3.114 \(\int \frac{\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=649 \[ \frac{3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right )}{128 c^{11/2}}+\frac{2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2} \left (16 c^2 f \left (20 a e f+21 b \left (d f+e^2\right )\right )-4 b c f^2 (73 a f+114 b e)+187 b^3 f^3-64 c^3 \left (6 d e f+e^3\right )\right )}{64 c^5}+\frac{f x \sqrt{a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{32 c^4}+\frac{f^2 x^2 \sqrt{a+b x+c x^2} (8 c e-5 b f)}{8 c^3}+\frac{f^3 x^3 \sqrt{a+b x+c x^2}}{4 c^2} \]
[Out]
(2*(3*a*b^4*c*e*f^2 - a*b^5*f^3 + a*b^3*c*f*(5*a*f^2 - 3*c*(e^2 + d*f)) - b*c^2*
(c^3*d^3 + 5*a^3*f^3 + 3*a*c^2*d*(e^2 + d*f) - 9*a^2*c*f*(e^2 + d*f)) - a*b^2*c^
2*e*(12*a*f^2 - c*(e^2 + 6*d*f)) + 2*a*c^3*e*(3*c^2*d^2 + 3*a^2*f^2 - a*c*(e^2 +
6*d*f)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*(c^4*d^2 - b*c^3*d*e + b^2*c^2*e^
2 - 3*a*c^3*e^2 + b^2*c^2*d*f - 2*a*c^3*d*f - 2*b^3*c*e*f + 7*a*b*c^2*e*f + b^4*
f^2 - 4*a*b^2*c*f^2 + a^2*c^2*f^2)*x))/(c^5*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])
- ((187*b^3*f^3 - 4*b*c*f^2*(114*b*e + 73*a*f) - 64*c^3*(e^3 + 6*d*e*f) + 16*c^
2*f*(20*a*e*f + 21*b*(e^2 + d*f)))*Sqrt[a + b*x + c*x^2])/(64*c^5) + (f*(41*b^2*
f^2 - 4*c*f*(22*b*e + 7*a*f) + 48*c^2*(e^2 + d*f))*x*Sqrt[a + b*x + c*x^2])/(32*
c^4) + (f^2*(8*c*e - 5*b*f)*x^2*Sqrt[a + b*x + c*x^2])/(8*c^3) + (f^3*x^3*Sqrt[a
+ b*x + c*x^2])/(4*c^2) + (3*(105*b^4*f^3 - 280*b^2*c*f^2*(b*e + a*f) + 128*c^4
*d*(e^2 + d*f) + 80*c^2*f*(6*a*b*e*f + a^2*f^2 + 3*b^2*(e^2 + d*f)) - 64*c^3*(3*
a*f*(e^2 + d*f) + b*(e^3 + 6*d*e*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*
x + c*x^2])])/(128*c^(11/2))
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Rubi [A] time = 4.14001, antiderivative size = 649, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185
\[ \frac{3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right )}{128 c^{11/2}}+\frac{2 \left (-x \left (-c (2 a f+b e)+b^2 f+2 c^2 d\right ) \left (c^2 \left (a^2 f^2+7 a b e f+b^2 \left (d f+e^2\right )\right )-2 b^2 c f (2 a f+b e)-c^3 \left (2 a d f+3 a e^2+b d e\right )+b^4 f^2+c^4 d^2\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2} \left (16 c^2 f \left (20 a e f+21 b \left (d f+e^2\right )\right )-4 b c f^2 (73 a f+114 b e)+187 b^3 f^3-64 c^3 \left (6 d e f+e^3\right )\right )}{64 c^5}+\frac{f x \sqrt{a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{32 c^4}+\frac{f^2 x^2 \sqrt{a+b x+c x^2} (8 c e-5 b f)}{8 c^3}+\frac{f^3 x^3 \sqrt{a+b x+c x^2}}{4 c^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x + f*x^2)^3/(a + b*x + c*x^2)^(3/2),x]
[Out]
(2*(3*a*b^4*c*e*f^2 - a*b^5*f^3 + a*b^3*c*f*(5*a*f^2 - 3*c*(e^2 + d*f)) - b*c^2*
(c^3*d^3 + 5*a^3*f^3 + 3*a*c^2*d*(e^2 + d*f) - 9*a^2*c*f*(e^2 + d*f)) - a*b^2*c^
2*e*(12*a*f^2 - c*(e^2 + 6*d*f)) + 2*a*c^3*e*(3*c^2*d^2 + 3*a^2*f^2 - a*c*(e^2 +
6*d*f)) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(c^4*d^2 + b^4*f^2 - 2*b^2*c*f*(b
*e + 2*a*f) - c^3*(b*d*e + 3*a*e^2 + 2*a*d*f) + c^2*(7*a*b*e*f + a^2*f^2 + b^2*(
e^2 + d*f)))*x))/(c^5*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - ((187*b^3*f^3 - 4*b
*c*f^2*(114*b*e + 73*a*f) - 64*c^3*(e^3 + 6*d*e*f) + 16*c^2*f*(20*a*e*f + 21*b*(
e^2 + d*f)))*Sqrt[a + b*x + c*x^2])/(64*c^5) + (f*(41*b^2*f^2 - 4*c*f*(22*b*e +
7*a*f) + 48*c^2*(e^2 + d*f))*x*Sqrt[a + b*x + c*x^2])/(32*c^4) + (f^2*(8*c*e - 5
*b*f)*x^2*Sqrt[a + b*x + c*x^2])/(8*c^3) + (f^3*x^3*Sqrt[a + b*x + c*x^2])/(4*c^
2) + (3*(105*b^4*f^3 - 280*b^2*c*f^2*(b*e + a*f) + 128*c^4*d*(e^2 + d*f) + 80*c^
2*f*(6*a*b*e*f + a^2*f^2 + 3*b^2*(e^2 + d*f)) - 64*c^3*(3*a*f*(e^2 + d*f) + b*(e
^3 + 6*d*e*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(
11/2))
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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((f*x**2+e*x+d)**3/(c*x**2+b*x+a)**(3/2),x)
[Out]
Timed out
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Mathematica [A] time = 4.23258, size = 745, normalized size = 1.15 \[ \frac{3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right )}{128 c^{11/2}}+\frac{-8 b^3 c \left (210 a^2 f^3+a c f \left (f \left (77 f x^2-90 d\right )-90 e^2-530 e f x\right )-c^2 x \left (2 e f \left (7 f x^2-72 d\right )+3 f^2 x \left (10 d+f x^2\right )-24 e^3+30 e^2 f x\right )\right )-16 b^2 c^2 \left (-a^2 f^2 (230 e+169 f x)+a c \left (2 e f \left (36 d-43 f x^2\right )+f^2 x \left (186 d-13 f x^2\right )+12 e^3+186 e^2 f x\right )+c^2 x \left (-24 d^2 f+6 d \left (-4 e^2+4 e f x+f^2 x^2\right )+x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )\right )\right )+16 b c^2 \left (113 a^3 f^3+a^2 c f \left (f \left (49 f x^2-156 d\right )-156 e^2-244 e f x\right )+2 a c^2 \left (12 d^2 f+6 d \left (2 e^2+20 e f x-5 f^2 x^2\right )-x \left (-20 e^3+30 e^2 f x+14 e f^2 x^2+3 f^3 x^3\right )\right )+8 c^3 d^2 (d-3 e x)\right )+32 c^3 \left (a^3 \left (-f^2\right ) (64 e+15 f x)+a^2 c \left (-32 e f \left (f x^2-3 d\right )+f^2 x \left (36 d-5 f x^2\right )+16 e^3+36 e^2 f x\right )+2 a c^2 \left (-12 d^2 (e+f x)+6 d x \left (-2 e^2+4 e f x+f^2 x^2\right )+x^2 \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )\right )+8 c^3 d^3 x\right )+105 b^5 f^2 (3 a f+c x (f x-8 e))-2 b^4 c f \left (105 a f (4 e+9 f x)+c x \left (-360 d f-360 e^2+140 e f x+21 f^2 x^2\right )\right )+315 b^6 f^3 x}{64 c^5 \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x + f*x^2)^3/(a + b*x + c*x^2)^(3/2),x]
[Out]
(315*b^6*f^3*x + 105*b^5*f^2*(3*a*f + c*x*(-8*e + f*x)) - 2*b^4*c*f*(105*a*f*(4*
e + 9*f*x) + c*x*(-360*e^2 - 360*d*f + 140*e*f*x + 21*f^2*x^2)) - 8*b^3*c*(210*a
^2*f^3 - c^2*x*(-24*e^3 + 30*e^2*f*x + 3*f^2*x*(10*d + f*x^2) + 2*e*f*(-72*d + 7
*f*x^2)) + a*c*f*(-90*e^2 - 530*e*f*x + f*(-90*d + 77*f*x^2))) - 16*b^2*c^2*(-(a
^2*f^2*(230*e + 169*f*x)) + a*c*(12*e^3 + 186*e^2*f*x + 2*e*f*(36*d - 43*f*x^2)
+ f^2*x*(186*d - 13*f*x^2)) + c^2*x*(-24*d^2*f + 6*d*(-4*e^2 + 4*e*f*x + f^2*x^2
) + x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))) + 32*c^3*(8*c^3*d^3*x - a^3*
f^2*(64*e + 15*f*x) + a^2*c*(16*e^3 + 36*e^2*f*x + f^2*x*(36*d - 5*f*x^2) - 32*e
*f*(-3*d + f*x^2)) + 2*a*c^2*(-12*d^2*(e + f*x) + 6*d*x*(-2*e^2 + 4*e*f*x + f^2*
x^2) + x^2*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))) + 16*b*c^2*(113*a^3*f^3
+ 8*c^3*d^2*(d - 3*e*x) + a^2*c*f*(-156*e^2 - 244*e*f*x + f*(-156*d + 49*f*x^2)
) + 2*a*c^2*(12*d^2*f + 6*d*(2*e^2 + 20*e*f*x - 5*f^2*x^2) - x*(-20*e^3 + 30*e^2
*f*x + 14*e*f^2*x^2 + 3*f^3*x^3))))/(64*c^5*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)]
) + (3*(105*b^4*f^3 - 280*b^2*c*f^2*(b*e + a*f) + 128*c^4*d*(e^2 + d*f) + 80*c^2
*f*(6*a*b*e*f + a^2*f^2 + 3*b^2*(e^2 + d*f)) - 64*c^3*(3*a*f*(e^2 + d*f) + b*(e^
3 + 6*d*e*f)))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(128*c^(11/2))
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Maple [B] time = 0.037, size = 2827, normalized size = 4.4 \[ \text{output too large to display} \]
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)^(3/2),x)
[Out]
-105/32*e*f^2*b^6/c^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+115/8*e*f^2*b^2/c^4*a/(c*x
^2+b*x+a)^(1/2)+45/4*e*f^2*b/c^(7/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))-4*e*f^2/c^2*a*x^2/(c*x^2+b*x+a)^(1/2)-7/4*e*f^2*b/c^2*x^3/(c*x^2+b*x+a)^(1/2)
+35/8*e*f^2*b^2/c^3*x^2/(c*x^2+b*x+a)^(1/2)+105/16*e*f^2*b^3/c^4*x/(c*x^2+b*x+a)
^(1/2)+3/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*f*d^2+3/2*b^3/c^2/(4*a*c-b^2)
/(c*x^2+b*x+a)^(1/2)*e^2*d-45/8*b^2/c^3*x/(c*x^2+b*x+a)^(1/2)*d*f^2-45/8*b^2/c^3
*x/(c*x^2+b*x+a)^(1/2)*e^2*f+45/16*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d*f^2
+45/16*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*e^2*f-39/4*b/c^3*a/(c*x^2+b*x+a)^
(1/2)*d*f^2-39/4*b/c^3*a/(c*x^2+b*x+a)^(1/2)*e^2*f+9/2/c^2*a*x/(c*x^2+b*x+a)^(1/
2)*d*f^2+9/2/c^2*a*x/(c*x^2+b*x+a)^(1/2)*e^2*f-15/4*b/c^2*x^2/(c*x^2+b*x+a)^(1/2
)*d*f^2-15/4*b/c^2*x^2/(c*x^2+b*x+a)^(1/2)*e^2*f+2/c^2*a*b^2/(4*a*c-b^2)/(c*x^2+
b*x+a)^(1/2)*e^3+12/c^2*a/(c*x^2+b*x+a)^(1/2)*d*e*f+6*x^2/c/(c*x^2+b*x+a)^(1/2)*
d*e*f-9/2*b^2/c^3/(c*x^2+b*x+a)^(1/2)*d*e*f-3/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a
)^(1/2)*x*e^3-9*b/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e*f+113/
16*f^3*b^3/c^4*a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+315/128*f^3*b^6/c^5/(4*a*c-b^
2)/(c*x^2+b*x+a)^(1/2)*x-6*d^2*e*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-3*d^2*e*b^2
/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+12/c^2*a*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*
d*e*f+24/c*a*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*e*f+3/c^(3/2)*ln((1/2*b+c*x)/
c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2*d+x^2/c/(c*x^2+b*x+a)^(1/2)*e^3-3/4*b^2/c^3/(c*
x^2+b*x+a)^(1/2)*e^3-3/2*b/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e
^3+2/c^2*a/(c*x^2+b*x+a)^(1/2)*e^3+315/256*f^3*b^5/c^6/(c*x^2+b*x+a)^(1/2)+1/4*f
^3*x^5/c/(c*x^2+b*x+a)^(1/2)+15/8*f^3/c^(7/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))+315/128*f^3*b^4/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))+2*d^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-3*d^2*e/c/(c*x^2+b*x+a)^(1/2)
+3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*d^2-39/2*b^2/c^2*a/(4*a
*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*e^2*f+115/4*e*f^2*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x
+a)^(1/2)*x-16*e*f^2/c^2*a^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-39/2*b^2/c^2*a/
(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*f^2-9*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2
)*x*d*e*f+3*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*f*d^2+3*b^2/c/(4*a*c-b^2)/(c
*x^2+b*x+a)^(1/2)*x*e^2*d+9*b/c^2*x/(c*x^2+b*x+a)^(1/2)*d*e*f-9/2*b^4/c^3/(4*a*c
-b^2)/(c*x^2+b*x+a)^(1/2)*d*e*f+4/c*a*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*e^3+45
/8*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*f^2+45/8*b^4/c^3/(4*a*c-b^2)/(c*x
^2+b*x+a)^(1/2)*x*e^2*f+113/8*f^3*b^2/c^3*a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-
105/8*f^3*b^4/c^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+115/8*e*f^2*b^4/c^4*a/(4*a
*c-b^2)/(c*x^2+b*x+a)^(1/2)-45/4*e*f^2*b/c^3*a*x/(c*x^2+b*x+a)^(1/2)-8*e*f^2/c^3
*a^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-105/16*e*f^2*b^5/c^4/(4*a*c-b^2)/(c*x^2
+b*x+a)^(1/2)*x-39/4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d*f^2-39/4*b^3/c^
3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*e^2*f+45/8*b^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*d*f^2+105/16*f^3*b^2/c^4*a*x/(c*x^2+b*x+a)^(1/2)+49/16*f^
3*b/c^3*a*x^2/(c*x^2+b*x+a)^(1/2)+315/256*f^3*b^7/c^6/(4*a*c-b^2)/(c*x^2+b*x+a)^
(1/2)-3/8*f^3*b/c^2*x^4/(c*x^2+b*x+a)^(1/2)+e*f^2*x^4/c/(c*x^2+b*x+a)^(1/2)-105/
32*e*f^2*b^4/c^5/(c*x^2+b*x+a)^(1/2)-105/16*e*f^2*b^3/c^(9/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))-8*e*f^2/c^3*a^2/(c*x^2+b*x+a)^(1/2)-3*x/c/(c*x^2+b*x+a
)^(1/2)*f*d^2-3*x/c/(c*x^2+b*x+a)^(1/2)*e^2*d+3/2*b/c^2/(c*x^2+b*x+a)^(1/2)*f*d^
2+3/2*b/c^2/(c*x^2+b*x+a)^(1/2)*e^2*d-9/2/c^(5/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))*d*f^2-9/2/c^(5/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*
e^2*f-3/4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*e^3+21/32*f^3*b^2/c^3*x^3/(c*x
^2+b*x+a)^(1/2)-105/16*f^3*b^2/c^(9/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))-5/8*f^3/c^2*a*x^3/(c*x^2+b*x+a)^(1/2)-15/8*f^3/c^3*a^2*x/(c*x^2+b*x+a)^(1/2
)-105/16*f^3*b^3/c^5*a/(c*x^2+b*x+a)^(1/2)+113/16*f^3*b/c^4*a^2/(c*x^2+b*x+a)^(1
/2)-105/64*f^3*b^3/c^4*x^2/(c*x^2+b*x+a)^(1/2)-315/128*f^3*b^4/c^5*x/(c*x^2+b*x+
a)^(1/2)+45/16*b^3/c^4/(c*x^2+b*x+a)^(1/2)*e^2*f-105/16*f^3*b^5/c^5*a/(4*a*c-b^2
)/(c*x^2+b*x+a)^(1/2)+45/8*b^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))*e^2*f+3/2*x^3/c/(c*x^2+b*x+a)^(1/2)*e^2*f+45/16*b^3/c^4/(c*x^2+b*x+a)^(1/2)*d
*f^2+3/2*x^3/c/(c*x^2+b*x+a)^(1/2)*d*f^2+3/2*b/c^2*x/(c*x^2+b*x+a)^(1/2)*e^3
_______________________________________________________________________________________
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((f*x^2 + e*x + d)^3/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.43661, size = 1, normalized size = 0. \[ \text{result too large to display} \]
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)^(3/2),x, algorithm="fricas")
[Out]
[-1/256*(4*(128*b*c^5*d^3 - 768*a*c^5*d^2*e + 384*a*b*c^4*d*e^2 - 16*(b^2*c^4 -
4*a*c^5)*f^3*x^5 - 8*(8*(b^2*c^4 - 4*a*c^5)*e*f^2 - 3*(b^3*c^3 - 4*a*b*c^4)*f^3)
*x^4 - 64*(3*a*b^2*c^3 - 8*a^2*c^4)*e^3 + (315*a*b^5 - 1680*a^2*b^3*c + 1808*a^3
*b*c^2)*f^3 - 2*(48*(b^2*c^4 - 4*a*c^5)*e^2*f + (21*b^4*c^2 - 104*a*b^2*c^3 + 80
*a^2*c^4)*f^3 + 8*(6*(b^2*c^4 - 4*a*c^5)*d - 7*(b^3*c^3 - 4*a*b*c^4)*e)*f^2)*x^3
+ 8*(6*(15*a*b^3*c^2 - 52*a^2*b*c^3)*d - (105*a*b^4*c - 460*a^2*b^2*c^2 + 256*a
^3*c^3)*e)*f^2 - (64*(b^2*c^4 - 4*a*c^5)*e^3 - 7*(15*b^5*c - 88*a*b^3*c^2 + 112*
a^2*b*c^3)*f^3 - 8*(30*(b^3*c^3 - 4*a*b*c^4)*d - (35*b^4*c^2 - 172*a*b^2*c^3 + 1
28*a^2*c^4)*e)*f^2 + 48*(8*(b^2*c^4 - 4*a*c^5)*d*e - 5*(b^3*c^3 - 4*a*b*c^4)*e^2
)*f)*x^2 + 48*(8*a*b*c^4*d^2 - 8*(3*a*b^2*c^3 - 8*a^2*c^4)*d*e + (15*a*b^3*c^2 -
52*a^2*b*c^3)*e^2)*f + (256*c^6*d^3 - 384*b*c^5*d^2*e + 384*(b^2*c^4 - 2*a*c^5)
*d*e^2 - 64*(3*b^3*c^3 - 10*a*b*c^4)*e^3 + (315*b^6 - 1890*a*b^4*c + 2704*a^2*b^
2*c^2 - 480*a^3*c^3)*f^3 + 8*(6*(15*b^4*c^2 - 62*a*b^2*c^3 + 24*a^2*c^4)*d - (10
5*b^5*c - 530*a*b^3*c^2 + 488*a^2*b*c^3)*e)*f^2 + 48*(8*(b^2*c^4 - 2*a*c^5)*d^2
- 8*(3*b^3*c^3 - 10*a*b*c^4)*d*e + (15*b^4*c^2 - 62*a*b^2*c^3 + 24*a^2*c^4)*e^2)
*f)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 3*(128*(a*b^2*c^4 - 4*a^2*c^5)*d*e^2 - 64
*(a*b^3*c^3 - 4*a^2*b*c^4)*e^3 + 5*(21*a*b^6 - 140*a^2*b^4*c + 240*a^3*b^2*c^2 -
64*a^4*c^3)*f^3 + 8*(6*(5*a*b^4*c^2 - 24*a^2*b^2*c^3 + 16*a^3*c^4)*d - 5*(7*a*b
^5*c - 40*a^2*b^3*c^2 + 48*a^3*b*c^3)*e)*f^2 + (128*(b^2*c^5 - 4*a*c^6)*d*e^2 -
64*(b^3*c^4 - 4*a*b*c^5)*e^3 + 5*(21*b^6*c - 140*a*b^4*c^2 + 240*a^2*b^2*c^3 - 6
4*a^3*c^4)*f^3 + 8*(6*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*d - 5*(7*b^5*c^2 -
40*a*b^3*c^3 + 48*a^2*b*c^4)*e)*f^2 + 16*(8*(b^2*c^5 - 4*a*c^6)*d^2 - 24*(b^3*c
^4 - 4*a*b*c^5)*d*e + 3*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*e^2)*f)*x^2 + 16
*(8*(a*b^2*c^4 - 4*a^2*c^5)*d^2 - 24*(a*b^3*c^3 - 4*a^2*b*c^4)*d*e + 3*(5*a*b^4*
c^2 - 24*a^2*b^2*c^3 + 16*a^3*c^4)*e^2)*f + (128*(b^3*c^4 - 4*a*b*c^5)*d*e^2 - 6
4*(b^4*c^3 - 4*a*b^2*c^4)*e^3 + 5*(21*b^7 - 140*a*b^5*c + 240*a^2*b^3*c^2 - 64*a
^3*b*c^3)*f^3 + 8*(6*(5*b^5*c^2 - 24*a*b^3*c^3 + 16*a^2*b*c^4)*d - 5*(7*b^6*c -
40*a*b^4*c^2 + 48*a^2*b^2*c^3)*e)*f^2 + 16*(8*(b^3*c^4 - 4*a*b*c^5)*d^2 - 24*(b^
4*c^3 - 4*a*b^2*c^4)*d*e + 3*(5*b^5*c^2 - 24*a*b^3*c^3 + 16*a^2*b*c^4)*e^2)*f)*x
)*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)))/((a*b^2*c^5 - 4*a^2*c^6 + (b^2*c^6 - 4*a*c^7)*x^2 + (b^3*c^5 - 4*
a*b*c^6)*x)*sqrt(c)), -1/128*(2*(128*b*c^5*d^3 - 768*a*c^5*d^2*e + 384*a*b*c^4*d
*e^2 - 16*(b^2*c^4 - 4*a*c^5)*f^3*x^5 - 8*(8*(b^2*c^4 - 4*a*c^5)*e*f^2 - 3*(b^3*
c^3 - 4*a*b*c^4)*f^3)*x^4 - 64*(3*a*b^2*c^3 - 8*a^2*c^4)*e^3 + (315*a*b^5 - 1680
*a^2*b^3*c + 1808*a^3*b*c^2)*f^3 - 2*(48*(b^2*c^4 - 4*a*c^5)*e^2*f + (21*b^4*c^2
- 104*a*b^2*c^3 + 80*a^2*c^4)*f^3 + 8*(6*(b^2*c^4 - 4*a*c^5)*d - 7*(b^3*c^3 - 4
*a*b*c^4)*e)*f^2)*x^3 + 8*(6*(15*a*b^3*c^2 - 52*a^2*b*c^3)*d - (105*a*b^4*c - 46
0*a^2*b^2*c^2 + 256*a^3*c^3)*e)*f^2 - (64*(b^2*c^4 - 4*a*c^5)*e^3 - 7*(15*b^5*c
- 88*a*b^3*c^2 + 112*a^2*b*c^3)*f^3 - 8*(30*(b^3*c^3 - 4*a*b*c^4)*d - (35*b^4*c^
2 - 172*a*b^2*c^3 + 128*a^2*c^4)*e)*f^2 + 48*(8*(b^2*c^4 - 4*a*c^5)*d*e - 5*(b^3
*c^3 - 4*a*b*c^4)*e^2)*f)*x^2 + 48*(8*a*b*c^4*d^2 - 8*(3*a*b^2*c^3 - 8*a^2*c^4)*
d*e + (15*a*b^3*c^2 - 52*a^2*b*c^3)*e^2)*f + (256*c^6*d^3 - 384*b*c^5*d^2*e + 38
4*(b^2*c^4 - 2*a*c^5)*d*e^2 - 64*(3*b^3*c^3 - 10*a*b*c^4)*e^3 + (315*b^6 - 1890*
a*b^4*c + 2704*a^2*b^2*c^2 - 480*a^3*c^3)*f^3 + 8*(6*(15*b^4*c^2 - 62*a*b^2*c^3
+ 24*a^2*c^4)*d - (105*b^5*c - 530*a*b^3*c^2 + 488*a^2*b*c^3)*e)*f^2 + 48*(8*(b^
2*c^4 - 2*a*c^5)*d^2 - 8*(3*b^3*c^3 - 10*a*b*c^4)*d*e + (15*b^4*c^2 - 62*a*b^2*c
^3 + 24*a^2*c^4)*e^2)*f)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*(128*(a*b^2*c^4 -
4*a^2*c^5)*d*e^2 - 64*(a*b^3*c^3 - 4*a^2*b*c^4)*e^3 + 5*(21*a*b^6 - 140*a^2*b^4
*c + 240*a^3*b^2*c^2 - 64*a^4*c^3)*f^3 + 8*(6*(5*a*b^4*c^2 - 24*a^2*b^2*c^3 + 16
*a^3*c^4)*d - 5*(7*a*b^5*c - 40*a^2*b^3*c^2 + 48*a^3*b*c^3)*e)*f^2 + (128*(b^2*c
^5 - 4*a*c^6)*d*e^2 - 64*(b^3*c^4 - 4*a*b*c^5)*e^3 + 5*(21*b^6*c - 140*a*b^4*c^2
+ 240*a^2*b^2*c^3 - 64*a^3*c^4)*f^3 + 8*(6*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c
^5)*d - 5*(7*b^5*c^2 - 40*a*b^3*c^3 + 48*a^2*b*c^4)*e)*f^2 + 16*(8*(b^2*c^5 - 4*
a*c^6)*d^2 - 24*(b^3*c^4 - 4*a*b*c^5)*d*e + 3*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2
*c^5)*e^2)*f)*x^2 + 16*(8*(a*b^2*c^4 - 4*a^2*c^5)*d^2 - 24*(a*b^3*c^3 - 4*a^2*b*
c^4)*d*e + 3*(5*a*b^4*c^2 - 24*a^2*b^2*c^3 + 16*a^3*c^4)*e^2)*f + (128*(b^3*c^4
- 4*a*b*c^5)*d*e^2 - 64*(b^4*c^3 - 4*a*b^2*c^4)*e^3 + 5*(21*b^7 - 140*a*b^5*c +
240*a^2*b^3*c^2 - 64*a^3*b*c^3)*f^3 + 8*(6*(5*b^5*c^2 - 24*a*b^3*c^3 + 16*a^2*b*
c^4)*d - 5*(7*b^6*c - 40*a*b^4*c^2 + 48*a^2*b^2*c^3)*e)*f^2 + 16*(8*(b^3*c^4 - 4
*a*b*c^5)*d^2 - 24*(b^4*c^3 - 4*a*b^2*c^4)*d*e + 3*(5*b^5*c^2 - 24*a*b^3*c^3 + 1
6*a^2*b*c^4)*e^2)*f)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c
)))/((a*b^2*c^5 - 4*a^2*c^6 + (b^2*c^6 - 4*a*c^7)*x^2 + (b^3*c^5 - 4*a*b*c^6)*x)
*sqrt(-c))]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)**(3/2),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.287321, size = 1, normalized size = 0. \[ \mathit{Done} \]
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)^(3/2),x, algorithm="giac")
[Out]
Done