3.118 \(\int \frac{\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=891 \[ \frac{x \sqrt{c x^2+b x+a} f^3}{2 c^3}+\frac{(12 c e-11 b f) \sqrt{c x^2+b x+a} f^2}{4 c^4}+\frac{\left (24 \left (e^2+d f\right ) c^2-20 f (3 b e+a f) c+35 b^2 f^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) f}{8 c^{9/2}}-\frac{2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right )^2 \sqrt{c x^2+b x+a}}+\frac{2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2-c e b+2 c^2 d-2 a c f\right ) \left (f^2 b^4-2 c e f b^3+c^2 e^2 b^2-4 a c f^2 b^2+c^2 d f b^2-c^3 d e b+7 a c^2 e f b+c^4 d^2-3 a c^3 e^2+a^2 c^2 f^2-2 a c^3 d f\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}} \]

[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))/(3*c^5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3
/2)) - (2*(3*b^6*c*e*f^2 - b^7*f^3 + 3*b^5*c*f*(6*a*f^2 - c*(e^2 + d*f)) - 3*b^3
*c^2*(29*a^2*f^3 + c^2*d*(e^2 + d*f) - 10*a*c*f*(e^2 + d*f)) - 4*b*c^3*(2*c^3*d^
3 - 29*a^3*f^3 + 3*a*c^2*d*(e^2 + d*f) + 24*a^2*c*f*(e^2 + d*f)) - 24*a^2*c^4*e*
(6*a*f^2 - c*(e^2 + 6*d*f)) - b^4*c^2*e*(42*a*f^2 - c*(e^2 + 6*d*f)) + 6*b^2*c^3
*e*(2*c^2*d^2 + 28*a^2*f^2 - a*c*(e^2 + 6*d*f)) - c*(16*c^6*d^3 - 10*b^6*f^3 + 3
*b^4*c*f^2*(7*b*e + 26*a*f) - 24*c^5*d*(b*d*e - a*(e^2 + d*f)) - 6*b^2*c^2*f*(25
*a*b*e*f + 27*a^2*f^2 + 2*b^2*(e^2 + d*f)) + 6*c^4*(b^2*d*(e^2 + d*f) - 16*a^2*f
*(e^2 + d*f) - 2*a*b*e*(e^2 + 6*d*f)) + c^3*(240*a^2*b*e*f^2 + 56*a^3*f^3 + 84*a
*b^2*f*(e^2 + d*f) + b^3*(e^3 + 6*d*e*f)))*x))/(3*c^5*(b^2 - 4*a*c)^2*Sqrt[a + b
*x + c*x^2]) + (f^2*(12*c*e - 11*b*f)*Sqrt[a + b*x + c*x^2])/(4*c^4) + (f^3*x*Sq
rt[a + b*x + c*x^2])/(2*c^3) + (f*(35*b^2*f^2 - 20*c*f*(3*b*e + a*f) + 24*c^2*(e
^2 + d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(9/2))

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Rubi [A]  time = 5.04396, antiderivative size = 891, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{x \sqrt{c x^2+b x+a} f^3}{2 c^3}+\frac{(12 c e-11 b f) \sqrt{c x^2+b x+a} f^2}{4 c^4}+\frac{\left (24 \left (e^2+d f\right ) c^2-20 f (3 b e+a f) c+35 b^2 f^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) f}{8 c^{9/2}}-\frac{2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right )^2 \sqrt{c x^2+b x+a}}+\frac{2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2+2 c^2 d-c (b e+2 a f)\right ) \left (f^2 b^4-2 c f (b e+2 a f) b^2+c^4 d^2-c^3 \left (3 a e^2+b d e+2 a d f\right )+c^2 \left (\left (e^2+d f\right ) b^2+7 a e f b+a^2 f^2\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*x^2)^3/(a + b*x + c*x^2)^(5/2),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))/(3*c^5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - (2*(3*b^6*c*e*f
^2 - b^7*f^3 + 3*b^5*c*f*(6*a*f^2 - c*(e^2 + d*f)) - 3*b^3*c^2*(29*a^2*f^3 + c^2
*d*(e^2 + d*f) - 10*a*c*f*(e^2 + d*f)) - 4*b*c^3*(2*c^3*d^3 - 29*a^3*f^3 + 3*a*c
^2*d*(e^2 + d*f) + 24*a^2*c*f*(e^2 + d*f)) - 24*a^2*c^4*e*(6*a*f^2 - c*(e^2 + 6*
d*f)) - b^4*c^2*e*(42*a*f^2 - c*(e^2 + 6*d*f)) + 6*b^2*c^3*e*(2*c^2*d^2 + 28*a^2
*f^2 - a*c*(e^2 + 6*d*f)) - c*(16*c^6*d^3 - 10*b^6*f^3 + 3*b^4*c*f^2*(7*b*e + 26
*a*f) - 24*c^5*d*(b*d*e - a*(e^2 + d*f)) - 6*b^2*c^2*f*(25*a*b*e*f + 27*a^2*f^2
+ 2*b^2*(e^2 + d*f)) + 6*c^4*(b^2*d*(e^2 + d*f) - 16*a^2*f*(e^2 + d*f) - 2*a*b*e
*(e^2 + 6*d*f)) + c^3*(240*a^2*b*e*f^2 + 56*a^3*f^3 + 84*a*b^2*f*(e^2 + d*f) + b
^3*(e^3 + 6*d*e*f)))*x))/(3*c^5*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) + (f^2*(1
2*c*e - 11*b*f)*Sqrt[a + b*x + c*x^2])/(4*c^4) + (f^3*x*Sqrt[a + b*x + c*x^2])/(
2*c^3) + (f*(35*b^2*f^2 - 20*c*f*(3*b*e + a*f) + 24*c^2*(e^2 + d*f))*ArcTanh[(b
+ 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(9/2))

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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)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 4.8857, size = 872, normalized size = 0.98 \[ \frac{-105 f^3 x^2 b^7-10 f^2 x (21 a f+2 c x (7 f x-9 e)) b^6-3 f \left (35 a^2 f^2-10 a c x (12 e+23 f x) f+c^2 x^2 \left (24 e^2-80 f x e+7 f^2 x^2+24 d f\right )\right ) b^5+6 c f \left (c^2 \left (-16 e^2+6 f x e+f^2 x^2-16 d f\right ) x^3-6 a c \left (4 e^2+30 f x e-31 f^2 x^2+4 d f\right ) x+5 a^2 f (6 e+53 f x)\right ) b^4-8 c \left (\left (d^3+9 x (e-f x) d^2-3 e x^2 (3 e+2 f x) d-e^3 x^3\right ) c^3-3 a f x^2 \left (18 e^2-74 f x e+f \left (7 f x^2+18 d\right )\right ) c^2+3 a^2 f \left (3 e^2+105 f x e+f \left (29 f x^2+3 d\right )\right ) c-95 a^3 f^3\right ) b^3-48 c^2 \left (f^2 (25 e+63 f x) a^3+c f x \left (-21 e^2-12 f x e+7 f \left (7 f x^2-3 d\right )\right ) a^2+c^2 \left ((e-6 f x) d^2-2 x \left (3 e^2-3 f x e+7 f^2 x^2\right ) d+x^2 \left (e^3-14 f x e^2+6 f^2 x^2 e+f^3 x^3\right )\right ) a-c^3 d x \left (d^2+x (f x-6 e) d+e^2 x^2\right )\right ) b^2-48 c^2 \left (27 f^3 a^4-2 c f \left (5 e^2+39 f x e+f \left (5 d-14 f x^2\right )\right ) a^3+c^2 \left (7 f^3 x^4-64 e f^2 x^3+4 e^3 x-4 d^2 f-4 d e (e-6 f x)\right ) a^2-2 c^3 \left (d^3+3 x (f x-e) d^2+3 e x^2 (e-2 f x) d-e^3 x^3\right ) a-4 c^4 d^2 x^2 (d-e x)\right ) b+32 c^3 \left (3 f^2 (16 e+5 f x) a^4-2 c \left (2 e^3+9 f x e^2+12 f \left (d-3 f x^2\right ) e+f^2 x \left (9 d-10 f x^2\right )\right ) a^3-3 c^2 \left (2 e d^2+4 f x^2 (3 e+2 f x) d+x^2 \left (2 e^3+8 f x e^2-6 f^2 x^2 e-f^3 x^3\right )\right ) a^2+6 c^3 d x \left (d^2+f x^2 d+e^2 x^2\right ) a+4 c^4 d^3 x^3\right )}{12 c^4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}+\frac{f \left (24 \left (e^2+d f\right ) c^2-20 f (3 b e+a f) c+35 b^2 f^2\right ) \log \left (b+2 c x+2 \sqrt{c} \sqrt{a+x (b+c x)}\right )}{8 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*x^2)^3/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-105*b^7*f^3*x^2 - 10*b^6*f^2*x*(21*a*f + 2*c*x*(-9*e + 7*f*x)) + 6*b^4*c*f*(5*
a^2*f*(6*e + 53*f*x) - 6*a*c*x*(4*e^2 + 4*d*f + 30*e*f*x - 31*f^2*x^2) + c^2*x^3
*(-16*e^2 - 16*d*f + 6*e*f*x + f^2*x^2)) - 3*b^5*f*(35*a^2*f^2 - 10*a*c*f*x*(12*
e + 23*f*x) + c^2*x^2*(24*e^2 + 24*d*f - 80*e*f*x + 7*f^2*x^2)) - 48*b*c^2*(27*a
^4*f^3 - 4*c^4*d^2*x^2*(d - e*x) + a^2*c^2*(-4*d^2*f + 4*e^3*x - 64*e*f^2*x^3 +
7*f^3*x^4 - 4*d*e*(e - 6*f*x)) - 2*a*c^3*(d^3 - e^3*x^3 + 3*d*e*x^2*(e - 2*f*x)
+ 3*d^2*x*(-e + f*x)) - 2*a^3*c*f*(5*e^2 + 39*e*f*x + f*(5*d - 14*f*x^2))) - 8*b
^3*c*(-95*a^3*f^3 + c^3*(d^3 - e^3*x^3 + 9*d^2*x*(e - f*x) - 3*d*e*x^2*(3*e + 2*
f*x)) - 3*a*c^2*f*x^2*(18*e^2 - 74*e*f*x + f*(18*d + 7*f*x^2)) + 3*a^2*c*f*(3*e^
2 + 105*e*f*x + f*(3*d + 29*f*x^2))) + 32*c^3*(4*c^4*d^3*x^3 + 3*a^4*f^2*(16*e +
5*f*x) + 6*a*c^3*d*x*(d^2 + e^2*x^2 + d*f*x^2) - 2*a^3*c*(2*e^3 + 9*e^2*f*x + f
^2*x*(9*d - 10*f*x^2) + 12*e*f*(d - 3*f*x^2)) - 3*a^2*c^2*(2*d^2*e + 4*d*f*x^2*(
3*e + 2*f*x) + x^2*(2*e^3 + 8*e^2*f*x - 6*e*f^2*x^2 - f^3*x^3))) - 48*b^2*c^2*(a
^3*f^2*(25*e + 63*f*x) - c^3*d*x*(d^2 + e^2*x^2 + d*x*(-6*e + f*x)) + a^2*c*f*x*
(-21*e^2 - 12*e*f*x + 7*f*(-3*d + 7*f*x^2)) + a*c^2*(d^2*(e - 6*f*x) - 2*d*x*(3*
e^2 - 3*e*f*x + 7*f^2*x^2) + x^2*(e^3 - 14*e^2*f*x + 6*e*f^2*x^2 + f^3*x^3))))/(
12*c^4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) + (f*(35*b^2*f^2 - 20*c*f*(3*b*e
+ a*f) + 24*c^2*(e^2 + d*f))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/
(8*c^(9/2))

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Maple [B]  time = 0.045, size = 4635, normalized size = 5.2 \[ \text{output too large to display} \]

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)^(5/2),x)

[Out]

2/3*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*e^3-1/2*b^2/c^2*a/(4*a*c-b^2)/(c*x
^2+b*x+a)^(3/2)*e^3-4*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*e^3-4/c^2*a/(c*x
^2+b*x+a)^(3/2)*d*e*f-d^2*e*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-2*d^2*e*b/(4*a
*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+1/4*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*f*d^2+
1/4*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*e^2*d+4*b^2/(4*a*c-b^2)^2/(c*x^2+b*x
+a)^(1/2)*x*f*d^2-x^2/c/(c*x^2+b*x+a)^(3/2)*e^3+1/24*b^2/c^3/(c*x^2+b*x+a)^(3/2)
*e^3-2/3/c^2*a/(c*x^2+b*x+a)^(3/2)*e^3+3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*d*f^2+3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2*f+3
5/16*f^3*b^3/c^5/(c*x^2+b*x+a)^(1/2)+35/8*f^3*b^2/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))-5/2*f^3/c^(7/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))+1/2*f^3*x^5/c/(c*x^2+b*x+a)^(3/2)-35/384*f^3*b^5/c^6/(c*x^2+b*x+a)^(3/2)+2/3
*d^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b-d^2*e/c/(c*x^2+b*x+a)^(3/2)-6*b/c*a/(4*a*
c-b^2)/(c*x^2+b*x+a)^(3/2)*x*d*e*f+12*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*
x*e^2*f+12*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*d*f^2+3/2*b^2/c^2*a/(4*a*
c-b^2)/(c*x^2+b*x+a)^(3/2)*x*e^2*f+3/2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)
*x*d*f^2+12*e*f^2/c^2*a^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+96*e*f^2/c*a^2*b/(
4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-38*e*f^2*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a
)^(1/2)*x-19/4*e*f^2*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-48*b*a/(4*a*c-b
^2)^2/(c*x^2+b*x+a)^(1/2)*x*d*e*f+1/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*
d*e*f+4*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*d*e*f-3*b^2/c^2*a/(4*a*c-b^2)/
(c*x^2+b*x+a)^(3/2)*d*e*f-15/4*e*f^2*b^4/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+12*
e*f^2/c^2*a*x^2/(c*x^2+b*x+a)^(3/2)+5/2*e*f^2*b/c^2*x^3/(c*x^2+b*x+a)^(3/2)+1/c*
a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b*e^2*d+16*c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/
2)*x*f*d^2+16*c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*e^2*d+1/2*b^2/c/(4*a*c-b^2
)/(c*x^2+b*x+a)^(3/2)*x*f*d^2+1/2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*e^2*d+
1/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b*f*d^2-15/2*e*f^2*b^3/c^3/(4*a*c-b^2)/(c*
x^2+b*x+a)^(1/2)*x+3*e*f^2/c^3*a*b*x/(c*x^2+b*x+a)^(3/2)+6*e*f^2/c^3*a^2*b^2/(4*
a*c-b^2)/(c*x^2+b*x+a)^(3/2)+48*e*f^2/c^2*a^2*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1
/2)+5/16*e*f^2*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+5/2*e*f^2*b^5/c^3/(4*a*
c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-19/8*e*f^2*b^4/c^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3
/2)+3/4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*d*f^2+3/4*b^3/c^3*a/(4*a*c-b^2
)/(c*x^2+b*x+a)^(3/2)*e^2*f+6*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*d*f^2+
6*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*e^2*f+3/c^2*b^2/(4*a*c-b^2)/(c*x^2
+b*x+a)^(1/2)*x*d*f^2+3/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*e^2*f-1/8*b^4/
c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*e^2*f-b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^
(1/2)*x*d*f^2-b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*e^2*f-1/8*b^4/c^3/(4*a
*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*d*f^2+23*f^3*b^4/c^3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)
^(1/2)*x+23/8*f^3*b^4/c^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-5/2*f^3/c^3*a*b^2/
(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-33/4*f^3*b^2/c^3*a^2/(4*a*c-b^2)/(c*x^2+b*x+a)
^(3/2)*x-66*f^3*b^2/c^2*a^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-16*d^2*e*b*c/(4*
a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-b/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*e^3-3/2
*b/c^2*x/(c*x^2+b*x+a)^(3/2)*d*e*f+1/4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*d
*e*f+2*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*d*e*f-19*e*f^2*b^4/c^3*a/(4*a*c
-b^2)^2/(c*x^2+b*x+a)^(1/2)+4*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*e^2*d+2*b^
3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*f*d^2+2*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^
(1/2)*e^2*d+2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*f*d^2+2*a/(4*a*c-b^2)/(c*x^2+b
*x+a)^(3/2)*x*e^2*d+8*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b*f*d^2+8*a/(4*a*c-b^2
)^2/(c*x^2+b*x+a)^(1/2)*b*e^2*d+16/3*d^3*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b-1
/4*b/c^2*x/(c*x^2+b*x+a)^(3/2)*e^3+3/2/c^3*b/(c*x^2+b*x+a)^(1/2)*e^2*f-x^3/c/(c*
x^2+b*x+a)^(3/2)*d*f^2-x^3/c/(c*x^2+b*x+a)^(3/2)*e^2*f-5/4*f^3/c^4*a*b/(c*x^2+b*
x+a)^(1/2)-11/2*f^3*b/c^4*a^2/(c*x^2+b*x+a)^(3/2)-35/24*f^3*b^2/c^3*x^3/(c*x^2+b
*x+a)^(3/2)-35/8*f^3*b^2/c^4*x/(c*x^2+b*x+a)^(1/2)+35/16*f^3*b^5/c^5/(4*a*c-b^2)
/(c*x^2+b*x+a)^(1/2)-7/4*f^3*b/c^2*x^4/(c*x^2+b*x+a)^(3/2)+35/16*f^3*b^3/c^4*x^2
/(c*x^2+b*x+a)^(3/2)+35/64*f^3*b^4/c^5*x/(c*x^2+b*x+a)^(3/2)+5/6*f^3/c^2*a*x^3/(
c*x^2+b*x+a)^(3/2)+5/2*f^3/c^3*a*x/(c*x^2+b*x+a)^(1/2)-8*d^2*e*b^2/(4*a*c-b^2)^2
/(c*x^2+b*x+a)^(1/2)-35/384*f^3*b^7/c^6/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-35/48*f^
3*b^7/c^5/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+173/96*f^3*b^3/c^5*a/(c*x^2+b*x+a)^(
3/2)+1/24*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*e^3+1/3*b^4/c^2/(4*a*c-b^2)^2/
(c*x^2+b*x+a)^(1/2)*e^3+4/3*d^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*c*x+32/3*d^3*c^2
/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-3/2*x/c/(c*x^2+b*x+a)^(3/2)*f*d^2+1/4*b/c^2
/(c*x^2+b*x+a)^(3/2)*e^2*d+3*e*f^2*x^4/c/(c*x^2+b*x+a)^(3/2)+8*e*f^2/c^3*a^2/(c*
x^2+b*x+a)^(3/2)-15/2*e*f^2*b/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)+5/32*e*f^2*b^4/c^5/(c*x^2+b*x+a)^(3/2)-15/4*e*f^2*b^2/c^4/(c*x^2+b*x+a)^(1/2)-
1/16*b^3/c^4/(c*x^2+b*x+a)^(3/2)*d*f^2-1/16*b^3/c^4/(c*x^2+b*x+a)^(3/2)*e^2*f-3/
c^2*x/(c*x^2+b*x+a)^(1/2)*d*f^2-3/c^2*x/(c*x^2+b*x+a)^(1/2)*e^2*f+3/2/c^3*b/(c*x
^2+b*x+a)^(1/2)*d*f^2+1/4*b/c^2/(c*x^2+b*x+a)^(3/2)*f*d^2-24*b^2/c*a/(4*a*c-b^2)
^2/(c*x^2+b*x+a)^(1/2)*d*e*f+5/4*e*f^2*b^6/c^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)
-3/2*x/c/(c*x^2+b*x+a)^(3/2)*e^2*d-33/8*f^3*b^3/c^4*a^2/(4*a*c-b^2)/(c*x^2+b*x+a
)^(3/2)-33*f^3*b^3/c^3*a^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-5/4*f^3/c^4*a*b^3/(
4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-6*x^2/c/(c*x^2+b*x+a)^(3/2)*d*e*f-8*b*a/(4*a*c-b^
2)^2/(c*x^2+b*x+a)^(1/2)*x*e^3+1/4*b^2/c^3/(c*x^2+b*x+a)^(3/2)*d*e*f+1/12*b^3/c^
2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*e^3-3*e*f^2*b^2/c^4*a/(c*x^2+b*x+a)^(3/2)+15
/2*e*f^2*b/c^3*x/(c*x^2+b*x+a)^(1/2)-15/4*e*f^2*b^2/c^3*x^2/(c*x^2+b*x+a)^(3/2)-
15/16*e*f^2*b^3/c^4*x/(c*x^2+b*x+a)^(3/2)+5/32*e*f^2*b^6/c^5/(4*a*c-b^2)/(c*x^2+
b*x+a)^(3/2)+3/2/c^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d*f^2-1/16*b^5/c^4/(4*a
*c-b^2)/(c*x^2+b*x+a)^(3/2)*d*f^2-1/16*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*e
^2*f-1/2*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*d*f^2-1/2*b^5/c^3/(4*a*c-b^2)
^2/(c*x^2+b*x+a)^(1/2)*e^2*f+3/8*b^2/c^3*x/(c*x^2+b*x+a)^(3/2)*e^2*f+3/2*b/c^2*x
^2/(c*x^2+b*x+a)^(3/2)*e^2*f+3/8*b^2/c^3*x/(c*x^2+b*x+a)^(3/2)*d*f^2+3/2*b/c^2*x
^2/(c*x^2+b*x+a)^(3/2)*d*f^2+3/2/c^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*e^2*f+b
/c^3*a/(c*x^2+b*x+a)^(3/2)*d*f^2+b/c^3*a/(c*x^2+b*x+a)^(3/2)*e^2*f+23/2*f^3*b^5/
c^4*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+35/8*f^3*b^4/c^4/(4*a*c-b^2)/(c*x^2+b*x+
a)^(1/2)*x-33/4*f^3*b/c^3*a*x^2/(c*x^2+b*x+a)^(3/2)-35/192*f^3*b^6/c^5/(4*a*c-b^
2)/(c*x^2+b*x+a)^(3/2)*x-35/24*f^3*b^6/c^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+2
3/16*f^3*b^5/c^5*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-33/16*f^3*b^2/c^4*a*x/(c*x^2+
b*x+a)^(3/2)

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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((f*x^2 + e*x + d)^3/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43224, 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)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(4*(192*a^2*b*c^4*d*e^2 - 128*a^3*c^4*e^3 + 6*(b^4*c^3 - 8*a*b^2*c^4 + 16*
a^2*c^5)*f^3*x^5 + 3*(12*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e*f^2 - 7*(b^5*c^2
- 8*a*b^3*c^3 + 16*a^2*b*c^4)*f^3)*x^4 - 8*(b^3*c^4 - 12*a*b*c^5)*d^3 - 48*(a*b
^2*c^4 + 4*a^2*c^5)*d^2*e - (105*a^2*b^5 - 760*a^3*b^3*c + 1296*a^4*b*c^2)*f^3 +
4*(32*c^7*d^3 - 48*b*c^6*d^2*e + 12*(b^2*c^5 + 4*a*c^6)*d*e^2 + 2*(b^3*c^4 - 12
*a*b*c^5)*e^3 - (35*b^6*c - 279*a*b^4*c^2 + 588*a^2*b^2*c^3 - 160*a^3*c^4)*f^3 -
12*(2*(b^4*c^3 - 7*a*b^2*c^4 + 8*a^2*c^5)*d - (5*b^5*c^2 - 37*a*b^3*c^3 + 64*a^
2*b*c^4)*e)*f^2 + 12*((b^2*c^5 + 4*a*c^6)*d^2 + (b^3*c^4 - 12*a*b*c^5)*d*e - 2*(
b^4*c^3 - 7*a*b^2*c^4 + 8*a^2*c^5)*e^2)*f)*x^3 - 12*(2*(3*a^2*b^3*c^2 - 20*a^3*b
*c^3)*d - (15*a^2*b^4*c - 100*a^3*b^2*c^2 + 128*a^4*c^3)*e)*f^2 + 3*(64*b*c^6*d^
3 - 96*b^2*c^5*d^2*e + 24*(b^3*c^4 + 4*a*b*c^5)*d*e^2 - 16*(a*b^2*c^4 + 4*a^2*c^
5)*e^3 - (35*b^7 - 230*a*b^5*c + 232*a^2*b^3*c^2 + 448*a^3*b*c^3)*f^3 - 12*(2*(b
^5*c^2 - 6*a*b^3*c^3)*d - (5*b^6*c - 30*a*b^4*c^2 + 16*a^2*b^2*c^3 + 64*a^3*c^4)
*e)*f^2 + 24*((b^3*c^4 + 4*a*b*c^5)*d^2 - 4*(a*b^2*c^4 + 4*a^2*c^5)*d*e - (b^5*c
^2 - 6*a*b^3*c^3)*e^2)*f)*x^2 + 24*(8*a^2*b*c^4*d^2 - 32*a^3*c^4*d*e - (3*a^2*b^
3*c^2 - 20*a^3*b*c^3)*e^2)*f + 6*(48*a*b^2*c^4*d*e^2 - 32*a^2*b*c^4*e^3 + 8*(b^2
*c^5 + 4*a*c^6)*d^3 - 12*(b^3*c^4 + 4*a*b*c^5)*d^2*e - (35*a*b^6 - 265*a^2*b^4*c
+ 504*a^3*b^2*c^2 - 80*a^4*c^3)*f^3 - 12*(2*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 4*a^3*
c^4)*d - (5*a*b^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^3)*e)*f^2 + 24*(2*a*b^2*c^4*d^
2 - 8*a^2*b*c^4*d*e - (a*b^4*c^2 - 7*a^2*b^2*c^3 + 4*a^3*c^4)*e^2)*f)*x)*sqrt(c*
x^2 + b*x + a)*sqrt(c) - 3*((24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*e^2*f + 5*(
7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*f^3 + 12*(2*(b^4*c^4 -
8*a*b^2*c^5 + 16*a^2*c^6)*d - 5*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*e)*f^2)*x
^4 + 24*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*e^2*f + 5*(7*a^2*b^6 - 60*a^3
*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*f^3 + 2*(24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a
^2*b*c^5)*e^2*f + 5*(7*b^7*c - 60*a*b^5*c^2 + 144*a^2*b^3*c^3 - 64*a^3*b*c^4)*f^
3 + 12*(2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d - 5*(b^6*c^2 - 8*a*b^4*c^3 +
16*a^2*b^2*c^4)*e)*f^2)*x^3 + 12*(2*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d
- 5*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*e)*f^2 + (24*(b^6*c^2 - 6*a*b^4*
c^3 + 32*a^3*c^5)*e^2*f + 5*(7*b^8 - 46*a*b^6*c + 24*a^2*b^4*c^2 + 224*a^3*b^2*c
^3 - 128*a^4*c^4)*f^3 + 12*(2*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d - 5*(b^7*c
- 6*a*b^5*c^2 + 32*a^3*b*c^4)*e)*f^2)*x^2 + 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 1
6*a^3*b*c^4)*e^2*f + 5*(7*a*b^7 - 60*a^2*b^5*c + 144*a^3*b^3*c^2 - 64*a^4*b*c^3)
*f^3 + 12*(2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d - 5*(a*b^6*c - 8*a^2*b
^4*c^2 + 16*a^3*b^2*c^3)*e)*f^2)*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^2*b^4*c^4 - 8*a^3*b^2*c^5 +
16*a^4*c^6 + (b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*x^4 + 2*(b^5*c^5 - 8*a*b^3*c^6
+ 16*a^2*b*c^7)*x^3 + (b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*x^2 + 2*(a*b^5*c^4 -
8*a^2*b^3*c^5 + 16*a^3*b*c^6)*x)*sqrt(c)), 1/24*(2*(192*a^2*b*c^4*d*e^2 - 128*a
^3*c^4*e^3 + 6*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*f^3*x^5 + 3*(12*(b^4*c^3 - 8
*a*b^2*c^4 + 16*a^2*c^5)*e*f^2 - 7*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*f^3)*x
^4 - 8*(b^3*c^4 - 12*a*b*c^5)*d^3 - 48*(a*b^2*c^4 + 4*a^2*c^5)*d^2*e - (105*a^2*
b^5 - 760*a^3*b^3*c + 1296*a^4*b*c^2)*f^3 + 4*(32*c^7*d^3 - 48*b*c^6*d^2*e + 12*
(b^2*c^5 + 4*a*c^6)*d*e^2 + 2*(b^3*c^4 - 12*a*b*c^5)*e^3 - (35*b^6*c - 279*a*b^4
*c^2 + 588*a^2*b^2*c^3 - 160*a^3*c^4)*f^3 - 12*(2*(b^4*c^3 - 7*a*b^2*c^4 + 8*a^2
*c^5)*d - (5*b^5*c^2 - 37*a*b^3*c^3 + 64*a^2*b*c^4)*e)*f^2 + 12*((b^2*c^5 + 4*a*
c^6)*d^2 + (b^3*c^4 - 12*a*b*c^5)*d*e - 2*(b^4*c^3 - 7*a*b^2*c^4 + 8*a^2*c^5)*e^
2)*f)*x^3 - 12*(2*(3*a^2*b^3*c^2 - 20*a^3*b*c^3)*d - (15*a^2*b^4*c - 100*a^3*b^2
*c^2 + 128*a^4*c^3)*e)*f^2 + 3*(64*b*c^6*d^3 - 96*b^2*c^5*d^2*e + 24*(b^3*c^4 +
4*a*b*c^5)*d*e^2 - 16*(a*b^2*c^4 + 4*a^2*c^5)*e^3 - (35*b^7 - 230*a*b^5*c + 232*
a^2*b^3*c^2 + 448*a^3*b*c^3)*f^3 - 12*(2*(b^5*c^2 - 6*a*b^3*c^3)*d - (5*b^6*c -
30*a*b^4*c^2 + 16*a^2*b^2*c^3 + 64*a^3*c^4)*e)*f^2 + 24*((b^3*c^4 + 4*a*b*c^5)*d
^2 - 4*(a*b^2*c^4 + 4*a^2*c^5)*d*e - (b^5*c^2 - 6*a*b^3*c^3)*e^2)*f)*x^2 + 24*(8
*a^2*b*c^4*d^2 - 32*a^3*c^4*d*e - (3*a^2*b^3*c^2 - 20*a^3*b*c^3)*e^2)*f + 6*(48*
a*b^2*c^4*d*e^2 - 32*a^2*b*c^4*e^3 + 8*(b^2*c^5 + 4*a*c^6)*d^3 - 12*(b^3*c^4 + 4
*a*b*c^5)*d^2*e - (35*a*b^6 - 265*a^2*b^4*c + 504*a^3*b^2*c^2 - 80*a^4*c^3)*f^3
- 12*(2*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 4*a^3*c^4)*d - (5*a*b^5*c - 35*a^2*b^3*c^2
+ 52*a^3*b*c^3)*e)*f^2 + 24*(2*a*b^2*c^4*d^2 - 8*a^2*b*c^4*d*e - (a*b^4*c^2 - 7*
a^2*b^2*c^3 + 4*a^3*c^4)*e^2)*f)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 3*((24*(b^4
*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*e^2*f + 5*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b
^2*c^4 - 64*a^3*c^5)*f^3 + 12*(2*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d - 5*(b^5
*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*e)*f^2)*x^4 + 24*(a^2*b^4*c^2 - 8*a^3*b^2*c^3
+ 16*a^4*c^4)*e^2*f + 5*(7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^
3)*f^3 + 2*(24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*e^2*f + 5*(7*b^7*c - 60*a*
b^5*c^2 + 144*a^2*b^3*c^3 - 64*a^3*b*c^4)*f^3 + 12*(2*(b^5*c^3 - 8*a*b^3*c^4 + 1
6*a^2*b*c^5)*d - 5*(b^6*c^2 - 8*a*b^4*c^3 + 16*a^2*b^2*c^4)*e)*f^2)*x^3 + 12*(2*
(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d - 5*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16
*a^4*b*c^3)*e)*f^2 + (24*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*e^2*f + 5*(7*b^8 -
46*a*b^6*c + 24*a^2*b^4*c^2 + 224*a^3*b^2*c^3 - 128*a^4*c^4)*f^3 + 12*(2*(b^6*c
^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d - 5*(b^7*c - 6*a*b^5*c^2 + 32*a^3*b*c^4)*e)*f^2
)*x^2 + 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*e^2*f + 5*(7*a*b^7 - 60
*a^2*b^5*c + 144*a^3*b^3*c^2 - 64*a^4*b*c^3)*f^3 + 12*(2*(a*b^5*c^2 - 8*a^2*b^3*
c^3 + 16*a^3*b*c^4)*d - 5*(a*b^6*c - 8*a^2*b^4*c^2 + 16*a^3*b^2*c^3)*e)*f^2)*x)*
arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((a^2*b^4*c^4 - 8*a^
3*b^2*c^5 + 16*a^4*c^6 + (b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*x^4 + 2*(b^5*c^5 -
8*a*b^3*c^6 + 16*a^2*b*c^7)*x^3 + (b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*x^2 + 2*
(a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*x)*sqrt(-c))]

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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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287527, 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)^(5/2),x, algorithm="giac")

[Out]

Done