3.3 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{d+f x^2} \, dx\)

Optimal. Leaf size=441 \[ \frac{\log \left (d+f x^2\right ) \left (A b f \left (-f \left (b^2 d-3 a^2 f\right )-6 a c d f+3 c^2 d^2\right )-B (c d-a f) \left (-f \left (3 b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )\right )}{2 f^4}-\frac{x^2 \left (A b f \left (-6 a c f+b^2 (-f)+3 c^2 d\right )-B \left (-3 c f \left (b^2 d-a^2 f\right )+3 a b^2 f^2-3 a c^2 d f+c^3 d^2\right )\right )}{2 f^3}-\frac{x \left (-A c \left (3 a^2 f^2-3 a c d f+c^2 d^2\right )+3 A b^2 f (c d-a f)-3 b B (c d-a f)^2+b^3 B d f\right )}{f^3}+\frac{c x^4 \left (3 A b c f-B \left (-3 a c f-3 b^2 f+c^2 d\right )\right )}{4 f^2}+\frac{x^3 \left (-A c^2 (c d-3 a f)-3 b B c (c d-2 a f)+3 A b^2 c f+b^3 B f\right )}{3 f^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) \left (3 A b^2 d f (c d-a f)-A (c d-a f)^3-3 b B d (c d-a f)^2+b^3 B d^2 f\right )}{\sqrt{d} f^{7/2}}+\frac{c^2 x^5 (A c+3 b B)}{5 f}+\frac{B c^3 x^6}{6 f} \]

[Out]

-(((b^3*B*d*f + 3*A*b^2*f*(c*d - a*f) - 3*b*B*(c*d - a*f)^2 - A*c*(c^2*d^2 - 3*a
*c*d*f + 3*a^2*f^2))*x)/f^3) - ((A*b*f*(3*c^2*d - b^2*f - 6*a*c*f) - B*(c^3*d^2
- 3*a*c^2*d*f + 3*a*b^2*f^2 - 3*c*f*(b^2*d - a^2*f)))*x^2)/(2*f^3) + ((b^3*B*f +
3*A*b^2*c*f - A*c^2*(c*d - 3*a*f) - 3*b*B*c*(c*d - 2*a*f))*x^3)/(3*f^2) + (c*(3
*A*b*c*f - B*(c^2*d - 3*b^2*f - 3*a*c*f))*x^4)/(4*f^2) + (c^2*(3*b*B + A*c)*x^5)
/(5*f) + (B*c^3*x^6)/(6*f) + ((b^3*B*d^2*f + 3*A*b^2*d*f*(c*d - a*f) - 3*b*B*d*(
c*d - a*f)^2 - A*(c*d - a*f)^3)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*f^(7/2)) +
((A*b*f*(3*c^2*d^2 - 6*a*c*d*f - f*(b^2*d - 3*a^2*f)) - B*(c*d - a*f)*(c^2*d^2
- 2*a*c*d*f - f*(3*b^2*d - a^2*f)))*Log[d + f*x^2])/(2*f^4)

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Rubi [A]  time = 1.58091, antiderivative size = 441, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\log \left (d+f x^2\right ) \left (A b f \left (-f \left (b^2 d-3 a^2 f\right )-6 a c d f+3 c^2 d^2\right )-B (c d-a f) \left (-f \left (3 b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )\right )}{2 f^4}-\frac{x^2 \left (A b f \left (-6 a c f+b^2 (-f)+3 c^2 d\right )-B \left (-3 c f \left (b^2 d-a^2 f\right )+3 a b^2 f^2-3 a c^2 d f+c^3 d^2\right )\right )}{2 f^3}-\frac{x \left (-A c \left (3 a^2 f^2-3 a c d f+c^2 d^2\right )+3 A b^2 f (c d-a f)-3 b B (c d-a f)^2+b^3 B d f\right )}{f^3}+\frac{c x^4 \left (3 A b c f-B \left (-3 a c f-3 b^2 f+c^2 d\right )\right )}{4 f^2}+\frac{x^3 \left (-A c^2 (c d-3 a f)-3 b B c (c d-2 a f)+3 A b^2 c f+b^3 B f\right )}{3 f^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) \left (3 A b^2 d f (c d-a f)-A (c d-a f)^3-3 b B d (c d-a f)^2+b^3 B d^2 f\right )}{\sqrt{d} f^{7/2}}+\frac{c^2 x^5 (A c+3 b B)}{5 f}+\frac{B c^3 x^6}{6 f} \]

Antiderivative was successfully verified.

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

[Out]

-(((b^3*B*d*f + 3*A*b^2*f*(c*d - a*f) - 3*b*B*(c*d - a*f)^2 - A*c*(c^2*d^2 - 3*a
*c*d*f + 3*a^2*f^2))*x)/f^3) - ((A*b*f*(3*c^2*d - b^2*f - 6*a*c*f) - B*(c^3*d^2
- 3*a*c^2*d*f + 3*a*b^2*f^2 - 3*c*f*(b^2*d - a^2*f)))*x^2)/(2*f^3) + ((b^3*B*f +
3*A*b^2*c*f - A*c^2*(c*d - 3*a*f) - 3*b*B*c*(c*d - 2*a*f))*x^3)/(3*f^2) + (c*(3
*A*b*c*f - B*(c^2*d - 3*b^2*f - 3*a*c*f))*x^4)/(4*f^2) + (c^2*(3*b*B + A*c)*x^5)
/(5*f) + (B*c^3*x^6)/(6*f) + ((b^3*B*d^2*f + 3*A*b^2*d*f*(c*d - a*f) - 3*b*B*d*(
c*d - a*f)^2 - A*(c*d - a*f)^3)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*f^(7/2)) +
((A*b*f*(3*c^2*d^2 - 6*a*c*d*f - f*(b^2*d - 3*a^2*f)) - B*(c*d - a*f)*(c^2*d^2
- 2*a*c*d*f - f*(3*b^2*d - a^2*f)))*Log[d + f*x^2])/(2*f^4)

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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((B*x+A)*(c*x**2+b*x+a)**3/(f*x**2+d),x)

[Out]

Timed out

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Mathematica [A]  time = 1.81564, size = 422, normalized size = 0.96 \[ \frac{f x \left (3 b \left (4 B \left (15 a^2 f^2+10 a c f \left (f x^2-3 d\right )+c^2 \left (15 d^2-5 d f x^2+3 f^2 x^4\right )\right )+15 A c f x \left (4 a f-2 c d+c f x^2\right )\right )+c \left (4 A \left (45 a^2 f^2+15 a c f \left (f x^2-3 d\right )+c^2 \left (15 d^2-5 d f x^2+3 f^2 x^4\right )\right )+5 B x \left (18 a^2 f^2+9 a c f \left (f x^2-2 d\right )+c^2 \left (6 d^2-3 d f x^2+2 f^2 x^4\right )\right )\right )+15 b^2 f \left (4 A \left (3 a f-3 c d+c f x^2\right )+3 B x \left (2 a f-2 c d+c f x^2\right )\right )+10 b^3 f \left (3 A f x-6 B d+2 B f x^2\right )\right )-30 \log \left (d+f x^2\right ) \left (A b f \left (-3 a^2 f^2+6 a c d f+b^2 d f-3 c^2 d^2\right )+B (c d-a f) \left (a^2 f^2-2 a c d f-3 b^2 d f+c^2 d^2\right )\right )}{60 f^4}+\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) \left (3 A b^2 d f (c d-a f)-A (c d-a f)^3-3 b B d (c d-a f)^2+b^3 B d^2 f\right )}{\sqrt{d} f^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((b^3*B*d^2*f + 3*A*b^2*d*f*(c*d - a*f) - 3*b*B*d*(c*d - a*f)^2 - A*(c*d - a*f)^
3)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*f^(7/2)) + (f*x*(10*b^3*f*(-6*B*d + 3*A
*f*x + 2*B*f*x^2) + 15*b^2*f*(3*B*x*(-2*c*d + 2*a*f + c*f*x^2) + 4*A*(-3*c*d + 3
*a*f + c*f*x^2)) + 3*b*(15*A*c*f*x*(-2*c*d + 4*a*f + c*f*x^2) + 4*B*(15*a^2*f^2
+ 10*a*c*f*(-3*d + f*x^2) + c^2*(15*d^2 - 5*d*f*x^2 + 3*f^2*x^4))) + c*(5*B*x*(1
8*a^2*f^2 + 9*a*c*f*(-2*d + f*x^2) + c^2*(6*d^2 - 3*d*f*x^2 + 2*f^2*x^4)) + 4*A*
(45*a^2*f^2 + 15*a*c*f*(-3*d + f*x^2) + c^2*(15*d^2 - 5*d*f*x^2 + 3*f^2*x^4))))
- 30*(A*b*f*(-3*c^2*d^2 + b^2*d*f + 6*a*c*d*f - 3*a^2*f^2) + B*(c*d - a*f)*(c^2*
d^2 - 3*b^2*d*f - 2*a*c*d*f + a^2*f^2))*Log[d + f*x^2])/(60*f^4)

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Maple [A]  time = 0.012, size = 822, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^3/(f*x^2+d),x)

[Out]

6/f^2/(d*f)^(1/2)*arctan(x*f/(d*f)^(1/2))*B*a*b*c*d^2-3/f^2*A*b^2*c*d*x-3/2/f^2*
B*x^2*b^2*c*d-3/2/f^2*B*x^2*a*c^2*d-3/2/f^2*A*x^2*b*c^2*d-3/f^2*A*a*c^2*d*x+2/f*
B*x^3*a*b*c+3/f*A*x^2*a*b*c+3/f^3*B*b*c^2*d^2*x-3/f^3/(d*f)^(1/2)*arctan(x*f/(d*
f)^(1/2))*B*b*c^2*d^3-6/f^2*B*a*b*c*d*x+3/f^2/(d*f)^(1/2)*arctan(x*f/(d*f)^(1/2)
)*A*a*c^2*d^2+3/f^2/(d*f)^(1/2)*arctan(x*f/(d*f)^(1/2))*A*b^2*c*d^2-3/f/(d*f)^(1
/2)*arctan(x*f/(d*f)^(1/2))*B*a^2*b*d-3/f^2*ln(f*x^2+d)*A*a*b*c*d-3/f/(d*f)^(1/2
)*arctan(x*f/(d*f)^(1/2))*A*a^2*c*d-3/f/(d*f)^(1/2)*arctan(x*f/(d*f)^(1/2))*A*a*
b^2*d-3/2/f^2*ln(f*x^2+d)*B*a^2*c*d-3/2/f^2*ln(f*x^2+d)*B*a*b^2*d+3/2/f^3*ln(f*x
^2+d)*B*a*c^2*d^2-1/f^2*B*x^3*b*c^2*d+3/2/f^3*ln(f*x^2+d)*A*b*c^2*d^2+3/2/f^3*ln
(f*x^2+d)*B*b^2*c*d^2-1/f^3/(d*f)^(1/2)*arctan(x*f/(d*f)^(1/2))*A*c^3*d^3+1/f^2/
(d*f)^(1/2)*arctan(x*f/(d*f)^(1/2))*b^3*B*d^2+1/5/f*A*x^5*c^3+1/2/f*A*x^2*b^3+1/
2/f*ln(f*x^2+d)*B*a^3+1/(d*f)^(1/2)*arctan(x*f/(d*f)^(1/2))*A*a^3+1/3/f*B*x^3*b^
3+1/f^3*A*c^3*d^2*x+3/f*b*B*a^2*x+3/f*A*c*a^2*x+3/f*b^2*A*a*x+3/4/f*B*x^4*b^2*c-
1/4/f^2*B*x^4*c^3*d-1/3/f^2*A*x^3*c^3*d+3/2/f*B*x^2*a^2*c+3/2/f*B*x^2*a*b^2-1/2/
f^2*ln(f*x^2+d)*A*b^3*d-1/2/f^4*ln(f*x^2+d)*B*c^3*d^3+3/2/f*ln(f*x^2+d)*A*a^2*b-
1/f^2*b^3*B*d*x+3/4/f*B*x^4*a*c^2+3/4/f*A*x^4*b*c^2+3/5/f*B*x^5*b*c^2+1/2/f^3*B*
x^2*c^3*d^2+1/f*A*x^3*b^2*c+1/f*A*x^3*a*c^2+1/6*B*c^3*x^6/f

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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*(B*x + A)/(f*x^2 + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282428, 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*(B*x + A)/(f*x^2 + d),x, algorithm="fricas")

[Out]

[1/60*(30*(A*a^3*f^4 - (3*B*b*c^2 + A*c^3)*d^3*f + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a
*b + A*b^2)*c)*d^2*f^2 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*f^3)*log((2*d*f*x + (
f*x^2 - d)*sqrt(-d*f))/(f*x^2 + d)) + (10*B*c^3*f^3*x^6 + 12*(3*B*b*c^2 + A*c^3)
*f^3*x^5 - 15*(B*c^3*d*f^2 - 3*(B*b^2*c + (B*a + A*b)*c^2)*f^3)*x^4 - 20*((3*B*b
*c^2 + A*c^3)*d*f^2 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*f^3)*x^3 + 30*
(B*c^3*d^2*f - 3*(B*b^2*c + (B*a + A*b)*c^2)*d*f^2 + (3*B*a*b^2 + A*b^3 + 3*(B*a
^2 + 2*A*a*b)*c)*f^3)*x^2 + 60*((3*B*b*c^2 + A*c^3)*d^2*f - (B*b^3 + 3*A*a*c^2 +
3*(2*B*a*b + A*b^2)*c)*d*f^2 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*f^3)*x - 30*(B*c
^3*d^3 - 3*(B*b^2*c + (B*a + A*b)*c^2)*d^2*f + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2
*A*a*b)*c)*d*f^2 - (B*a^3 + 3*A*a^2*b)*f^3)*log(f*x^2 + d))*sqrt(-d*f))/(sqrt(-d
*f)*f^4), 1/60*(60*(A*a^3*f^4 - (3*B*b*c^2 + A*c^3)*d^3*f + (B*b^3 + 3*A*a*c^2 +
3*(2*B*a*b + A*b^2)*c)*d^2*f^2 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*f^3)*arctan(
sqrt(d*f)*x/d) + (10*B*c^3*f^3*x^6 + 12*(3*B*b*c^2 + A*c^3)*f^3*x^5 - 15*(B*c^3*
d*f^2 - 3*(B*b^2*c + (B*a + A*b)*c^2)*f^3)*x^4 - 20*((3*B*b*c^2 + A*c^3)*d*f^2 -
(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*f^3)*x^3 + 30*(B*c^3*d^2*f - 3*(B*b
^2*c + (B*a + A*b)*c^2)*d*f^2 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*f^3)
*x^2 + 60*((3*B*b*c^2 + A*c^3)*d^2*f - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*
c)*d*f^2 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*f^3)*x - 30*(B*c^3*d^3 - 3*(B*b^2*c +
(B*a + A*b)*c^2)*d^2*f + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*f^2 - (B
*a^3 + 3*A*a^2*b)*f^3)*log(f*x^2 + d))*sqrt(d*f))/(sqrt(d*f)*f^4)]

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Sympy [A]  time = 30.0664, size = 1940, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**3/(f*x**2+d),x)

[Out]

B*c**3*x**6/(6*f) + ((3*A*a**2*b*f**3 - 6*A*a*b*c*d*f**2 - A*b**3*d*f**2 + 3*A*b
*c**2*d**2*f + B*a**3*f**3 - 3*B*a**2*c*d*f**2 - 3*B*a*b**2*d*f**2 + 3*B*a*c**2*
d**2*f + 3*B*b**2*c*d**2*f - B*c**3*d**3)/(2*f**4) - sqrt(-d*f**9)*(A*a**3*f**3
- 3*A*a**2*c*d*f**2 - 3*A*a*b**2*d*f**2 + 3*A*a*c**2*d**2*f + 3*A*b**2*c*d**2*f
- A*c**3*d**3 - 3*B*a**2*b*d*f**2 + 6*B*a*b*c*d**2*f + B*b**3*d**2*f - 3*B*b*c**
2*d**3)/(2*d*f**8))*log(x + (-3*A*a**2*b*d*f**3 + 6*A*a*b*c*d**2*f**2 + A*b**3*d
**2*f**2 - 3*A*b*c**2*d**3*f - B*a**3*d*f**3 + 3*B*a**2*c*d**2*f**2 + 3*B*a*b**2
*d**2*f**2 - 3*B*a*c**2*d**3*f - 3*B*b**2*c*d**3*f + B*c**3*d**4 + 2*d*f**4*((3*
A*a**2*b*f**3 - 6*A*a*b*c*d*f**2 - A*b**3*d*f**2 + 3*A*b*c**2*d**2*f + B*a**3*f*
*3 - 3*B*a**2*c*d*f**2 - 3*B*a*b**2*d*f**2 + 3*B*a*c**2*d**2*f + 3*B*b**2*c*d**2
*f - B*c**3*d**3)/(2*f**4) - sqrt(-d*f**9)*(A*a**3*f**3 - 3*A*a**2*c*d*f**2 - 3*
A*a*b**2*d*f**2 + 3*A*a*c**2*d**2*f + 3*A*b**2*c*d**2*f - A*c**3*d**3 - 3*B*a**2
*b*d*f**2 + 6*B*a*b*c*d**2*f + B*b**3*d**2*f - 3*B*b*c**2*d**3)/(2*d*f**8)))/(A*
a**3*f**4 - 3*A*a**2*c*d*f**3 - 3*A*a*b**2*d*f**3 + 3*A*a*c**2*d**2*f**2 + 3*A*b
**2*c*d**2*f**2 - A*c**3*d**3*f - 3*B*a**2*b*d*f**3 + 6*B*a*b*c*d**2*f**2 + B*b*
*3*d**2*f**2 - 3*B*b*c**2*d**3*f)) + ((3*A*a**2*b*f**3 - 6*A*a*b*c*d*f**2 - A*b*
*3*d*f**2 + 3*A*b*c**2*d**2*f + B*a**3*f**3 - 3*B*a**2*c*d*f**2 - 3*B*a*b**2*d*f
**2 + 3*B*a*c**2*d**2*f + 3*B*b**2*c*d**2*f - B*c**3*d**3)/(2*f**4) + sqrt(-d*f*
*9)*(A*a**3*f**3 - 3*A*a**2*c*d*f**2 - 3*A*a*b**2*d*f**2 + 3*A*a*c**2*d**2*f + 3
*A*b**2*c*d**2*f - A*c**3*d**3 - 3*B*a**2*b*d*f**2 + 6*B*a*b*c*d**2*f + B*b**3*d
**2*f - 3*B*b*c**2*d**3)/(2*d*f**8))*log(x + (-3*A*a**2*b*d*f**3 + 6*A*a*b*c*d**
2*f**2 + A*b**3*d**2*f**2 - 3*A*b*c**2*d**3*f - B*a**3*d*f**3 + 3*B*a**2*c*d**2*
f**2 + 3*B*a*b**2*d**2*f**2 - 3*B*a*c**2*d**3*f - 3*B*b**2*c*d**3*f + B*c**3*d**
4 + 2*d*f**4*((3*A*a**2*b*f**3 - 6*A*a*b*c*d*f**2 - A*b**3*d*f**2 + 3*A*b*c**2*d
**2*f + B*a**3*f**3 - 3*B*a**2*c*d*f**2 - 3*B*a*b**2*d*f**2 + 3*B*a*c**2*d**2*f
+ 3*B*b**2*c*d**2*f - B*c**3*d**3)/(2*f**4) + sqrt(-d*f**9)*(A*a**3*f**3 - 3*A*a
**2*c*d*f**2 - 3*A*a*b**2*d*f**2 + 3*A*a*c**2*d**2*f + 3*A*b**2*c*d**2*f - A*c**
3*d**3 - 3*B*a**2*b*d*f**2 + 6*B*a*b*c*d**2*f + B*b**3*d**2*f - 3*B*b*c**2*d**3)
/(2*d*f**8)))/(A*a**3*f**4 - 3*A*a**2*c*d*f**3 - 3*A*a*b**2*d*f**3 + 3*A*a*c**2*
d**2*f**2 + 3*A*b**2*c*d**2*f**2 - A*c**3*d**3*f - 3*B*a**2*b*d*f**3 + 6*B*a*b*c
*d**2*f**2 + B*b**3*d**2*f**2 - 3*B*b*c**2*d**3*f)) + x**5*(A*c**3 + 3*B*b*c**2)
/(5*f) + x**4*(3*A*b*c**2*f + 3*B*a*c**2*f + 3*B*b**2*c*f - B*c**3*d)/(4*f**2) +
x**3*(3*A*a*c**2*f + 3*A*b**2*c*f - A*c**3*d + 6*B*a*b*c*f + B*b**3*f - 3*B*b*c
**2*d)/(3*f**2) + x**2*(6*A*a*b*c*f**2 + A*b**3*f**2 - 3*A*b*c**2*d*f + 3*B*a**2
*c*f**2 + 3*B*a*b**2*f**2 - 3*B*a*c**2*d*f - 3*B*b**2*c*d*f + B*c**3*d**2)/(2*f*
*3) + x*(3*A*a**2*c*f**2 + 3*A*a*b**2*f**2 - 3*A*a*c**2*d*f - 3*A*b**2*c*d*f + A
*c**3*d**2 + 3*B*a**2*b*f**2 - 6*B*a*b*c*d*f - B*b**3*d*f + 3*B*b*c**2*d**2)/f**
3

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GIAC/XCAS [A]  time = 0.266579, size = 841, normalized size = 1.91 \[ -\frac{{\left (3 \, B b c^{2} d^{3} + A c^{3} d^{3} - B b^{3} d^{2} f - 6 \, B a b c d^{2} f - 3 \, A b^{2} c d^{2} f - 3 \, A a c^{2} d^{2} f + 3 \, B a^{2} b d f^{2} + 3 \, A a b^{2} d f^{2} + 3 \, A a^{2} c d f^{2} - A a^{3} f^{3}\right )} \arctan \left (\frac{f x}{\sqrt{d f}}\right )}{\sqrt{d f} f^{3}} - \frac{{\left (B c^{3} d^{3} - 3 \, B b^{2} c d^{2} f - 3 \, B a c^{2} d^{2} f - 3 \, A b c^{2} d^{2} f + 3 \, B a b^{2} d f^{2} + A b^{3} d f^{2} + 3 \, B a^{2} c d f^{2} + 6 \, A a b c d f^{2} - B a^{3} f^{3} - 3 \, A a^{2} b f^{3}\right )}{\rm ln}\left (f x^{2} + d\right )}{2 \, f^{4}} + \frac{10 \, B c^{3} f^{5} x^{6} + 36 \, B b c^{2} f^{5} x^{5} + 12 \, A c^{3} f^{5} x^{5} - 15 \, B c^{3} d f^{4} x^{4} + 45 \, B b^{2} c f^{5} x^{4} + 45 \, B a c^{2} f^{5} x^{4} + 45 \, A b c^{2} f^{5} x^{4} - 60 \, B b c^{2} d f^{4} x^{3} - 20 \, A c^{3} d f^{4} x^{3} + 20 \, B b^{3} f^{5} x^{3} + 120 \, B a b c f^{5} x^{3} + 60 \, A b^{2} c f^{5} x^{3} + 60 \, A a c^{2} f^{5} x^{3} + 30 \, B c^{3} d^{2} f^{3} x^{2} - 90 \, B b^{2} c d f^{4} x^{2} - 90 \, B a c^{2} d f^{4} x^{2} - 90 \, A b c^{2} d f^{4} x^{2} + 90 \, B a b^{2} f^{5} x^{2} + 30 \, A b^{3} f^{5} x^{2} + 90 \, B a^{2} c f^{5} x^{2} + 180 \, A a b c f^{5} x^{2} + 180 \, B b c^{2} d^{2} f^{3} x + 60 \, A c^{3} d^{2} f^{3} x - 60 \, B b^{3} d f^{4} x - 360 \, B a b c d f^{4} x - 180 \, A b^{2} c d f^{4} x - 180 \, A a c^{2} d f^{4} x + 180 \, B a^{2} b f^{5} x + 180 \, A a b^{2} f^{5} x + 180 \, A a^{2} c f^{5} x}{60 \, f^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)/(f*x^2 + d),x, algorithm="giac")

[Out]

-(3*B*b*c^2*d^3 + A*c^3*d^3 - B*b^3*d^2*f - 6*B*a*b*c*d^2*f - 3*A*b^2*c*d^2*f -
3*A*a*c^2*d^2*f + 3*B*a^2*b*d*f^2 + 3*A*a*b^2*d*f^2 + 3*A*a^2*c*d*f^2 - A*a^3*f^
3)*arctan(f*x/sqrt(d*f))/(sqrt(d*f)*f^3) - 1/2*(B*c^3*d^3 - 3*B*b^2*c*d^2*f - 3*
B*a*c^2*d^2*f - 3*A*b*c^2*d^2*f + 3*B*a*b^2*d*f^2 + A*b^3*d*f^2 + 3*B*a^2*c*d*f^
2 + 6*A*a*b*c*d*f^2 - B*a^3*f^3 - 3*A*a^2*b*f^3)*ln(f*x^2 + d)/f^4 + 1/60*(10*B*
c^3*f^5*x^6 + 36*B*b*c^2*f^5*x^5 + 12*A*c^3*f^5*x^5 - 15*B*c^3*d*f^4*x^4 + 45*B*
b^2*c*f^5*x^4 + 45*B*a*c^2*f^5*x^4 + 45*A*b*c^2*f^5*x^4 - 60*B*b*c^2*d*f^4*x^3 -
20*A*c^3*d*f^4*x^3 + 20*B*b^3*f^5*x^3 + 120*B*a*b*c*f^5*x^3 + 60*A*b^2*c*f^5*x^
3 + 60*A*a*c^2*f^5*x^3 + 30*B*c^3*d^2*f^3*x^2 - 90*B*b^2*c*d*f^4*x^2 - 90*B*a*c^
2*d*f^4*x^2 - 90*A*b*c^2*d*f^4*x^2 + 90*B*a*b^2*f^5*x^2 + 30*A*b^3*f^5*x^2 + 90*
B*a^2*c*f^5*x^2 + 180*A*a*b*c*f^5*x^2 + 180*B*b*c^2*d^2*f^3*x + 60*A*c^3*d^2*f^3
*x - 60*B*b^3*d*f^4*x - 360*B*a*b*c*d*f^4*x - 180*A*b^2*c*d*f^4*x - 180*A*a*c^2*
d*f^4*x + 180*B*a^2*b*f^5*x + 180*A*a*b^2*f^5*x + 180*A*a^2*c*f^5*x)/f^6