d*f) + 2*a*b*e*(e^2 + 6*d*f)) - 320*c^3*(9*a^2*b*e*f^2 + a^3*f^3 + 9*a*b^2*f*(e^2 + d*f) +
b^3*(e^3 + 6*d*e*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(1024*c^(13/2))

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fricas [A]  time = 3.20, size = 1583, normalized size = 2.21 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/30720*(15*(1024*c^6*d^3 - 1536*b*c^5*d^2*e + 384*(3*b^2*c^4 - 4*a*c^5)*d*e^2 - 64*(5*b^3*c^3 - 12*a*b*c^4)
*e^3 + (231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*f^3 + 12*(2*(35*b^4*c^2 - 120*a*b^2*c^3 + 48*
a^2*c^4)*d - (63*b^5*c - 280*a*b^3*c^2 + 240*a^2*b*c^3)*e)*f^2 + 24*(16*(3*b^2*c^4 - 4*a*c^5)*d^2 - 16*(5*b^3*
c^3 - 12*a*b*c^4)*d*e + (35*b^4*c^2 - 120*a*b^2*c^3 + 48*a^2*c^4)*e^2)*f)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b
^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(1280*c^6*f^3*x^5 + 23040*c^6*d^2*e - 17280*b*c^
5*d*e^2 + 128*(36*c^6*e*f^2 - 11*b*c^5*f^3)*x^4 + 320*(15*b^2*c^4 - 16*a*c^5)*e^3 - 21*(165*b^5*c - 680*a*b^3*
c^2 + 528*a^2*b*c^3)*f^3 + 16*(360*c^6*e^2*f + (99*b^2*c^4 - 100*a*c^5)*f^3 + 36*(10*c^6*d - 9*b*c^5*e)*f^2)*x
^3 - 12*(50*(21*b^3*c^3 - 44*a*b*c^4)*d - (945*b^4*c^2 - 2940*a*b^2*c^3 + 1024*a^2*c^4)*e)*f^2 + 8*(320*c^6*e^
3 - 3*(77*b^3*c^3 - 156*a*b*c^4)*f^3 - 12*(70*b*c^5*d - (63*b^2*c^4 - 64*a*c^5)*e)*f^2 + 120*(16*c^6*d*e - 7*b
*c^5*e^2)*f)*x^2 - 120*(144*b*c^5*d^2 - 16*(15*b^2*c^4 - 16*a*c^5)*d*e + 5*(21*b^3*c^3 - 44*a*b*c^4)*e^2)*f +
2*(5760*c^6*d*e^2 - 1600*b*c^5*e^3 + 3*(385*b^4*c^2 - 1176*a*b^2*c^3 + 400*a^2*c^4)*f^3 + 12*(10*(35*b^2*c^4 -
36*a*c^5)*d - 7*(45*b^3*c^3 - 92*a*b*c^4)*e)*f^2 + 120*(48*c^6*d^2 - 80*b*c^5*d*e + (35*b^2*c^4 - 36*a*c^5)*e
^2)*f)*x)*sqrt(c*x^2 + b*x + a))/c^7, -1/15360*(15*(1024*c^6*d^3 - 1536*b*c^5*d^2*e + 384*(3*b^2*c^4 - 4*a*c^5
)*d*e^2 - 64*(5*b^3*c^3 - 12*a*b*c^4)*e^3 + (231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*f^3 + 12
*(2*(35*b^4*c^2 - 120*a*b^2*c^3 + 48*a^2*c^4)*d - (63*b^5*c - 280*a*b^3*c^2 + 240*a^2*b*c^3)*e)*f^2 + 24*(16*(
3*b^2*c^4 - 4*a*c^5)*d^2 - 16*(5*b^3*c^3 - 12*a*b*c^4)*d*e + (35*b^4*c^2 - 120*a*b^2*c^3 + 48*a^2*c^4)*e^2)*f)
*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(1280*c^6*f^3*x^5
+ 23040*c^6*d^2*e - 17280*b*c^5*d*e^2 + 128*(36*c^6*e*f^2 - 11*b*c^5*f^3)*x^4 + 320*(15*b^2*c^4 - 16*a*c^5)*e
^3 - 21*(165*b^5*c - 680*a*b^3*c^2 + 528*a^2*b*c^3)*f^3 + 16*(360*c^6*e^2*f + (99*b^2*c^4 - 100*a*c^5)*f^3 + 3
6*(10*c^6*d - 9*b*c^5*e)*f^2)*x^3 - 12*(50*(21*b^3*c^3 - 44*a*b*c^4)*d - (945*b^4*c^2 - 2940*a*b^2*c^3 + 1024*
a^2*c^4)*e)*f^2 + 8*(320*c^6*e^3 - 3*(77*b^3*c^3 - 156*a*b*c^4)*f^3 - 12*(70*b*c^5*d - (63*b^2*c^4 - 64*a*c^5)
*e)*f^2 + 120*(16*c^6*d*e - 7*b*c^5*e^2)*f)*x^2 - 120*(144*b*c^5*d^2 - 16*(15*b^2*c^4 - 16*a*c^5)*d*e + 5*(21*
b^3*c^3 - 44*a*b*c^4)*e^2)*f + 2*(5760*c^6*d*e^2 - 1600*b*c^5*e^3 + 3*(385*b^4*c^2 - 1176*a*b^2*c^3 + 400*a^2*
c^4)*f^3 + 12*(10*(35*b^2*c^4 - 36*a*c^5)*d - 7*(45*b^3*c^3 - 92*a*b*c^4)*e)*f^2 + 120*(48*c^6*d^2 - 80*b*c^5*
d*e + (35*b^2*c^4 - 36*a*c^5)*e^2)*f)*x)*sqrt(c*x^2 + b*x + a))/c^7]

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giac [A]  time = 0.53, size = 824, normalized size = 1.15 \[ \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (\frac {10 \, f^{3} x}{c} - \frac {11 \, b c^{4} f^{3} - 36 \, c^{5} f^{2} e}{c^{6}}\right )} x + \frac {360 \, c^{5} d f^{2} + 99 \, b^{2} c^{3} f^{3} - 100 \, a c^{4} f^{3} - 324 \, b c^{4} f^{2} e + 360 \, c^{5} f e^{2}}{c^{6}}\right )} x - \frac {840 \, b c^{4} d f^{2} + 231 \, b^{3} c^{2} f^{3} - 468 \, a b c^{3} f^{3} - 1920 \, c^{5} d f e - 756 \, b^{2} c^{3} f^{2} e + 768 \, a c^{4} f^{2} e + 840 \, b c^{4} f e^{2} - 320 \, c^{5} e^{3}}{c^{6}}\right )} x + \frac {5760 \, c^{5} d^{2} f + 4200 \, b^{2} c^{3} d f^{2} - 4320 \, a c^{4} d f^{2} + 1155 \, b^{4} c f^{3} - 3528 \, a b^{2} c^{2} f^{3} + 1200 \, a^{2} c^{3} f^{3} - 9600 \, b c^{4} d f e - 3780 \, b^{3} c^{2} f^{2} e + 7728 \, a b c^{3} f^{2} e + 5760 \, c^{5} d e^{2} + 4200 \, b^{2} c^{3} f e^{2} - 4320 \, a c^{4} f e^{2} - 1600 \, b c^{4} e^{3}}{c^{6}}\right )} x - \frac {17280 \, b c^{4} d^{2} f + 12600 \, b^{3} c^{2} d f^{2} - 26400 \, a b c^{3} d f^{2} + 3465 \, b^{5} f^{3} - 14280 \, a b^{3} c f^{3} + 11088 \, a^{2} b c^{2} f^{3} - 23040 \, c^{5} d^{2} e - 28800 \, b^{2} c^{3} d f e + 30720 \, a c^{4} d f e - 11340 \, b^{4} c f^{2} e + 35280 \, a b^{2} c^{2} f^{2} e - 12288 \, a^{2} c^{3} f^{2} e + 17280 \, b c^{4} d e^{2} + 12600 \, b^{3} c^{2} f e^{2} - 26400 \, a b c^{3} f e^{2} - 4800 \, b^{2} c^{3} e^{3} + 5120 \, a c^{4} e^{3}}{c^{6}}\right )} - \frac {{\left (1024 \, c^{6} d^{3} + 1152 \, b^{2} c^{4} d^{2} f - 1536 \, a c^{5} d^{2} f + 840 \, b^{4} c^{2} d f^{2} - 2880 \, a b^{2} c^{3} d f^{2} + 1152 \, a^{2} c^{4} d f^{2} + 231 \, b^{6} f^{3} - 1260 \, a b^{4} c f^{3} + 1680 \, a^{2} b^{2} c^{2} f^{3} - 320 \, a^{3} c^{3} f^{3} - 1536 \, b c^{5} d^{2} e - 1920 \, b^{3} c^{3} d f e + 4608 \, a b c^{4} d f e - 756 \, b^{5} c f^{2} e + 3360 \, a b^{3} c^{2} f^{2} e - 2880 \, a^{2} b c^{3} f^{2} e + 1152 \, b^{2} c^{4} d e^{2} - 1536 \, a c^{5} d e^{2} + 840 \, b^{4} c^{2} f e^{2} - 2880 \, a b^{2} c^{3} f e^{2} + 1152 \, a^{2} c^{4} f e^{2} - 320 \, b^{3} c^{3} e^{3} + 768 \, a b c^{4} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*f^3*x/c - (11*b*c^4*f^3 - 36*c^5*f^2*e)/c^6)*x + (360*c^5*d*f^2 +
99*b^2*c^3*f^3 - 100*a*c^4*f^3 - 324*b*c^4*f^2*e + 360*c^5*f*e^2)/c^6)*x - (840*b*c^4*d*f^2 + 231*b^3*c^2*f^3
- 468*a*b*c^3*f^3 - 1920*c^5*d*f*e - 756*b^2*c^3*f^2*e + 768*a*c^4*f^2*e + 840*b*c^4*f*e^2 - 320*c^5*e^3)/c^6
)*x + (5760*c^5*d^2*f + 4200*b^2*c^3*d*f^2 - 4320*a*c^4*d*f^2 + 1155*b^4*c*f^3 - 3528*a*b^2*c^2*f^3 + 1200*a^2
*c^3*f^3 - 9600*b*c^4*d*f*e - 3780*b^3*c^2*f^2*e + 7728*a*b*c^3*f^2*e + 5760*c^5*d*e^2 + 4200*b^2*c^3*f*e^2 -
4320*a*c^4*f*e^2 - 1600*b*c^4*e^3)/c^6)*x - (17280*b*c^4*d^2*f + 12600*b^3*c^2*d*f^2 - 26400*a*b*c^3*d*f^2 + 3
465*b^5*f^3 - 14280*a*b^3*c*f^3 + 11088*a^2*b*c^2*f^3 - 23040*c^5*d^2*e - 28800*b^2*c^3*d*f*e + 30720*a*c^4*d*
f*e - 11340*b^4*c*f^2*e + 35280*a*b^2*c^2*f^2*e - 12288*a^2*c^3*f^2*e + 17280*b*c^4*d*e^2 + 12600*b^3*c^2*f*e^
2 - 26400*a*b*c^3*f*e^2 - 4800*b^2*c^3*e^3 + 5120*a*c^4*e^3)/c^6) - 1/1024*(1024*c^6*d^3 + 1152*b^2*c^4*d^2*f
- 1536*a*c^5*d^2*f + 840*b^4*c^2*d*f^2 - 2880*a*b^2*c^3*d*f^2 + 1152*a^2*c^4*d*f^2 + 231*b^6*f^3 - 1260*a*b^4*
c*f^3 + 1680*a^2*b^2*c^2*f^3 - 320*a^3*c^3*f^3 - 1536*b*c^5*d^2*e - 1920*b^3*c^3*d*f*e + 4608*a*b*c^4*d*f*e -
756*b^5*c*f^2*e + 3360*a*b^3*c^2*f^2*e - 2880*a^2*b*c^3*f^2*e + 1152*b^2*c^4*d*e^2 - 1536*a*c^5*d*e^2 + 840*b^
4*c^2*f*e^2 - 2880*a*b^2*c^3*f*e^2 + 1152*a^2*c^4*f*e^2 - 320*b^3*c^3*e^3 + 768*a*b*c^4*e^3)*log(abs(-2*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)

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maple [B]  time = 0.03, size = 1930, normalized size = 2.69 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*d^2*e/c*(c*x^2+b*x+a)^(1/2)+1/3*x^2/c*(c*x^2+b*x+a)^(1/2)*e^3+5/8*b^2/c^3*(c*x^2+b*x+a)^(1/2)*e^3-5/16*b^3/c
^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^3-2/3*a/c^2*(c*x^2+b*x+a)^(1/2)*e^3-231/512*f^3*b^5/c^6*(
c*x^2+b*x+a)^(1/2)+231/1024*f^3*b^6/c^(13/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/16*f^3*a^3/c^(7/2)*
ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+d^3*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+161/80*e*f
^2*b/c^3*a*x*(c*x^2+b*x+a)^(1/2)-5/2*b/c^2*x*(c*x^2+b*x+a)^(1/2)*d*e*f+9/2*b/c^(5/2)*a*ln((c*x+1/2*b)/c^(1/2)+
(c*x^2+b*x+a)^(1/2))*d*e*f-45/16*b^2/c^(7/2)*a*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2*f+55/16*b/c^3*a
*(c*x^2+b*x+a)^(1/2)*d*f^2+55/16*b/c^3*a*(c*x^2+b*x+a)^(1/2)*e^2*f-9/8*a/c^2*x*(c*x^2+b*x+a)^(1/2)*d*f^2-147/1
60*f^3*b^2/c^4*a*x*(c*x^2+b*x+a)^(1/2)+39/80*f^3*b/c^3*a*x^2*(c*x^2+b*x+a)^(1/2)-9/8*a/c^2*x*(c*x^2+b*x+a)^(1/
2)*e^2*f+2*x^2/c*(c*x^2+b*x+a)^(1/2)*d*e*f+15/4*b^2/c^3*(c*x^2+b*x+a)^(1/2)*d*e*f-15/8*b^3/c^(7/2)*ln((c*x+1/2
*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e*f-4*a/c^2*(c*x^2+b*x+a)^(1/2)*d*e*f-27/40*e*f^2*b/c^2*x^3*(c*x^2+b*x+a)^(
1/2)+63/80*e*f^2*b^2/c^3*x^2*(c*x^2+b*x+a)^(1/2)-63/64*e*f^2*b^3/c^4*x*(c*x^2+b*x+a)^(1/2)+105/32*e*f^2*b^3/c^
(9/2)*a*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-147/32*e*f^2*b^2/c^4*a*(c*x^2+b*x+a)^(1/2)-45/16*e*f^2*b/c
^(7/2)*a^2*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-4/5*e*f^2*a/c^2*x^2*(c*x^2+b*x+a)^(1/2)-7/8*b/c^2*x^2*(
c*x^2+b*x+a)^(1/2)*d*f^2-7/8*b/c^2*x^2*(c*x^2+b*x+a)^(1/2)*e^2*f+35/32*b^2/c^3*x*(c*x^2+b*x+a)^(1/2)*d*f^2+35/
32*b^2/c^3*x*(c*x^2+b*x+a)^(1/2)*e^2*f-45/16*b^2/c^(7/2)*a*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f^2+1
/6*f^3*x^5*(c*x^2+b*x+a)^(1/2)/c+9/8*a^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2*f-5/12*b/c^2*
x*(c*x^2+b*x+a)^(1/2)*e^3+3/4*b/c^(5/2)*a*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^3-105/64*b^3/c^4*(c*x^
2+b*x+a)^(1/2)*e^2*f+105/128*b^4/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f^2+105/128*b^4/c^(9/2)
*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2*f+9/8*a^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
*d*f^2+3/4*x^3/c*(c*x^2+b*x+a)^(1/2)*d*f^2+3/4*x^3/c*(c*x^2+b*x+a)^(1/2)*e^2*f-105/64*b^3/c^4*(c*x^2+b*x+a)^(1
/2)*d*f^2+3/5*e*f^2*x^4/c*(c*x^2+b*x+a)^(1/2)+189/128*e*f^2*b^4/c^5*(c*x^2+b*x+a)^(1/2)-189/256*e*f^2*b^5/c^(1
1/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+8/5*e*f^2*a^2/c^3*(c*x^2+b*x+a)^(1/2)-315/256*f^3*b^4/c^(11/2
)*a*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+119/64*f^3*b^3/c^5*a*(c*x^2+b*x+a)^(1/2)+105/64*f^3*b^2/c^(9/2
)*a^2*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-231/160*f^3*b/c^4*a^2*(c*x^2+b*x+a)^(1/2)-5/24*f^3*a/c^2*x^3
*(c*x^2+b*x+a)^(1/2)+5/16*f^3*a^2/c^3*x*(c*x^2+b*x+a)^(1/2)-11/60*f^3*b/c^2*x^4*(c*x^2+b*x+a)^(1/2)+33/160*f^3
*b^2/c^3*x^3*(c*x^2+b*x+a)^(1/2)-77/320*f^3*b^3/c^4*x^2*(c*x^2+b*x+a)^(1/2)+77/256*f^3*b^4/c^5*x*(c*x^2+b*x+a)
^(1/2)-9/4*b/c^2*(c*x^2+b*x+a)^(1/2)*f*d^2-9/4*b/c^2*(c*x^2+b*x+a)^(1/2)*e^2*d+9/8*b^2/c^(5/2)*ln((c*x+1/2*b)/
c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*d^2+9/8*b^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2*d-3/2*a/c^(
3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*d^2-3/2*a/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))*e^2*d-3/2*d^2*e*b/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/2*x/c*(c*x^2+b*x+a)^(1/2)*f*d^2+3/2
*x/c*(c*x^2+b*x+a)^(1/2)*e^2*d

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f\,x^2+e\,x+d\right )}^3}{\sqrt {c\,x^2+b\,x+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x + f x^{2}\right )^{3}}{\sqrt {a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2+e*x+d)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((d + e*x + f*x**2)**3/sqrt(a + b*x + c*x**2), x)

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