3.2327 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=289 \[ \frac{2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (d+e x)}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac{\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{3 e^6 (d+e x)^3}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac{c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac{B c^2 x}{e^5} \]

[Out]

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

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Rubi [A]  time = 1.01216, antiderivative size = 287, 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 \[ \frac{2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (d+e x)}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac{c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac{B c^2 x}{e^5} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^5,x]

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.951731, size = 391, normalized size = 1.35 \[ -\frac{A e \left (e^2 \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )+2 c e \left (a e \left (d^2+4 d e x+6 e^2 x^2\right )+3 b \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+c^2 (-d) \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+B \left (e^2 \left (a^2 e^2 (d+4 e x)+2 a b e \left (d^2+4 d e x+6 e^2 x^2\right )+3 b^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+2 c e \left (3 a e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-b d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+c^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )+12 c (d+e x)^4 \log (d+e x) (-A c e-2 b B e+5 B c d)}{12 e^6 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(A*e*(-(c^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + e^2*(3*a^2*
e^2 + 2*a*b*e*(d + 4*e*x) + b^2*(d^2 + 4*d*e*x + 6*e^2*x^2)) + 2*c*e*(a*e*(d^2 +
4*d*e*x + 6*e^2*x^2) + 3*b*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3))) + B*(c
^2*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*
e^5*x^5) + e^2*(a^2*e^2*(d + 4*e*x) + 2*a*b*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*b^
2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) + 2*c*e*(3*a*e*(d^3 + 4*d^2*e*x +
6*d*e^2*x^2 + 4*e^3*x^3) - b*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^
3))) + 12*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^4*Log[d + e*x])/(12*e^6*(d + e
*x)^4)

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Maple [B]  time = 0.016, size = 710, normalized size = 2.5 \[ \text{result too large to display} \]

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)^5,x)

[Out]

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

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Maxima [A]  time = 0.709442, size = 563, normalized size = 1.95 \[ -\frac{77 \, B c^{2} d^{5} + 3 \, A a^{2} e^{5} - 25 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} +{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 12 \,{\left (10 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 6 \,{\left (50 \, B c^{2} d^{3} e^{2} - 18 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 4 \,{\left (65 \, B c^{2} d^{4} e - 22 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} +{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac{B c^{2} x}{e^{5}} - \frac{{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )} \log \left (e x + d\right )}{e^{6}} \]

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)^5,x, algorithm="maxima")

[Out]

-1/12*(77*B*c^2*d^5 + 3*A*a^2*e^5 - 25*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*(B
*a + A*b)*c)*d^3*e^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 + 12*(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 + 6*(50*B*c^2*d^3*e^2 - 18*(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 + 4*(65*B*c^2*d^4*e
- 22*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 + (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^10*x^4 + 4*d*e^9*x^3 +
6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6) + B*c^2*x/e^5 - (5*B*c^2*d - (2*B*b*c + A
*c^2)*e)*log(e*x + d)/e^6

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Fricas [A]  time = 0.262483, size = 772, normalized size = 2.67 \[ \frac{12 \, B c^{2} e^{5} x^{5} + 48 \, B c^{2} d e^{4} x^{4} - 77 \, B c^{2} d^{5} - 3 \, A a^{2} e^{5} + 25 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} -{\left (B a^{2} + 2 \, A a b\right )} d e^{4} - 12 \,{\left (4 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} - 6 \,{\left (42 \, B c^{2} d^{3} e^{2} - 18 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} - 4 \,{\left (62 \, B c^{2} d^{4} e - 22 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} +{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x - 12 \,{\left (5 \, B c^{2} d^{5} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (5 \, B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 4 \,{\left (5 \, B c^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} + 6 \,{\left (5 \, B c^{2} d^{3} e^{2} -{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (5 \, B c^{2} d^{4} e -{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

1/12*(12*B*c^2*e^5*x^5 + 48*B*c^2*d*e^4*x^4 - 77*B*c^2*d^5 - 3*A*a^2*e^5 + 25*(2
*B*b*c + A*c^2)*d^4*e - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^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 - 12*(4*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 - 6*(42*B*c^2*d^3*e^2 - 18*(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 - 4*(62*B*c^2*d^4*e - 22*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2
+ 2*(B*a + A*b)*c)*d^2*e^3 + (2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + (B*a^2 + 2*A*a
*b)*e^5)*x - 12*(5*B*c^2*d^5 - (2*B*b*c + A*c^2)*d^4*e + (5*B*c^2*d*e^4 - (2*B*b
*c + A*c^2)*e^5)*x^4 + 4*(5*B*c^2*d^2*e^3 - (2*B*b*c + A*c^2)*d*e^4)*x^3 + 6*(5*
B*c^2*d^3*e^2 - (2*B*b*c + A*c^2)*d^2*e^3)*x^2 + 4*(5*B*c^2*d^4*e - (2*B*b*c + A
*c^2)*d^3*e^2)*x)*log(e*x + d))/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*
e^7*x + d^4*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)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.29352, 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)^2*(B*x + A)/(e*x + d)^5,x, algorithm="giac")

[Out]

Done