3.2326 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=286 \[ -\frac{\log (d+e x) \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^6 (d+e x)}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^3}+\frac{\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{2 e^6 (d+e x)^2}-\frac{c x (-A c e-2 b B e+4 B c d)}{e^5}+\frac{B c^2 x^2}{2 e^4} \]

[Out]

-((c*(4*B*c*d - 2*b*B*e - A*c*e)*x)/e^5) + (B*c^2*x^2)/(2*e^4) + ((B*d - A*e)*(c
*d^2 - b*d*e + a*e^2)^2)/(3*e^6*(d + e*x)^3) + ((c*d^2 - b*d*e + a*e^2)*(2*A*e*(
2*c*d - b*e) - B*(5*c*d^2 - e*(3*b*d - a*e))))/(2*e^6*(d + e*x)^2) + (B*(10*c^2*
d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d - a*e)) - A*e*(6*c^2*d^2 + b^2*e^2
- 2*c*e*(3*b*d - a*e)))/(e^6*(d + e*x)) - ((2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^
2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))*Log[d + e*x])/e^6

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Rubi [A]  time = 1.08121, antiderivative size = 284, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{\log (d+e x) \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^6 (d+e x)}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{2 e^6 (d+e x)^2}-\frac{c x (-A c e-2 b B e+4 B c d)}{e^5}+\frac{B c^2 x^2}{2 e^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^4,x]

[Out]

-((c*(4*B*c*d - 2*b*B*e - A*c*e)*x)/e^5) + (B*c^2*x^2)/(2*e^4) + ((B*d - A*e)*(c
*d^2 - b*d*e + a*e^2)^2)/(3*e^6*(d + e*x)^3) - ((c*d^2 - b*d*e + a*e^2)*(5*B*c*d
^2 - B*e*(3*b*d - a*e) - 2*A*e*(2*c*d - b*e)))/(2*e^6*(d + e*x)^2) + (B*(10*c^2*
d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d - a*e)) - A*e*(6*c^2*d^2 + b^2*e^2
- 2*c*e*(3*b*d - a*e)))/(e^6*(d + e*x)) - ((2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^
2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))*Log[d + e*x])/e^6

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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)**2/(e*x+d)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 0.314626, size = 263, normalized size = 0.92 \[ \frac{6 \log (d+e x) \left (B \left (2 c e (a e-4 b d)+b^2 e^2+10 c^2 d^2\right )+2 A c e (b e-2 c d)\right )-\frac{6 \left (A e \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+B \left (6 c d e (2 b d-a e)+b e^2 (2 a e-3 b d)-10 c^2 d^3\right )\right )}{d+e x}+\frac{2 (B d-A e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^3}-\frac{3 \left (e (a e-b d)+c d^2\right ) \left (B e (a e-3 b d)+2 A e (b e-2 c d)+5 B c d^2\right )}{(d+e x)^2}+6 c e x (A c e+2 b B e-4 B c d)+3 B c^2 e^2 x^2}{6 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^4,x]

[Out]

(6*c*e*(-4*B*c*d + 2*b*B*e + A*c*e)*x + 3*B*c^2*e^2*x^2 + (2*(B*d - A*e)*(c*d^2
+ e*(-(b*d) + a*e))^2)/(d + e*x)^3 - (3*(c*d^2 + e*(-(b*d) + a*e))*(5*B*c*d^2 +
B*e*(-3*b*d + a*e) + 2*A*e*(-2*c*d + b*e)))/(d + e*x)^2 - (6*(A*e*(6*c^2*d^2 + b
^2*e^2 + 2*c*e*(-3*b*d + a*e)) + B*(-10*c^2*d^3 + 6*c*d*e*(2*b*d - a*e) + b*e^2*
(-3*b*d + 2*a*e))))/(d + e*x) + 6*(2*A*c*e*(-2*c*d + b*e) + B*(10*c^2*d^2 + b^2*
e^2 + 2*c*e*(-4*b*d + a*e)))*Log[d + e*x])/(6*e^6)

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Maple [B]  time = 0.017, size = 690, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^4,x)

[Out]

1/2*B*c^2*x^2/e^4+b^2*B*ln(e*x+d)/e^4+c^2/e^4*A*x-1/3/e/(e*x+d)^3*A*a^2-1/e^3/(e
*x+d)*A*b^2-1/2/e^2/(e*x+d)^2*B*a^2-12/e^5/(e*x+d)*B*b*c*d^2+2/e^3/(e*x+d)^2*A*a
*c*d-3/e^4/(e*x+d)^2*A*b*c*d^2+2/e^3/(e*x+d)^2*B*a*b*d-3/e^4/(e*x+d)^2*B*a*c*d^2
+4/e^5/(e*x+d)^2*B*b*c*d^3-2/3/e^5/(e*x+d)^3*B*b*c*d^4-8/e^5*ln(e*x+d)*B*b*c*d+6
/e^4/(e*x+d)*A*b*c*d+6/e^4/(e*x+d)*B*a*c*d+2/3/e^2/(e*x+d)^3*A*d*a*b-2/3/e^3/(e*
x+d)^3*A*d^2*a*c+2/3/e^4/(e*x+d)^3*A*d^3*b*c-2/3/e^3/(e*x+d)^3*B*d^2*a*b+2/3/e^4
/(e*x+d)^3*B*d^3*a*c+1/3/e^2/(e*x+d)^3*B*d*a^2+1/3/e^4/(e*x+d)^3*B*d^3*b^2+1/3/e
^6/(e*x+d)^3*B*c^2*d^5+2/e^4*ln(e*x+d)*A*b*c-4/e^5*ln(e*x+d)*A*c^2*d+2/e^4*ln(e*
x+d)*B*a*c+10/e^6*ln(e*x+d)*B*c^2*d^2-2/e^3/(e*x+d)*A*a*c-6/e^5/(e*x+d)*A*c^2*d^
2-2/e^3/(e*x+d)*B*a*b+3/e^4/(e*x+d)*B*b^2*d+2*c/e^4*b*B*x-4*c^2/e^5*B*d*x+1/e^3/
(e*x+d)^2*A*b^2*d+2/e^5/(e*x+d)^2*A*c^2*d^3-3/2/e^4/(e*x+d)^2*B*b^2*d^2-5/2/e^6/
(e*x+d)^2*B*c^2*d^4-1/3/e^3/(e*x+d)^3*A*d^2*b^2-1/3/e^5/(e*x+d)^3*A*c^2*d^4+10/e
^6/(e*x+d)*B*c^2*d^3-1/e^2/(e*x+d)^2*A*a*b

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Maxima [A]  time = 0.705616, size = 552, normalized size = 1.93 \[ \frac{47 \, B c^{2} d^{5} - 2 \, A a^{2} e^{5} - 26 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 11 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} -{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 6 \,{\left (10 \, B c^{2} d^{3} e^{2} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 3 \,{\left (35 \, B c^{2} d^{4} e - 20 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 9 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} -{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac{B c^{2} e x^{2} - 2 \,{\left (4 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )} x}{2 \, e^{5}} + \frac{{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(B*x + A)/(e*x + d)^4,x, algorithm="maxima")

[Out]

1/6*(47*B*c^2*d^5 - 2*A*a^2*e^5 - 26*(2*B*b*c + A*c^2)*d^4*e + 11*(B*b^2 + 2*(B*
a + A*b)*c)*d^3*e^2 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 - (B*a^2 + 2*A*a*b)*
d*e^4 + 6*(10*B*c^2*d^3*e^2 - 6*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*(B*a +
A*b)*c)*d*e^4 - (2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 + 3*(35*B*c^2*d^4*e - 20*(2
*B*b*c + A*c^2)*d^3*e^2 + 9*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 - 2*(2*B*a*b + A*b
^2 + 2*A*a*c)*d*e^4 - (B*a^2 + 2*A*a*b)*e^5)*x)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e
^7*x + d^3*e^6) + 1/2*(B*c^2*e*x^2 - 2*(4*B*c^2*d - (2*B*b*c + A*c^2)*e)*x)/e^5
+ (10*B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*(B*a + A*b)*c)*e^2)*log(e
*x + d)/e^6

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Fricas [A]  time = 0.266914, size = 834, normalized size = 2.92 \[ \frac{3 \, B c^{2} e^{5} x^{5} + 47 \, B c^{2} d^{5} - 2 \, A a^{2} e^{5} - 26 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 11 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} -{\left (B a^{2} + 2 \, A a b\right )} d e^{4} - 3 \,{\left (5 \, B c^{2} d e^{4} - 2 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} - 9 \,{\left (7 \, B c^{2} d^{2} e^{3} - 2 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} - 3 \,{\left (3 \, B c^{2} d^{3} e^{2} + 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 6 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} + 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 3 \,{\left (27 \, B c^{2} d^{4} e - 18 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 9 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} -{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x + 6 \,{\left (10 \, B c^{2} d^{5} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} +{\left (10 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 3 \,{\left (10 \, B c^{2} d^{3} e^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4}\right )} x^{2} + 3 \,{\left (10 \, B c^{2} d^{4} e - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(B*x + A)/(e*x + d)^4,x, algorithm="fricas")

[Out]

1/6*(3*B*c^2*e^5*x^5 + 47*B*c^2*d^5 - 2*A*a^2*e^5 - 26*(2*B*b*c + A*c^2)*d^4*e +
11*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 -
(B*a^2 + 2*A*a*b)*d*e^4 - 3*(5*B*c^2*d*e^4 - 2*(2*B*b*c + A*c^2)*e^5)*x^4 - 9*(7
*B*c^2*d^2*e^3 - 2*(2*B*b*c + A*c^2)*d*e^4)*x^3 - 3*(3*B*c^2*d^3*e^2 + 6*(2*B*b*
c + A*c^2)*d^2*e^3 - 6*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4 + 2*(2*B*a*b + A*b^2 + 2*
A*a*c)*e^5)*x^2 + 3*(27*B*c^2*d^4*e - 18*(2*B*b*c + A*c^2)*d^3*e^2 + 9*(B*b^2 +
2*(B*a + A*b)*c)*d^2*e^3 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 - (B*a^2 + 2*A*a*
b)*e^5)*x + 6*(10*B*c^2*d^5 - 4*(2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*(B*a + A*b)
*c)*d^3*e^2 + (10*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*(B*a +
A*b)*c)*e^5)*x^3 + 3*(10*B*c^2*d^3*e^2 - 4*(2*B*b*c + A*c^2)*d^2*e^3 + (B*b^2 +
2*(B*a + A*b)*c)*d*e^4)*x^2 + 3*(10*B*c^2*d^4*e - 4*(2*B*b*c + A*c^2)*d^3*e^2 +
(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3)*x)*log(e*x + d))/(e^9*x^3 + 3*d*e^8*x^2 + 3*d
^2*e^7*x + d^3*e^6)

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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((B*x+A)*(c*x**2+b*x+a)**2/(e*x+d)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.259238, size = 572, normalized size = 2. \[{\left (10 \, B c^{2} d^{2} - 8 \, B b c d e - 4 \, A c^{2} d e + B b^{2} e^{2} + 2 \, B a c e^{2} + 2 \, A b c e^{2}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B c^{2} x^{2} e^{4} - 8 \, B c^{2} d x e^{3} + 4 \, B b c x e^{4} + 2 \, A c^{2} x e^{4}\right )} e^{\left (-8\right )} + \frac{{\left (47 \, B c^{2} d^{5} - 52 \, B b c d^{4} e - 26 \, A c^{2} d^{4} e + 11 \, B b^{2} d^{3} e^{2} + 22 \, B a c d^{3} e^{2} + 22 \, A b c d^{3} e^{2} - 4 \, B a b d^{2} e^{3} - 2 \, A b^{2} d^{2} e^{3} - 4 \, A a c d^{2} e^{3} - B a^{2} d e^{4} - 2 \, A a b d e^{4} - 2 \, A a^{2} e^{5} + 6 \,{\left (10 \, B c^{2} d^{3} e^{2} - 12 \, B b c d^{2} e^{3} - 6 \, A c^{2} d^{2} e^{3} + 3 \, B b^{2} d e^{4} + 6 \, B a c d e^{4} + 6 \, A b c d e^{4} - 2 \, B a b e^{5} - A b^{2} e^{5} - 2 \, A a c e^{5}\right )} x^{2} + 3 \,{\left (35 \, B c^{2} d^{4} e - 40 \, B b c d^{3} e^{2} - 20 \, A c^{2} d^{3} e^{2} + 9 \, B b^{2} d^{2} e^{3} + 18 \, B a c d^{2} e^{3} + 18 \, A b c d^{2} e^{3} - 4 \, B a b d e^{4} - 2 \, A b^{2} d e^{4} - 4 \, A a c d e^{4} - B a^{2} e^{5} - 2 \, A a b e^{5}\right )} x\right )} e^{\left (-6\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(B*x + A)/(e*x + d)^4,x, algorithm="giac")

[Out]

(10*B*c^2*d^2 - 8*B*b*c*d*e - 4*A*c^2*d*e + B*b^2*e^2 + 2*B*a*c*e^2 + 2*A*b*c*e^
2)*e^(-6)*ln(abs(x*e + d)) + 1/2*(B*c^2*x^2*e^4 - 8*B*c^2*d*x*e^3 + 4*B*b*c*x*e^
4 + 2*A*c^2*x*e^4)*e^(-8) + 1/6*(47*B*c^2*d^5 - 52*B*b*c*d^4*e - 26*A*c^2*d^4*e
+ 11*B*b^2*d^3*e^2 + 22*B*a*c*d^3*e^2 + 22*A*b*c*d^3*e^2 - 4*B*a*b*d^2*e^3 - 2*A
*b^2*d^2*e^3 - 4*A*a*c*d^2*e^3 - B*a^2*d*e^4 - 2*A*a*b*d*e^4 - 2*A*a^2*e^5 + 6*(
10*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 - 6*A*c^2*d^2*e^3 + 3*B*b^2*d*e^4 + 6*B*a*c*
d*e^4 + 6*A*b*c*d*e^4 - 2*B*a*b*e^5 - A*b^2*e^5 - 2*A*a*c*e^5)*x^2 + 3*(35*B*c^2
*d^4*e - 40*B*b*c*d^3*e^2 - 20*A*c^2*d^3*e^2 + 9*B*b^2*d^2*e^3 + 18*B*a*c*d^2*e^
3 + 18*A*b*c*d^2*e^3 - 4*B*a*b*d*e^4 - 2*A*b^2*d*e^4 - 4*A*a*c*d*e^4 - B*a^2*e^5
- 2*A*a*b*e^5)*x)*e^(-6)/(x*e + d)^3