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

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]

-((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) + ((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)*(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) + ((9
9*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))

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Rubi [A]  time = 2.0149, antiderivative size = 564, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )}{32768 c^{13/2}}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )}{16384 c^6}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )}{6144 c^5}+\frac{\left (a+b x+c x^2\right )^{5/2} \left (-32 c^2 \left (48 a e f+49 b \left (2 d f+e^2\right )\right )+36 b c f (31 a f+56 b e)-693 b^3 f^2+5376 c^3 d e\right )}{13440 c^4}+\frac{x \left (a+b x+c x^2\right )^{5/2} \left (-12 c f (7 a f+24 b e)+99 b^2 f^2+224 c^2 \left (2 d f+e^2\right )\right )}{1344 c^3}+\frac{f x^2 \left (a+b x+c x^2\right )^{5/2} (32 c e-11 b f)}{112 c^2}+\frac{f^2 x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c} \]

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) + ((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)*(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) + ((9
9*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))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 3.71328, size = 766, normalized size = 1.36 \[ \frac{105 \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (16 c^2 \left (3 a^2 f^2+24 a b e f+14 b^2 \left (2 d f+e^2\right )\right )-72 b^2 c f (3 a f+4 b e)-128 c^3 \left (a \left (2 d f+e^2\right )+6 b d e\right )+99 b^4 f^2+768 c^4 d^2\right )-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b^3 c^2 \left (15309 a^2 f^2-4 a c \left (5320 d f+2660 e^2+2184 e f x+585 f^2 x^2\right )+8 c^2 \left (630 d^2+28 d x (15 e+7 f x)+x^2 \left (98 e^2+108 e f x+33 f^2 x^2\right )\right )\right )-96 b^2 c^3 \left (a^2 f (5488 e+1181 f x)-4 a c \left (56 d (25 e+9 f x)+x \left (252 e^2+248 e f x+71 f^2 x^2\right )\right )+8 c^2 x \left (70 d^2+28 d x (2 e+f x)+x^2 \left (14 e^2+16 e f x+5 f^2 x^2\right )\right )\right )-64 b c^3 \left (2757 a^3 f^2-6 a^2 c \left (f \left (1512 d+151 f x^2\right )+756 e^2+584 e f x\right )+24 a c^2 \left (350 d^2+28 d x (7 e+3 f x)+x^2 \left (42 e^2+44 e f x+13 f^2 x^2\right )\right )+16 c^3 x^2 \left (630 d^2+28 d x (33 e+26 f x)+x^2 \left (364 e^2+600 e f x+255 f^2 x^2\right )\right )\right )-128 c^4 \left (-3 a^3 f (512 e+105 f x)+6 a^2 c \left (56 d (16 e+5 f x)+x \left (140 e^2+128 e f x+35 f^2 x^2\right )\right )+8 a c^2 x \left (1050 d^2+28 d x (48 e+35 f x)+x^2 \left (490 e^2+768 e f x+315 f^2 x^2\right )\right )+16 c^3 x^3 \left (210 d^2+56 d x (6 e+5 f x)+5 x^2 \left (28 e^2+48 e f x+21 f^2 x^2\right )\right )\right )+84 b^5 c \left (c \left (560 d f+280 e^2+240 e f x+66 f^2 x^2\right )-1095 a f^2\right )-8 b^4 c^2 \left (-63 a f (480 e+107 f x)+560 c d (18 e+7 f x)+2 c x \left (980 e^2+1008 e f x+297 f^2 x^2\right )\right )+10395 b^7 f^2-630 b^6 c f (48 e+11 f x)\right )}{3440640 c^{13/2}} \]

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*(280*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 + 9
9*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)) + 1
6*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))

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Maple [B]  time = 0.031, size = 2458, normalized size = 4.4 \[ \text{result too large to display} \]

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]

3/16*e*f*b/c^2*a^2*(c*x^2+b*x+a)^(1/2)*x-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*(c*x^2+b*x+a)^(3/2)*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+1/8*f^2*x^3*(c*x^2+b*x+a)^(5/2)/c-15/64*e*f*
b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+21/256*e*f*b^5/c^(9/
2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-3/14*e*f*b/c^2*x*(c*x^2+b*x+a)^
(5/2)-3/32*e*f*b^3/c^3*(c*x^2+b*x+a)^(3/2)*x+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+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)-9/128*f^2*b^2/c^3*a*(c*x^2+b*x+a)^(3
/2)*x-57/512*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+7/48*b^2/c^2*(c*x^2+b*x+a)^(3/2)*x*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+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((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d*
f-15/128*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*f-1/12/c*a*
(c*x^2+b*x+a)^(3/2)*x*d*f-1/24/c^2*a*(c*x^2+b*x+a)^(3/2)*b*d*f-1/8/c*a^2*(c*x^2+
b*x+a)^(1/2)*x*d*f-1/16/c^2*a^2*(c*x^2+b*x+a)^(1/2)*b*d*f-1/4*d*e*b/c*(c*x^2+b*x
+a)^(3/2)*x+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((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+3/16
*d*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/16*e*f*b/c^(5/2
)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/8*d^2/c^(1/2)*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+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^2-7/60*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((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2-1/16/c^(3/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))*e^2-99/16384*f^2*b^7/c^6*(c*x^2+b*x+a)^(1/2)+99/32768*f^2*b^8/c^(
13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/128*f^2/c^(5/2)*a^4*ln((1/2*
b+c*x)/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)+33/20
48*f^2*b^5/c^5*(c*x^2+b*x+a)^(3/2)+3/128*d^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))*b^4+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^3-15/128*f^2*b^2/c^(7/2)*a^3*ln((1/2*b+c
*x)/c^(1/2)+(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)+33/102
4*f^2*b^4/c^4*(c*x^2+b*x+a)^(3/2)*x+105/1024*f^2*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))*a^2-63/2048*f^2*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))*a+3/128*f^2/c^2*a^3*(c*x^2+b*x+a)^(1/2)*x+3/256*f^2/c^3*a^3*(c
*x^2+b*x+a)^(1/2)*b-1/16*f^2/c^2*a*x*(c*x^2+b*x+a)^(5/2)+1/64*f^2/c^2*a^2*(c*x^2
+b*x+a)^(3/2)*x+1/128*f^2/c^3*a^2*(c*x^2+b*x+a)^(3/2)*b-99/8192*f^2*b^6/c^5*(c*x
^2+b*x+a)^(1/2)*x+153/4096*f^2*b^5/c^5*(c*x^2+b*x+a)^(1/2)*a-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)+93/1120*f^2*b/c
^3*a*(c*x^2+b*x+a)^(5/2)-7/30*b/c^2*(c*x^2+b*x+a)^(5/2)*d*f+7/96*b^2/c^2*(c*x^2+
b*x+a)^(3/2)*x*e^2-11/112*f^2*b/c^2*x^2*(c*x^2+b*x+a)^(5/2)-1/48/c^2*a*(c*x^2+b*
x+a)^(3/2)*b*e^2-1/16/c*a^2*(c*x^2+b*x+a)^(1/2)*x*e^2-1/32/c^2*a^2*(c*x^2+b*x+a)
^(1/2)*b*e^2-1/8/c^(3/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f+1/3
*x*(c*x^2+b*x+a)^(5/2)/c*d*f-15/256*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))*a*e^2+7/512*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*
d*f-1/24/c*a*(c*x^2+b*x+a)^(3/2)*x*e^2+7/96*b^3/c^3*(c*x^2+b*x+a)^(3/2)*d*f-7/25
6*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^2-7/256
*b^5/c^4*(c*x^2+b*x+a)^(1/2)*d*f+9/64*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))*a^2*e^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((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/8*d*e*b^2/c^2*(c*x^2+b*x+a)^(3/2)
+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)+3/16*d^2/c*(c*x^2+b*x+a)^(1/2)*b*a+2/7*e*f*x^2*
(c*x^2+b*x+a)^(5/2)/c-9/1024*e*f*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))-4/35*e*f/c^2*a*(c*x^2+b*x+a)^(5/2)+1/4*d^2*(c*x^2+b*x+a)^(3/2)*x-3/32*
d^2/c*(c*x^2+b*x+a)^(1/2)*x*b^2-3/16*d^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*b^2*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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

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Fricas [A]  time = 1.15616, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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*(4*(215040*c^7*f^2*x^7 + 15360*(32*c^7*e*f + 17*b*c^6*f^2)*x^6 + 1280
*(224*c^7*e^2 + 3*(b^2*c^5 + 84*a*c^6)*f^2 + 32*(14*c^7*d + 15*b*c^6*e)*f)*x^5 +
128*(5376*c^7*d*e + 2912*b*c^6*e^2 - 3*(11*b^3*c^4 - 52*a*b*c^5)*f^2 + 32*(182*
b*c^6*d + 3*(b^2*c^5 + 64*a*c^6)*e)*f)*x^4 + 16*(26880*c^7*d^2 + 59136*b*c^6*d*e
+ 224*(3*b^2*c^5 + 140*a*c^6)*e^2 + 3*(99*b^4*c^3 - 568*a*b^2*c^4 + 560*a^2*c^5
)*f^2 + 32*(14*(3*b^2*c^5 + 140*a*c^6)*d - 3*(9*b^3*c^4 - 44*a*b*c^5)*e)*f)*x^3
- 26880*(3*b^3*c^4 - 20*a*b*c^5)*d^2 + 5376*(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^
2*c^5)*d*e - 224*(105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*e^2 - 3*(3465*b^
7 - 30660*a*b^5*c + 81648*a^2*b^3*c^2 - 58816*a^3*b*c^3)*f^2 + 8*(80640*b*c^6*d^
2 + 5376*(b^2*c^5 + 32*a*c^6)*d*e - 224*(7*b^3*c^4 - 36*a*b*c^5)*e^2 - 3*(231*b^
5*c^2 - 1560*a*b^3*c^3 + 2416*a^2*b*c^4)*f^2 - 32*(14*(7*b^3*c^4 - 36*a*b*c^5)*d
- 3*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*e)*f)*x^2 - 32*(14*(105*b^5*c^2
- 760*a*b^3*c^3 + 1296*a^2*b*c^4)*d - 3*(315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b
^2*c^3 - 2048*a^3*c^4)*e)*f + 2*(26880*(b^2*c^5 + 20*a*c^6)*d^2 - 5376*(5*b^3*c^
4 - 28*a*b*c^5)*d*e + 224*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*e^2 + 3*(11
55*b^6*c - 8988*a*b^4*c^2 + 18896*a^2*b^2*c^3 - 6720*a^3*c^4)*f^2 + 32*(14*(35*b
^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*d - 3*(105*b^5*c^2 - 728*a*b^3*c^3 + 1168*
a^2*b*c^4)*e)*f)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 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 + 8
0*a^2*b^3*c^3 - 64*a^3*b*c^4)*e)*f)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a)
- (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/c^(13/2), 1/3440640*(2*(215040*
c^7*f^2*x^7 + 15360*(32*c^7*e*f + 17*b*c^6*f^2)*x^6 + 1280*(224*c^7*e^2 + 3*(b^2
*c^5 + 84*a*c^6)*f^2 + 32*(14*c^7*d + 15*b*c^6*e)*f)*x^5 + 128*(5376*c^7*d*e + 2
912*b*c^6*e^2 - 3*(11*b^3*c^4 - 52*a*b*c^5)*f^2 + 32*(182*b*c^6*d + 3*(b^2*c^5 +
64*a*c^6)*e)*f)*x^4 + 16*(26880*c^7*d^2 + 59136*b*c^6*d*e + 224*(3*b^2*c^5 + 14
0*a*c^6)*e^2 + 3*(99*b^4*c^3 - 568*a*b^2*c^4 + 560*a^2*c^5)*f^2 + 32*(14*(3*b^2*
c^5 + 140*a*c^6)*d - 3*(9*b^3*c^4 - 44*a*b*c^5)*e)*f)*x^3 - 26880*(3*b^3*c^4 - 2
0*a*b*c^5)*d^2 + 5376*(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d*e - 224*(105*
b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*e^2 - 3*(3465*b^7 - 30660*a*b^5*c + 81
648*a^2*b^3*c^2 - 58816*a^3*b*c^3)*f^2 + 8*(80640*b*c^6*d^2 + 5376*(b^2*c^5 + 32
*a*c^6)*d*e - 224*(7*b^3*c^4 - 36*a*b*c^5)*e^2 - 3*(231*b^5*c^2 - 1560*a*b^3*c^3
+ 2416*a^2*b*c^4)*f^2 - 32*(14*(7*b^3*c^4 - 36*a*b*c^5)*d - 3*(21*b^4*c^3 - 124
*a*b^2*c^4 + 128*a^2*c^5)*e)*f)*x^2 - 32*(14*(105*b^5*c^2 - 760*a*b^3*c^3 + 1296
*a^2*b*c^4)*d - 3*(315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3*c^4)
*e)*f + 2*(26880*(b^2*c^5 + 20*a*c^6)*d^2 - 5376*(5*b^3*c^4 - 28*a*b*c^5)*d*e +
224*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*e^2 + 3*(1155*b^6*c - 8988*a*b^4*
c^2 + 18896*a^2*b^2*c^3 - 6720*a^3*c^4)*f^2 + 32*(14*(35*b^4*c^3 - 216*a*b^2*c^4
+ 240*a^2*c^5)*d - 3*(105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*e)*f)*x)*sq
rt(c*x^2 + b*x + a)*sqrt(-c) + 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)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt
(-c)*c^6)]

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

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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GIAC/XCAS [A]  time = 0.283909, size = 1, normalized size = 0. \[ \mathit{Done} \]

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]

Done