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

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

[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

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Rubi [A]  time = 0.93, 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 = {1661, 640, 612, 621, 206} \[ \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )}{6144 c^5}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )}{16384 c^6}+\frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )}{32768 c^{13/2}}+\frac {x \left (a+b x+c x^2\right )^{5/2} \left (-12 c f (7 a f+24 b e)+99 b^2 f^2+224 c^2 \left (2 d f+e^2\right )\right )}{1344 c^3}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-32 c^2 \left (48 a e f+49 b \left (2 d f+e^2\right )\right )+36 b c f (31 a f+56 b e)-693 b^3 f^2+5376 c^3 d e\right )}{13440 c^4}+\frac {f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{112 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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])/(16384*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 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 612

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 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 \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 ) \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 )}{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*}

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Mathematica [A]  time = 1.76, size = 829, normalized size = 1.47 \[ \frac {430080 f^2 (a+x (b+c x))^{5/2} x^3+983040 e f (a+x (b+c x))^{5/2} x^2+573440 \left (e^2+2 d f\right ) (a+x (b+c x))^{5/2} x+1376256 d e (a+x (b+c x))^{5/2}+430080 d^2 (b+2 c x) (a+x (b+c x))^{3/2}+\frac {80640 \left (b^2-4 a c\right ) d^2 \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}\right )}{c^{3/2}}-\frac {26880 b d e \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )}{c^{5/2}}+\frac {96 e f \left (-256 c^{5/2} \left (-21 b^2+30 c x b+16 a c\right ) (a+x (b+c x))^{5/2}-35 b \left (3 b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{c^{9/2}}-\frac {224 \left (e^2+2 d f\right ) \left (1792 b c^{5/2} (a+x (b+c x))^{5/2}-5 \left (7 b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{c^{7/2}}-\frac {3 f^2 \left (112640 b x^2 (a+x (b+c x))^{5/2} c^{9/2}+256 \left (231 b^3-330 c x b^2-372 a c b+280 a c^2 x\right ) (a+x (b+c x))^{5/2} c^{5/2}-35 \left (33 b^4-72 a c b^2+16 a^2 c^2\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{c^{11/2}}}{3440640 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(430080*d^2*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) + 1376256*d*e*(a + x*(b + c*x))^(5/2) + 573440*(e^2 + 2*d*f)*x
*(a + x*(b + c*x))^(5/2) + 983040*e*f*x^2*(a + x*(b + c*x))^(5/2) + 430080*f^2*x^3*(a + x*(b + c*x))^(5/2) + (
80640*(b^2 - 4*a*c)*d^2*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*S
qrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(3/2) - (26880*b*d*e*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b
^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ x*(b + c*x)])])))/c^(5/2) + (96*e*f*(-256*c^(5/2)*(-21*b^2 + 16*a*c + 30*b*c*x)*(a + x*(b + c*x))^(5/2) - 3
5*b*(3*b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*S
qrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))))/c^(9/2) - (224
*(e^2 + 2*d*f)*(1792*b*c^(5/2)*(a + x*(b + c*x))^(5/2) - 5*(7*b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b +
c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x
)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))))/c^(7/2) - (3*f^2*(112640*b*c^(9/2)*x^2*(a + x*(b + c*x))^(5/2) + 256*
c^(5/2)*(231*b^3 - 372*a*b*c - 330*b^2*c*x + 280*a*c^2*x)*(a + x*(b + c*x))^(5/2) - 35*(33*b^4 - 72*a*b^2*c +
16*a^2*c^2)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a +
x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))))/c^(11/2))/(3440640*c)

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fricas [B]  time = 3.79, size = 2179, normalized size = 3.86 \[ \text {result too large to display} \]

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 - 768*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d*e
+ 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 + 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*(2*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*d -
3*(3*b^7*c - 28*a*b^5*c^2 + 80*a^2*b^3*c^3 - 64*a^3*b*c^4)*e)*f)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*s
qrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(215040*c^8*f^2*x^7 + 15360*(32*c^8*e*f + 17*b*c^7*f^2)*
x^6 + 1280*(224*c^8*e^2 + 3*(b^2*c^6 + 84*a*c^7)*f^2 + 32*(14*c^8*d + 15*b*c^7*e)*f)*x^5 + 128*(5376*c^8*d*e +
2912*b*c^7*e^2 - 3*(11*b^3*c^5 - 52*a*b*c^6)*f^2 + 32*(182*b*c^7*d + 3*(b^2*c^6 + 64*a*c^7)*e)*f)*x^4 + 16*(2
6880*c^8*d^2 + 59136*b*c^7*d*e + 224*(3*b^2*c^6 + 140*a*c^7)*e^2 + 3*(99*b^4*c^4 - 568*a*b^2*c^5 + 560*a^2*c^6
)*f^2 + 32*(14*(3*b^2*c^6 + 140*a*c^7)*d - 3*(9*b^3*c^5 - 44*a*b*c^6)*e)*f)*x^3 - 26880*(3*b^3*c^5 - 20*a*b*c^
6)*d^2 + 5376*(15*b^4*c^4 - 100*a*b^2*c^5 + 128*a^2*c^6)*d*e - 224*(105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c
^5)*e^2 - 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 + 53
76*(b^2*c^6 + 32*a*c^7)*d*e - 224*(7*b^3*c^5 - 36*a*b*c^6)*e^2 - 3*(231*b^5*c^3 - 1560*a*b^3*c^4 + 2416*a^2*b*
c^5)*f^2 - 32*(14*(7*b^3*c^5 - 36*a*b*c^6)*d - 3*(21*b^4*c^4 - 124*a*b^2*c^5 + 128*a^2*c^6)*e)*f)*x^2 - 32*(14
*(105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*d - 3*(315*b^6*c^2 - 2520*a*b^4*c^3 + 5488*a^2*b^2*c^4 - 2048*
a^3*c^5)*e)*f + 2*(26880*(b^2*c^6 + 20*a*c^7)*d^2 - 5376*(5*b^3*c^5 - 28*a*b*c^6)*d*e + 224*(35*b^4*c^4 - 216*
a*b^2*c^5 + 240*a^2*c^6)*e^2 + 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 + 32*(
14*(35*b^4*c^4 - 216*a*b^2*c^5 + 240*a^2*c^6)*d - 3*(105*b^5*c^3 - 728*a*b^3*c^4 + 1168*a^2*b*c^5)*e)*f)*x)*sq
rt(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 - 768*(b^5*c^3 - 8*a*b
^3*c^4 + 16*a^2*b*c^5)*d*e + 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 + 3*(33*b^8 - 33
6*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*(2*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2
*b^2*c^4 - 64*a^3*c^5)*d - 3*(3*b^7*c - 28*a*b^5*c^2 + 80*a^2*b^3*c^3 - 64*a^3*b*c^4)*e)*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*(215040*c^8*f^2*x^7 + 15360*(32*c^8*
e*f + 17*b*c^7*f^2)*x^6 + 1280*(224*c^8*e^2 + 3*(b^2*c^6 + 84*a*c^7)*f^2 + 32*(14*c^8*d + 15*b*c^7*e)*f)*x^5 +
128*(5376*c^8*d*e + 2912*b*c^7*e^2 - 3*(11*b^3*c^5 - 52*a*b*c^6)*f^2 + 32*(182*b*c^7*d + 3*(b^2*c^6 + 64*a*c^
7)*e)*f)*x^4 + 16*(26880*c^8*d^2 + 59136*b*c^7*d*e + 224*(3*b^2*c^6 + 140*a*c^7)*e^2 + 3*(99*b^4*c^4 - 568*a*b
^2*c^5 + 560*a^2*c^6)*f^2 + 32*(14*(3*b^2*c^6 + 140*a*c^7)*d - 3*(9*b^3*c^5 - 44*a*b*c^6)*e)*f)*x^3 - 26880*(3
*b^3*c^5 - 20*a*b*c^6)*d^2 + 5376*(15*b^4*c^4 - 100*a*b^2*c^5 + 128*a^2*c^6)*d*e - 224*(105*b^5*c^3 - 760*a*b^
3*c^4 + 1296*a^2*b*c^5)*e^2 - 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 + 5376*(b^2*c^6 + 32*a*c^7)*d*e - 224*(7*b^3*c^5 - 36*a*b*c^6)*e^2 - 3*(231*b^5*c^3 - 1560*a*b
^3*c^4 + 2416*a^2*b*c^5)*f^2 - 32*(14*(7*b^3*c^5 - 36*a*b*c^6)*d - 3*(21*b^4*c^4 - 124*a*b^2*c^5 + 128*a^2*c^6
)*e)*f)*x^2 - 32*(14*(105*b^5*c^3 - 760*a*b^3*c^4 + 1296*a^2*b*c^5)*d - 3*(315*b^6*c^2 - 2520*a*b^4*c^3 + 5488
*a^2*b^2*c^4 - 2048*a^3*c^5)*e)*f + 2*(26880*(b^2*c^6 + 20*a*c^7)*d^2 - 5376*(5*b^3*c^5 - 28*a*b*c^6)*d*e + 22
4*(35*b^4*c^4 - 216*a*b^2*c^5 + 240*a^2*c^6)*e^2 + 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 + 32*(14*(35*b^4*c^4 - 216*a*b^2*c^5 + 240*a^2*c^6)*d - 3*(105*b^5*c^3 - 728*a*b^3*c^4 + 1168*a^
2*b*c^5)*e)*f)*x)*sqrt(c*x^2 + b*x + a))/c^7]

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giac [B]  time = 0.60, size = 1150, normalized size = 2.04 \[ \text {result too large to display} \]

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)

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maple [B]  time = 0.02, size = 2458, normalized size = 4.36 \[ \text {result too large to display} \]

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)

[Out]

1/8*f^2*x^3*(c*x^2+b*x+a)^(5/2)/c+1/8*d^2/c*(c*x^2+b*x+a)^(3/2)*b+3/8*d^2*(c*x^2+b*x+a)^(1/2)*x*a-3/64*d^2/c^2
*(c*x^2+b*x+a)^(1/2)*b^3+3/8*d^2/c^(1/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+3/128*d^2/c^(5/2)*ln(
(c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4+2/5*d*e*(c*x^2+b*x+a)^(5/2)/c+1/6*x*(c*x^2+b*x+a)^(5/2)/c*e^2-7/6
0*b/c^2*(c*x^2+b*x+a)^(5/2)*e^2+7/192*b^3/c^3*(c*x^2+b*x+a)^(3/2)*e^2-7/512*b^5/c^4*(c*x^2+b*x+a)^(1/2)*e^2+7/
1024*b^6/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2-1/16*a^3/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))*e^2+3/128*f^2*a^4/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+99/32768*f^2*b^8/c^(13/2
)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-33/640*f^2*b^3/c^4*(c*x^2+b*x+a)^(5/2)+33/2048*f^2*b^5/c^5*(c*x^
2+b*x+a)^(3/2)-99/16384*f^2*b^7/c^6*(c*x^2+b*x+a)^(1/2)-3/16*e*f*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x*a+1/8*e*f*b/c^2
*a*x*(c*x^2+b*x+a)^(3/2)+3/16*e*f*b/c^2*a^2*(c*x^2+b*x+a)^(1/2)*x+1/4*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a*d*f-3/8*
d*e*b/c*(c*x^2+b*x+a)^(1/2)*x*a+3/16*d*e*b^3/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/16*e*f*b^
2/c^3*a*(c*x^2+b*x+a)^(3/2)+3/32*e*f*b^2/c^3*a^2*(c*x^2+b*x+a)^(1/2)-3/14*e*f*b/c^2*x*(c*x^2+b*x+a)^(5/2)-3/32
*e*f*b^3/c^3*x*(c*x^2+b*x+a)^(3/2)+9/256*e*f*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x-3/32*e*f*b^4/c^4*(c*x^2+b*x+a)^(1/2
)*a-15/64*e*f*b^3/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+21/256*e*f*b^5/c^(9/2)*ln((c*x+1/2*b
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/16*e*f*b/c^(5/2)*a^3*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+7/48*b^2/c
^2*x*(c*x^2+b*x+a)^(3/2)*d*f+1/8*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a*e^2-7/128*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*d*f-5
7/512*f^2*b^2/c^3*a^2*(c*x^2+b*x+a)^(1/2)*x+153/2048*f^2*b^4/c^4*(c*x^2+b*x+a)^(1/2)*x*a-9/128*f^2*b^2/c^3*a*x
*(c*x^2+b*x+a)^(3/2)+1/8*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a*d*f+9/32*b^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*a^2*d*f-15/128*b^4/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*f-1/12*a/c*x*(c*x^2+b*x+a
)^(3/2)*d*f-1/24*a/c^2*(c*x^2+b*x+a)^(3/2)*b*d*f-1/8*a^2/c*(c*x^2+b*x+a)^(1/2)*x*d*f-1/16*a^2/c^2*(c*x^2+b*x+a
)^(1/2)*b*d*f-1/4*d*e*b/c*x*(c*x^2+b*x+a)^(3/2)+3/32*d*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x-3/16*d*e*b^2/c^2*(c*x^2
+b*x+a)^(1/2)*a-3/8*d*e*b/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+1/4*d^2*x*(c*x^2+b*x+a)^(3/2
)+153/4096*f^2*b^5/c^5*(c*x^2+b*x+a)^(1/2)*a+93/1120*f^2*b/c^3*a*(c*x^2+b*x+a)^(5/2)-11/112*f^2*b/c^2*x^2*(c*x
^2+b*x+a)^(5/2)+105/1024*f^2*b^4/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-1/8*a^3/c^(3/2)*ln((c
*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f-7/30*b/c^2*(c*x^2+b*x+a)^(5/2)*d*f+7/96*b^2/c^2*x*(c*x^2+b*x+a)^(3/
2)*e^2+7/96*b^3/c^3*(c*x^2+b*x+a)^(3/2)*d*f-7/256*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*e^2+1/16*b^3/c^3*(c*x^2+b*x+a)
^(1/2)*a*e^2-9/1024*e*f*b^7/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-4/35*e*f*a/c^2*(c*x^2+b*x+a)^
(5/2)+3/20*e*f*b^2/c^3*(c*x^2+b*x+a)^(5/2)-3/64*e*f*b^4/c^4*(c*x^2+b*x+a)^(3/2)+9/512*e*f*b^6/c^5*(c*x^2+b*x+a
)^(1/2)+2/7*e*f*x^2*(c*x^2+b*x+a)^(5/2)/c-9/256*f^2*b^3/c^4*a*(c*x^2+b*x+a)^(3/2)-57/1024*f^2*b^3/c^4*a^2*(c*x
^2+b*x+a)^(1/2)+33/448*f^2*b^2/c^3*x*(c*x^2+b*x+a)^(5/2)-63/2048*f^2*b^6/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))*a-15/128*f^2*b^2/c^(7/2)*a^3*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/64*f^2*a^2/c^2*x*(c
*x^2+b*x+a)^(3/2)+1/128*f^2*a^2/c^3*(c*x^2+b*x+a)^(3/2)*b+3/128*f^2*a^3/c^2*(c*x^2+b*x+a)^(1/2)*x+3/256*f^2*a^
3/c^3*(c*x^2+b*x+a)^(1/2)*b-1/16*f^2*a/c^2*x*(c*x^2+b*x+a)^(5/2)+33/1024*f^2*b^4/c^4*x*(c*x^2+b*x+a)^(3/2)-99/
8192*f^2*b^6/c^5*(c*x^2+b*x+a)^(1/2)*x-15/256*b^4/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e^2+7/
512*b^6/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f-1/24*a/c*x*(c*x^2+b*x+a)^(3/2)*e^2-1/48*a/c^2*
(c*x^2+b*x+a)^(3/2)*b*e^2-1/16*a^2/c*(c*x^2+b*x+a)^(1/2)*x*e^2+1/3*x*(c*x^2+b*x+a)^(5/2)/c*d*f-1/8*d*e*b^2/c^2
*(c*x^2+b*x+a)^(3/2)+3/64*d*e*b^4/c^3*(c*x^2+b*x+a)^(1/2)-3/128*d*e*b^5/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))-3/32*d^2/c*(c*x^2+b*x+a)^(1/2)*x*b^2+3/16*d^2/c*(c*x^2+b*x+a)^(1/2)*b*a-3/16*d^2/c^(3/2)*ln((c*x
+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a-1/32*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b*e^2-7/256*b^5/c^4*(c*x^2+b*x+a)^
(1/2)*d*f+9/64*b^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*e^2

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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((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 details)Is 4*a*c-b^2 positive, negative or zero?

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

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)

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

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)

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