3.2.4 \(\int (a+b x+c x^2)^{3/2} (d+e x+f x^2)^2 \, dx\) [104]

Optimal. Leaf size=564 \[ -\frac {\left (b^2-4 a c\right ) \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^6}+\frac {\left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}+\frac {\left (5376 c^3 d e-693 b^3 f^2+36 b c f (56 b e+31 a f)-32 c^2 \left (48 a e f+49 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4}+\frac {\left (99 b^2 f^2-12 c f (24 b e+7 a f)+224 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{5/2}}{1344 c^3}+\frac {f (32 c e-11 b f) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\left (b^2-4 a c\right )^2 \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2}} \]

[Out]

1/6144*(768*c^4*d^2+99*b^4*f^2-72*b^2*c*f*(3*a*f+4*b*e)-128*c^3*(6*b*d*e+a*(2*d*f+e^2))+16*c^2*(24*a*b*e*f+3*a
^2*f^2+14*b^2*(2*d*f+e^2)))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^5+1/13440*(5376*c^3*d*e-693*b^3*f^2+36*b*c*f*(31*a
*f+56*b*e)-32*c^2*(48*a*e*f+49*b*(2*d*f+e^2)))*(c*x^2+b*x+a)^(5/2)/c^4+1/1344*(99*b^2*f^2-12*c*f*(7*a*f+24*b*e
)+224*c^2*(2*d*f+e^2))*x*(c*x^2+b*x+a)^(5/2)/c^3+1/112*f*(-11*b*f+32*c*e)*x^2*(c*x^2+b*x+a)^(5/2)/c^2+1/8*f^2*
x^3*(c*x^2+b*x+a)^(5/2)/c+1/32768*(-4*a*c+b^2)^2*(768*c^4*d^2+99*b^4*f^2-72*b^2*c*f*(3*a*f+4*b*e)-128*c^3*(6*b
*d*e+a*(2*d*f+e^2))+16*c^2*(24*a*b*e*f+3*a^2*f^2+14*b^2*(2*d*f+e^2)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x
+a)^(1/2))/c^(13/2)-1/16384*(-4*a*c+b^2)*(768*c^4*d^2+99*b^4*f^2-72*b^2*c*f*(3*a*f+4*b*e)-128*c^3*(6*b*d*e+a*(
2*d*f+e^2))+16*c^2*(24*a*b*e*f+3*a^2*f^2+14*b^2*(2*d*f+e^2)))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^6

________________________________________________________________________________________

Rubi [A]
time = 0.61, antiderivative size = 564, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1675, 654, 626, 635, 212} \begin {gather*} \frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )}{32768 c^{13/2}}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )}{16384 c^6}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )}{6144 c^5}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-32 c^2 \left (48 a e f+49 b \left (2 d f+e^2\right )\right )+36 b c f (31 a f+56 b e)-693 b^3 f^2+5376 c^3 d e\right )}{13440 c^4}+\frac {x \left (a+b x+c x^2\right )^{5/2} \left (-12 c f (7 a f+24 b e)+99 b^2 f^2+224 c^2 \left (2 d f+e^2\right )\right )}{1344 c^3}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16384*((b^2 - 4*a*c)*(768*c^4*d^2 + 99*b^4*f^2 - 72*b^2*c*f*(4*b*e + 3*a*f) - 128*c^3*(6*b*d*e + a*(e^2 + 2
*d*f)) + 16*c^2*(24*a*b*e*f + 3*a^2*f^2 + 14*b^2*(e^2 + 2*d*f)))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/c^6 + ((76
8*c^4*d^2 + 99*b^4*f^2 - 72*b^2*c*f*(4*b*e + 3*a*f) - 128*c^3*(6*b*d*e + a*(e^2 + 2*d*f)) + 16*c^2*(24*a*b*e*f
+ 3*a^2*f^2 + 14*b^2*(e^2 + 2*d*f)))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(6144*c^5) + ((5376*c^3*d*e - 693*b
^3*f^2 + 36*b*c*f*(56*b*e + 31*a*f) - 32*c^2*(48*a*e*f + 49*b*(e^2 + 2*d*f)))*(a + b*x + c*x^2)^(5/2))/(13440*
c^4) + ((99*b^2*f^2 - 12*c*f*(24*b*e + 7*a*f) + 224*c^2*(e^2 + 2*d*f))*x*(a + b*x + c*x^2)^(5/2))/(1344*c^3) +
(f*(32*c*e - 11*b*f)*x^2*(a + b*x + c*x^2)^(5/2))/(112*c^2) + (f^2*x^3*(a + b*x + c*x^2)^(5/2))/(8*c) + ((b^2
- 4*a*c)^2*(768*c^4*d^2 + 99*b^4*f^2 - 72*b^2*c*f*(4*b*e + 3*a*f) - 128*c^3*(6*b*d*e + a*(e^2 + 2*d*f)) + 16*
c^2*(24*a*b*e*f + 3*a^2*f^2 + 14*b^2*(e^2 + 2*d*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(
32768*c^(13/2))

Rule 212

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

Rule 626

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

Rule 635

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 654

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 1675

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 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )^2 \, dx &=\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\int \left (a+b x+c x^2\right )^{3/2} \left (8 c d^2+16 c d e x-\left (3 a f^2-8 c \left (e^2+2 d f\right )\right ) x^2+\frac {1}{2} f (32 c e-11 b f) x^3\right ) \, dx}{8 c}\\ &=\frac {f (32 c e-11 b f) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\int \left (a+b x+c x^2\right )^{3/2} \left (56 c^2 d^2+\left (112 c^2 d e-32 a c e f+11 a b f^2\right ) x+\frac {1}{4} \left (99 b^2 f^2-12 c f (24 b e+7 a f)+224 c^2 \left (e^2+2 d f\right )\right ) x^2\right ) \, dx}{56 c^2}\\ &=\frac {\left (99 b^2 f^2-12 c f (24 b e+7 a f)+224 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{5/2}}{1344 c^3}+\frac {f (32 c e-11 b f) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\int \left (\frac {1}{4} \left (1344 c^3 d^2-99 a b^2 f^2+12 a c f (24 b e+7 a f)-224 a c^2 \left (e^2+2 d f\right )\right )+\frac {1}{8} \left (5376 c^3 d e-693 b^3 f^2+36 b c f (56 b e+31 a f)-32 c^2 \left (48 a e f+49 b \left (e^2+2 d f\right )\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{336 c^3}\\ &=\frac {\left (5376 c^3 d e-693 b^3 f^2+36 b c f (56 b e+31 a f)-32 c^2 \left (48 a e f+49 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4}+\frac {\left (99 b^2 f^2-12 c f (24 b e+7 a f)+224 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{5/2}}{1344 c^3}+\frac {f (32 c e-11 b f) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{768 c^4}\\ &=\frac {\left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}+\frac {\left (5376 c^3 d e-693 b^3 f^2+36 b c f (56 b e+31 a f)-32 c^2 \left (48 a e f+49 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4}+\frac {\left (99 b^2 f^2-12 c f (24 b e+7 a f)+224 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{5/2}}{1344 c^3}+\frac {f (32 c e-11 b f) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\left (\left (b^2-4 a c\right ) \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{4096 c^5}\\ &=-\frac {\left (b^2-4 a c\right ) \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^6}+\frac {\left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}+\frac {\left (5376 c^3 d e-693 b^3 f^2+36 b c f (56 b e+31 a f)-32 c^2 \left (48 a e f+49 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4}+\frac {\left (99 b^2 f^2-12 c f (24 b e+7 a f)+224 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{5/2}}{1344 c^3}+\frac {f (32 c e-11 b f) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{32768 c^6}\\ &=-\frac {\left (b^2-4 a c\right ) \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^6}+\frac {\left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}+\frac {\left (5376 c^3 d e-693 b^3 f^2+36 b c f (56 b e+31 a f)-32 c^2 \left (48 a e f+49 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4}+\frac {\left (99 b^2 f^2-12 c f (24 b e+7 a f)+224 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{5/2}}{1344 c^3}+\frac {f (32 c e-11 b f) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{16384 c^6}\\ &=-\frac {\left (b^2-4 a c\right ) \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^6}+\frac {\left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^5}+\frac {\left (5376 c^3 d e-693 b^3 f^2+36 b c f (56 b e+31 a f)-32 c^2 \left (48 a e f+49 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{13440 c^4}+\frac {\left (99 b^2 f^2-12 c f (24 b e+7 a f)+224 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{5/2}}{1344 c^3}+\frac {f (32 c e-11 b f) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\left (b^2-4 a c\right )^2 \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 5.08, size = 766, normalized size = 1.36 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-10395 b^7 f^2+630 b^6 c f (48 e+11 f x)-84 b^5 c \left (-1095 a f^2+c \left (280 e^2+560 d f+240 e f x+66 f^2 x^2\right )\right )+8 b^4 c^2 \left (560 c d (18 e+7 f x)-63 a f (480 e+107 f x)+2 c x \left (980 e^2+1008 e f x+297 f^2 x^2\right )\right )-16 b^3 c^2 \left (15309 a^2 f^2-4 a c \left (2660 e^2+5320 d f+2184 e f x+585 f^2 x^2\right )+8 c^2 \left (630 d^2+28 d x (15 e+7 f x)+x^2 \left (98 e^2+108 e f x+33 f^2 x^2\right )\right )\right )+96 b^2 c^3 \left (a^2 f (5488 e+1181 f x)+8 c^2 x \left (70 d^2+28 d x (2 e+f x)+x^2 \left (14 e^2+16 e f x+5 f^2 x^2\right )\right )-4 a c \left (56 d (25 e+9 f x)+x \left (252 e^2+248 e f x+71 f^2 x^2\right )\right )\right )+64 b c^3 \left (2757 a^3 f^2-6 a^2 c \left (756 e^2+584 e f x+f \left (1512 d+151 f x^2\right )\right )+24 a c^2 \left (350 d^2+28 d x (7 e+3 f x)+x^2 \left (42 e^2+44 e f x+13 f^2 x^2\right )\right )+16 c^3 x^2 \left (630 d^2+28 d x (33 e+26 f x)+x^2 \left (364 e^2+600 e f x+255 f^2 x^2\right )\right )\right )+128 c^4 \left (-3 a^3 f (512 e+105 f x)+16 c^3 x^3 \left (210 d^2+56 d x (6 e+5 f x)+5 x^2 \left (28 e^2+48 e f x+21 f^2 x^2\right )\right )+6 a^2 c \left (56 d (16 e+5 f x)+x \left (140 e^2+128 e f x+35 f^2 x^2\right )\right )+8 a c^2 x \left (1050 d^2+28 d x (48 e+35 f x)+x^2 \left (490 e^2+768 e f x+315 f^2 x^2\right )\right )\right )\right )-105 \left (b^2-4 a c\right )^2 \left (768 c^4 d^2+99 b^4 f^2-72 b^2 c f (4 b e+3 a f)-128 c^3 \left (6 b d e+a \left (e^2+2 d f\right )\right )+16 c^2 \left (24 a b e f+3 a^2 f^2+14 b^2 \left (e^2+2 d f\right )\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{3440640 c^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-10395*b^7*f^2 + 630*b^6*c*f*(48*e + 11*f*x) - 84*b^5*c*(-1095*a*f^2 + c*(28
0*e^2 + 560*d*f + 240*e*f*x + 66*f^2*x^2)) + 8*b^4*c^2*(560*c*d*(18*e + 7*f*x) - 63*a*f*(480*e + 107*f*x) + 2*
c*x*(980*e^2 + 1008*e*f*x + 297*f^2*x^2)) - 16*b^3*c^2*(15309*a^2*f^2 - 4*a*c*(2660*e^2 + 5320*d*f + 2184*e*f*
x + 585*f^2*x^2) + 8*c^2*(630*d^2 + 28*d*x*(15*e + 7*f*x) + x^2*(98*e^2 + 108*e*f*x + 33*f^2*x^2))) + 96*b^2*c
^3*(a^2*f*(5488*e + 1181*f*x) + 8*c^2*x*(70*d^2 + 28*d*x*(2*e + f*x) + x^2*(14*e^2 + 16*e*f*x + 5*f^2*x^2)) -
4*a*c*(56*d*(25*e + 9*f*x) + x*(252*e^2 + 248*e*f*x + 71*f^2*x^2))) + 64*b*c^3*(2757*a^3*f^2 - 6*a^2*c*(756*e^
2 + 584*e*f*x + f*(1512*d + 151*f*x^2)) + 24*a*c^2*(350*d^2 + 28*d*x*(7*e + 3*f*x) + x^2*(42*e^2 + 44*e*f*x +
13*f^2*x^2)) + 16*c^3*x^2*(630*d^2 + 28*d*x*(33*e + 26*f*x) + x^2*(364*e^2 + 600*e*f*x + 255*f^2*x^2))) + 128*
c^4*(-3*a^3*f*(512*e + 105*f*x) + 16*c^3*x^3*(210*d^2 + 56*d*x*(6*e + 5*f*x) + 5*x^2*(28*e^2 + 48*e*f*x + 21*f
^2*x^2)) + 6*a^2*c*(56*d*(16*e + 5*f*x) + x*(140*e^2 + 128*e*f*x + 35*f^2*x^2)) + 8*a*c^2*x*(1050*d^2 + 28*d*x
*(48*e + 35*f*x) + x^2*(490*e^2 + 768*e*f*x + 315*f^2*x^2)))) - 105*(b^2 - 4*a*c)^2*(768*c^4*d^2 + 99*b^4*f^2
- 72*b^2*c*f*(4*b*e + 3*a*f) - 128*c^3*(6*b*d*e + a*(e^2 + 2*d*f)) + 16*c^2*(24*a*b*e*f + 3*a^2*f^2 + 14*b^2*(
e^2 + 2*d*f)))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(3440640*c^(13/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1653\) vs. \(2(530)=1060\).
time = 0.15, size = 1654, normalized size = 2.93

method	result	size
default	\(\text {Expression too large to display}\)	\(1654\)
risch	\(\text {Expression too large to display}\)	\(1751\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^2*(1/8*x^3*(c*x^2+b*x+a)^(5/2)/c-11/16*b/c*(1/7*x^2*(c*x^2+b*x+a)^(5/2)/c-9/14*b/c*(1/6*x*(c*x^2+b*x+a)^(5/2
)/c-7/12*b/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(
2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*
(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c
^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-2/7*a/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b
)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))-3/8*a/c*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x+a)^(5/
2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*
(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2
)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))))))+2*e*f*(1/7*x^2*(c*x^2+b*x+a)^(5/2)/c-9/14*b/c*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(
c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b
*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)/c*(c
*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-2/7*a/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(
3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))))))+(2*d*f+e^2)*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1
/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(
3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b
^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
)))+2*d*e*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c
*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+d^2*(1/8*(2
*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*
ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)^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 deta

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1086 vs. \(2 (536) = 1072\).
time = 4.58, size = 2175, normalized size = 3.86 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6881280*(105*(768*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^2 + 64*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^
4 - 64*a^3*c^5)*d*f + 3*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280*a^3*b^2*c^3 + 256*a^4*c^4)*f^2 + 32*(7
*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*e^2 - 96*(8*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d +
(3*b^7*c - 28*a*b^5*c^2 + 80*a^2*b^3*c^3 - 64*a^3*b*c^4)*f)*e)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqr
t(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(215040*c^8*f^2*x^7 + 261120*b*c^7*f^2*x^6 + 1280*(448*c^8
*d*f + 3*(b^2*c^6 + 84*a*c^7)*f^2)*x^5 + 128*(5824*b*c^7*d*f - 3*(11*b^3*c^5 - 52*a*b*c^6)*f^2)*x^4 + 16*(2688
0*c^8*d^2 + 448*(3*b^2*c^6 + 140*a*c^7)*d*f + 3*(99*b^4*c^4 - 568*a*b^2*c^5 + 560*a^2*c^6)*f^2)*x^3 - 26880*(3
*b^3*c^5 - 20*a*b*c^6)*d^2 - 448*(105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*d*f - 3*(3465*b^7*c - 30660*a*
b^5*c^2 + 81648*a^2*b^3*c^3 - 58816*a^3*b*c^4)*f^2 + 8*(80640*b*c^7*d^2 - 448*(7*b^3*c^5 - 36*a*b*c^6)*d*f - 3
*(231*b^5*c^3 - 1560*a*b^3*c^4 + 2416*a^2*b*c^5)*f^2)*x^2 + 2*(26880*(b^2*c^6 + 20*a*c^7)*d^2 + 448*(35*b^4*c^
4 - 216*a*b^2*c^5 + 240*a^2*c^6)*d*f + 3*(1155*b^6*c^2 - 8988*a*b^4*c^3 + 18896*a^2*b^2*c^4 - 6720*a^3*c^5)*f^
2)*x + 224*(1280*c^8*x^5 + 1664*b*c^7*x^4 - 105*b^5*c^3 + 760*a*b^3*c^4 - 1296*a^2*b*c^5 + 16*(3*b^2*c^6 + 140
*a*c^7)*x^3 - 8*(7*b^3*c^5 - 36*a*b*c^6)*x^2 + 2*(35*b^4*c^4 - 216*a*b^2*c^5 + 240*a^2*c^6)*x)*e^2 + 96*(5120*
c^8*f*x^6 + 6400*b*c^7*f*x^5 + 128*(56*c^8*d + (b^2*c^6 + 64*a*c^7)*f)*x^4 + 16*(616*b*c^7*d - (9*b^3*c^5 - 44
*a*b*c^6)*f)*x^3 + 8*(56*(b^2*c^6 + 32*a*c^7)*d + (21*b^4*c^4 - 124*a*b^2*c^5 + 128*a^2*c^6)*f)*x^2 + 56*(15*b
^4*c^4 - 100*a*b^2*c^5 + 128*a^2*c^6)*d + (315*b^6*c^2 - 2520*a*b^4*c^3 + 5488*a^2*b^2*c^4 - 2048*a^3*c^5)*f -
2*(56*(5*b^3*c^5 - 28*a*b*c^6)*d + (105*b^5*c^3 - 728*a*b^3*c^4 + 1168*a^2*b*c^5)*f)*x)*e)*sqrt(c*x^2 + b*x +
a))/c^7, -1/3440640*(105*(768*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^2 + 64*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a
^2*b^2*c^4 - 64*a^3*c^5)*d*f + 3*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280*a^3*b^2*c^3 + 256*a^4*c^4)*f^
2 + 32*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*e^2 - 96*(8*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b
*c^5)*d + (3*b^7*c - 28*a*b^5*c^2 + 80*a^2*b^3*c^3 - 64*a^3*b*c^4)*f)*e)*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*(215040*c^8*f^2*x^7 + 261120*b*c^7*f^2*x^6 + 1280*(448*
c^8*d*f + 3*(b^2*c^6 + 84*a*c^7)*f^2)*x^5 + 128*(5824*b*c^7*d*f - 3*(11*b^3*c^5 - 52*a*b*c^6)*f^2)*x^4 + 16*(2
6880*c^8*d^2 + 448*(3*b^2*c^6 + 140*a*c^7)*d*f + 3*(99*b^4*c^4 - 568*a*b^2*c^5 + 560*a^2*c^6)*f^2)*x^3 - 26880
*(3*b^3*c^5 - 20*a*b*c^6)*d^2 - 448*(105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*d*f - 3*(3465*b^7*c - 30660
*a*b^5*c^2 + 81648*a^2*b^3*c^3 - 58816*a^3*b*c^4)*f^2 + 8*(80640*b*c^7*d^2 - 448*(7*b^3*c^5 - 36*a*b*c^6)*d*f
- 3*(231*b^5*c^3 - 1560*a*b^3*c^4 + 2416*a^2*b*c^5)*f^2)*x^2 + 2*(26880*(b^2*c^6 + 20*a*c^7)*d^2 + 448*(35*b^4
*c^4 - 216*a*b^2*c^5 + 240*a^2*c^6)*d*f + 3*(1155*b^6*c^2 - 8988*a*b^4*c^3 + 18896*a^2*b^2*c^4 - 6720*a^3*c^5)
*f^2)*x + 224*(1280*c^8*x^5 + 1664*b*c^7*x^4 - 105*b^5*c^3 + 760*a*b^3*c^4 - 1296*a^2*b*c^5 + 16*(3*b^2*c^6 +
140*a*c^7)*x^3 - 8*(7*b^3*c^5 - 36*a*b*c^6)*x^2 + 2*(35*b^4*c^4 - 216*a*b^2*c^5 + 240*a^2*c^6)*x)*e^2 + 96*(51
20*c^8*f*x^6 + 6400*b*c^7*f*x^5 + 128*(56*c^8*d + (b^2*c^6 + 64*a*c^7)*f)*x^4 + 16*(616*b*c^7*d - (9*b^3*c^5 -
44*a*b*c^6)*f)*x^3 + 8*(56*(b^2*c^6 + 32*a*c^7)*d + (21*b^4*c^4 - 124*a*b^2*c^5 + 128*a^2*c^6)*f)*x^2 + 56*(1
5*b^4*c^4 - 100*a*b^2*c^5 + 128*a^2*c^6)*d + (315*b^6*c^2 - 2520*a*b^4*c^3 + 5488*a^2*b^2*c^4 - 2048*a^3*c^5)*
f - 2*(56*(5*b^3*c^5 - 28*a*b*c^6)*d + (105*b^5*c^3 - 728*a*b^3*c^4 + 1168*a^2*b*c^5)*f)*x)*e)*sqrt(c*x^2 + b*
x + a))/c^7]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1150 vs. \(2 (536) = 1072\).
time = 6.53, size = 1150, normalized size = 2.04 \begin {gather*} \frac {1}{1720320} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, {\left (14 \, c f^{2} x + \frac {17 \, b c^{7} f^{2} + 32 \, c^{8} f e}{c^{7}}\right )} x + \frac {448 \, c^{8} d f + 3 \, b^{2} c^{6} f^{2} + 252 \, a c^{7} f^{2} + 480 \, b c^{7} f e + 224 \, c^{8} e^{2}}{c^{7}}\right )} x + \frac {5824 \, b c^{7} d f - 33 \, b^{3} c^{5} f^{2} + 156 \, a b c^{6} f^{2} + 5376 \, c^{8} d e + 96 \, b^{2} c^{6} f e + 6144 \, a c^{7} f e + 2912 \, b c^{7} e^{2}}{c^{7}}\right )} x + \frac {26880 \, c^{8} d^{2} + 1344 \, b^{2} c^{6} d f + 62720 \, a c^{7} d f + 297 \, b^{4} c^{4} f^{2} - 1704 \, a b^{2} c^{5} f^{2} + 1680 \, a^{2} c^{6} f^{2} + 59136 \, b c^{7} d e - 864 \, b^{3} c^{5} f e + 4224 \, a b c^{6} f e + 672 \, b^{2} c^{6} e^{2} + 31360 \, a c^{7} e^{2}}{c^{7}}\right )} x + \frac {80640 \, b c^{7} d^{2} - 3136 \, b^{3} c^{5} d f + 16128 \, a b c^{6} d f - 693 \, b^{5} c^{3} f^{2} + 4680 \, a b^{3} c^{4} f^{2} - 7248 \, a^{2} b c^{5} f^{2} + 5376 \, b^{2} c^{6} d e + 172032 \, a c^{7} d e + 2016 \, b^{4} c^{4} f e - 11904 \, a b^{2} c^{5} f e + 12288 \, a^{2} c^{6} f e - 1568 \, b^{3} c^{5} e^{2} + 8064 \, a b c^{6} e^{2}}{c^{7}}\right )} x + \frac {26880 \, b^{2} c^{6} d^{2} + 537600 \, a c^{7} d^{2} + 15680 \, b^{4} c^{4} d f - 96768 \, a b^{2} c^{5} d f + 107520 \, a^{2} c^{6} d f + 3465 \, b^{6} c^{2} f^{2} - 26964 \, a b^{4} c^{3} f^{2} + 56688 \, a^{2} b^{2} c^{4} f^{2} - 20160 \, a^{3} c^{5} f^{2} - 26880 \, b^{3} c^{5} d e + 150528 \, a b c^{6} d e - 10080 \, b^{5} c^{3} f e + 69888 \, a b^{3} c^{4} f e - 112128 \, a^{2} b c^{5} f e + 7840 \, b^{4} c^{4} e^{2} - 48384 \, a b^{2} c^{5} e^{2} + 53760 \, a^{2} c^{6} e^{2}}{c^{7}}\right )} x - \frac {80640 \, b^{3} c^{5} d^{2} - 537600 \, a b c^{6} d^{2} + 47040 \, b^{5} c^{3} d f - 340480 \, a b^{3} c^{4} d f + 580608 \, a^{2} b c^{5} d f + 10395 \, b^{7} c f^{2} - 91980 \, a b^{5} c^{2} f^{2} + 244944 \, a^{2} b^{3} c^{3} f^{2} - 176448 \, a^{3} b c^{4} f^{2} - 80640 \, b^{4} c^{4} d e + 537600 \, a b^{2} c^{5} d e - 688128 \, a^{2} c^{6} d e - 30240 \, b^{6} c^{2} f e + 241920 \, a b^{4} c^{3} f e - 526848 \, a^{2} b^{2} c^{4} f e + 196608 \, a^{3} c^{5} f e + 23520 \, b^{5} c^{3} e^{2} - 170240 \, a b^{3} c^{4} e^{2} + 290304 \, a^{2} b c^{5} e^{2}}{c^{7}}\right )} - \frac {{\left (768 \, b^{4} c^{4} d^{2} - 6144 \, a b^{2} c^{5} d^{2} + 12288 \, a^{2} c^{6} d^{2} + 448 \, b^{6} c^{2} d f - 3840 \, a b^{4} c^{3} d f + 9216 \, a^{2} b^{2} c^{4} d f - 4096 \, a^{3} c^{5} d f + 99 \, b^{8} f^{2} - 1008 \, a b^{6} c f^{2} + 3360 \, a^{2} b^{4} c^{2} f^{2} - 3840 \, a^{3} b^{2} c^{3} f^{2} + 768 \, a^{4} c^{4} f^{2} - 768 \, b^{5} c^{3} d e + 6144 \, a b^{3} c^{4} d e - 12288 \, a^{2} b c^{5} d e - 288 \, b^{7} c f e + 2688 \, a b^{5} c^{2} f e - 7680 \, a^{2} b^{3} c^{3} f e + 6144 \, a^{3} b c^{4} f e + 224 \, b^{6} c^{2} e^{2} - 1920 \, a b^{4} c^{3} e^{2} + 4608 \, a^{2} b^{2} c^{4} e^{2} - 2048 \, a^{3} c^{5} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{32768 \, c^{\frac {13}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1720320*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*(14*c*f^2*x + (17*b*c^7*f^2 + 32*c^8*f*e)/c^7)*x + (448*c^
8*d*f + 3*b^2*c^6*f^2 + 252*a*c^7*f^2 + 480*b*c^7*f*e + 224*c^8*e^2)/c^7)*x + (5824*b*c^7*d*f - 33*b^3*c^5*f^2
+ 156*a*b*c^6*f^2 + 5376*c^8*d*e + 96*b^2*c^6*f*e + 6144*a*c^7*f*e + 2912*b*c^7*e^2)/c^7)*x + (26880*c^8*d^2
+ 1344*b^2*c^6*d*f + 62720*a*c^7*d*f + 297*b^4*c^4*f^2 - 1704*a*b^2*c^5*f^2 + 1680*a^2*c^6*f^2 + 59136*b*c^7*d
*e - 864*b^3*c^5*f*e + 4224*a*b*c^6*f*e + 672*b^2*c^6*e^2 + 31360*a*c^7*e^2)/c^7)*x + (80640*b*c^7*d^2 - 3136*
b^3*c^5*d*f + 16128*a*b*c^6*d*f - 693*b^5*c^3*f^2 + 4680*a*b^3*c^4*f^2 - 7248*a^2*b*c^5*f^2 + 5376*b^2*c^6*d*e
+ 172032*a*c^7*d*e + 2016*b^4*c^4*f*e - 11904*a*b^2*c^5*f*e + 12288*a^2*c^6*f*e - 1568*b^3*c^5*e^2 + 8064*a*b
*c^6*e^2)/c^7)*x + (26880*b^2*c^6*d^2 + 537600*a*c^7*d^2 + 15680*b^4*c^4*d*f - 96768*a*b^2*c^5*d*f + 107520*a^
2*c^6*d*f + 3465*b^6*c^2*f^2 - 26964*a*b^4*c^3*f^2 + 56688*a^2*b^2*c^4*f^2 - 20160*a^3*c^5*f^2 - 26880*b^3*c^5
*d*e + 150528*a*b*c^6*d*e - 10080*b^5*c^3*f*e + 69888*a*b^3*c^4*f*e - 112128*a^2*b*c^5*f*e + 7840*b^4*c^4*e^2
- 48384*a*b^2*c^5*e^2 + 53760*a^2*c^6*e^2)/c^7)*x - (80640*b^3*c^5*d^2 - 537600*a*b*c^6*d^2 + 47040*b^5*c^3*d*
f - 340480*a*b^3*c^4*d*f + 580608*a^2*b*c^5*d*f + 10395*b^7*c*f^2 - 91980*a*b^5*c^2*f^2 + 244944*a^2*b^3*c^3*f
^2 - 176448*a^3*b*c^4*f^2 - 80640*b^4*c^4*d*e + 537600*a*b^2*c^5*d*e - 688128*a^2*c^6*d*e - 30240*b^6*c^2*f*e
+ 241920*a*b^4*c^3*f*e - 526848*a^2*b^2*c^4*f*e + 196608*a^3*c^5*f*e + 23520*b^5*c^3*e^2 - 170240*a*b^3*c^4*e^
2 + 290304*a^2*b*c^5*e^2)/c^7) - 1/32768*(768*b^4*c^4*d^2 - 6144*a*b^2*c^5*d^2 + 12288*a^2*c^6*d^2 + 448*b^6*c
^2*d*f - 3840*a*b^4*c^3*d*f + 9216*a^2*b^2*c^4*d*f - 4096*a^3*c^5*d*f + 99*b^8*f^2 - 1008*a*b^6*c*f^2 + 3360*a
^2*b^4*c^2*f^2 - 3840*a^3*b^2*c^3*f^2 + 768*a^4*c^4*f^2 - 768*b^5*c^3*d*e + 6144*a*b^3*c^4*d*e - 12288*a^2*b*c
^5*d*e - 288*b^7*c*f*e + 2688*a*b^5*c^2*f*e - 7680*a^2*b^3*c^3*f*e + 6144*a^3*b*c^4*f*e + 224*b^6*c^2*e^2 - 19
20*a*b^4*c^3*e^2 + 4608*a^2*b^2*c^4*e^2 - 2048*a^3*c^5*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqr
t(c) - b))/c^(13/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________