3.2129 \(\int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx\)

Optimal. Leaf size=251 \[ \frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{c^3 x^2}{2 e^5} \]

[Out]

-((c^2*(5*c*d - 3*b*e)*x)/e^6) + (c^3*x^2)/(2*e^5) - (c*d^2 - b*d*e + a*e^2)^3/(
4*e^7*(d + e*x)^4) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^7*(d + e*x)^3)
- (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(2*e^7*
(d + e*x)^2) + ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(e
^7*(d + e*x)) + (3*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*Log[d + e*x])/e^7

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Rubi [A]  time = 0.789164, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{c^3 x^2}{2 e^5} \]

Antiderivative was successfully verified.

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

[Out]

-((c^2*(5*c*d - 3*b*e)*x)/e^6) + (c^3*x^2)/(2*e^5) - (c*d^2 - b*d*e + a*e^2)^3/(
4*e^7*(d + e*x)^4) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^7*(d + e*x)^3)
- (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(2*e^7*
(d + e*x)^2) + ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(e
^7*(d + e*x)) + (3*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*Log[d + e*x])/e^7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{c^{3} \int x\, dx}{e^{5}} + \frac{3 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} + \frac{\left (3 b e - 5 c d\right ) \int c^{2}\, dx}{e^{6}} - \frac{\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{e^{7} \left (d + e x\right )} - \frac{3 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{2 e^{7} \left (d + e x\right )^{2}} - \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{3}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{3}}{4 e^{7} \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**3*Integral(x, x)/e**5 + 3*c*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*
log(d + e*x)/e**7 + (3*b*e - 5*c*d)*Integral(c**2, x)/e**6 - (b*e - 2*c*d)*(6*a*
c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)/(e**7*(d + e*x)) - 3*(a*e**2 - b
*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(2*e**7*(d + e*x
)**2) - (b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(e**7*(d + e*x)**3) - (a*e**2
- b*d*e + c*d**2)**3/(4*e**7*(d + e*x)**4)

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Mathematica [A]  time = 0.355119, size = 402, normalized size = 1.6 \[ \frac{-c e^2 \left (a^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+6 a b e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+b^2 (-d) \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-e^3 \left (a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+12 c (d+e x)^4 \log (d+e x) \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )+c^2 e \left (a d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-b \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )+c^3 \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )}{4 e^7 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

(c^3*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 -
12*d*e^5*x^5 + 2*e^6*x^6) - e^3*(a^3*e^3 + a^2*b*e^2*(d + 4*e*x) + a*b^2*e*(d^2
+ 4*d*e*x + 6*e^2*x^2) + b^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) - c*e
^2*(a^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + 6*a*b*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2
+ 4*e^3*x^3) - b^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + c^2*
e*(a*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) - b*(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)) + 12*c*(5
*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(d + e*x)^4*Log[d + e*x])/(4*e^7*(d + e
*x)^4)

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Maple [B]  time = 0.016, size = 678, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/e^5/(e*x+d)^2*a*c^2*d^2-9/e^5/(e*x+d)^2*b^2*c*d^2+3/2/e^4/(e*x+d)^4*d^3*a*b*c
-5*c^3*d*x/e^6+9/e^4/(e*x+d)^2*a*b*c*d-6/e^4/(e*x+d)^3*a*b*c*d^2-1/e^4/(e*x+d)^3
*b^3*d^2+2/e^7/(e*x+d)^3*c^3*d^5+3*c^2/e^5*ln(e*x+d)*a+3*c/e^5*ln(e*x+d)*b^2+15*
c^3/e^7*ln(e*x+d)*d^2+20/e^7/(e*x+d)*c^3*d^3+1/4/e^4/(e*x+d)^4*d^3*b^3-1/4/e^7/(
e*x+d)^4*c^3*d^6+3*c^2/e^5*b*x-3/2/e^3/(e*x+d)^2*a*b^2+3/2/e^4/(e*x+d)^2*b^3*d-1
5/2/e^7/(e*x+d)^2*c^3*d^4-1/e^2/(e*x+d)^3*a^2*b+1/2*c^3*x^2/e^5-1/e^4/(e*x+d)*b^
3-1/4/e/(e*x+d)^4*a^3+15/e^6/(e*x+d)^2*d^3*b*c^2+2/e^3/(e*x+d)^3*a^2*c*d+2/e^3/(
e*x+d)^3*a*b^2*d+4/e^5/(e*x+d)^3*a*c^2*d^3+4/e^5/(e*x+d)^3*b^2*c*d^3-5/e^6/(e*x+
d)^3*b*c^2*d^4-15*c^2/e^6*ln(e*x+d)*b*d-6/e^4/(e*x+d)*a*b*c+12/e^5/(e*x+d)*a*c^2
*d+12/e^5/(e*x+d)*b^2*c*d-30/e^6/(e*x+d)*b*c^2*d^2+3/4/e^2/(e*x+d)^4*d*a^2*b-3/4
/e^3/(e*x+d)^4*a^2*c*d^2-3/4/e^3/(e*x+d)^4*d^2*a*b^2-3/4/e^5/(e*x+d)^4*c^2*d^4*a
-3/4/e^5/(e*x+d)^4*d^4*b^2*c+3/4/e^6/(e*x+d)^4*b*c^2*d^5-3/2/e^3/(e*x+d)^2*a^2*c

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Maxima [A]  time = 0.830928, size = 595, normalized size = 2.37 \[ \frac{57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e - a^{2} b d e^{5} - a^{3} e^{6} + 25 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} -{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 4 \,{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 6 \,{\left (35 \, c^{3} d^{4} e^{2} - 50 \, b c^{2} d^{3} e^{3} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} -{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 4 \,{\left (47 \, c^{3} d^{5} e - 65 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 22 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} -{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac{c^{3} e x^{2} - 2 \,{\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} x}{2 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(57*c^3*d^6 - 77*b*c^2*d^5*e - a^2*b*d*e^5 - a^3*e^6 + 25*(b^2*c + a*c^2)*d^
4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - (a*b^2 + a^2*c)*d^2*e^4 + 4*(20*c^3*d^3*e^3 -
30*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 - (b^3 + 6*a*b*c)*e^6)*x^3 + 6*(35*c
^3*d^4*e^2 - 50*b*c^2*d^3*e^3 + 18*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e
^5 - (a*b^2 + a^2*c)*e^6)*x^2 + 4*(47*c^3*d^5*e - 65*b*c^2*d^4*e^2 - a^2*b*e^6 +
22*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*e^4 - (a*b^2 + a^2*c)*d*e^5)*x
)/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7) + 1/2*(c^3*e
*x^2 - 2*(5*c^3*d - 3*b*c^2*e)*x)/e^6 + 3*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*
c^2)*e^2)*log(e*x + d)/e^7

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Fricas [A]  time = 0.220076, size = 873, normalized size = 3.48 \[ \frac{2 \, c^{3} e^{6} x^{6} + 57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e - a^{2} b d e^{5} - a^{3} e^{6} + 25 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} -{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 12 \,{\left (c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} - 4 \,{\left (17 \, c^{3} d^{2} e^{4} - 12 \, b c^{2} d e^{5}\right )} x^{4} - 4 \,{\left (8 \, c^{3} d^{3} e^{3} + 12 \, b c^{2} d^{2} e^{4} - 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 6 \,{\left (22 \, c^{3} d^{4} e^{2} - 42 \, b c^{2} d^{3} e^{3} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} -{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 4 \,{\left (42 \, c^{3} d^{5} e - 62 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 22 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} -{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e +{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} +{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} +{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 4 \,{\left (5 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} +{\left (b^{2} c + a c^{2}\right )} d e^{5}\right )} x^{3} + 6 \,{\left (5 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} +{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (5 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} +{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(e*x + d)^5,x, algorithm="fricas")

[Out]

1/4*(2*c^3*e^6*x^6 + 57*c^3*d^6 - 77*b*c^2*d^5*e - a^2*b*d*e^5 - a^3*e^6 + 25*(b
^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - (a*b^2 + a^2*c)*d^2*e^4 - 12*(
c^3*d*e^5 - b*c^2*e^6)*x^5 - 4*(17*c^3*d^2*e^4 - 12*b*c^2*d*e^5)*x^4 - 4*(8*c^3*
d^3*e^3 + 12*b*c^2*d^2*e^4 - 12*(b^2*c + a*c^2)*d*e^5 + (b^3 + 6*a*b*c)*e^6)*x^3
+ 6*(22*c^3*d^4*e^2 - 42*b*c^2*d^3*e^3 + 18*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*
a*b*c)*d*e^5 - (a*b^2 + a^2*c)*e^6)*x^2 + 4*(42*c^3*d^5*e - 62*b*c^2*d^4*e^2 - a
^2*b*e^6 + 22*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*e^4 - (a*b^2 + a^2*c
)*d*e^5)*x + 12*(5*c^3*d^6 - 5*b*c^2*d^5*e + (b^2*c + a*c^2)*d^4*e^2 + (5*c^3*d^
2*e^4 - 5*b*c^2*d*e^5 + (b^2*c + a*c^2)*e^6)*x^4 + 4*(5*c^3*d^3*e^3 - 5*b*c^2*d^
2*e^4 + (b^2*c + a*c^2)*d*e^5)*x^3 + 6*(5*c^3*d^4*e^2 - 5*b*c^2*d^3*e^3 + (b^2*c
+ a*c^2)*d^2*e^4)*x^2 + 4*(5*c^3*d^5*e - 5*b*c^2*d^4*e^2 + (b^2*c + a*c^2)*d^3*
e^3)*x)*log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d
^4*e^7)

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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((c*x**2+b*x+a)**3/(e*x+d)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.208162, 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)^3/(e*x + d)^5,x, algorithm="giac")

[Out]

Done