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

Optimal. Leaf size=120 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{e^5 (d+e x)}-\frac{d^2 (c d-b e)^2}{3 e^5 (d+e x)^3}+\frac{d (c d-b e) (2 c d-b e)}{e^5 (d+e x)^2}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]

[Out]

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

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Rubi [A]  time = 0.255894, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{e^5 (d+e x)}-\frac{d^2 (c d-b e)^2}{3 e^5 (d+e x)^3}+\frac{d (c d-b e) (2 c d-b e)}{e^5 (d+e x)^2}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**2/(e*x+d)**4,x)

[Out]

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

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Mathematica [A]  time = 0.097775, size = 134, normalized size = 1.12 \[ \frac{-b^2 e^2 \left (d^2+3 d e x+3 e^2 x^2\right )+b c d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-6 c (d+e x)^3 (2 c d-b e) \log (d+e x)+c^2 \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )}{3 e^5 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-(b^2*e^2*(d^2 + 3*d*e*x + 3*e^2*x^2)) + b*c*d*e*(11*d^2 + 27*d*e*x + 18*e^2*x^
2) + c^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) - 6*c*
(2*c*d - b*e)*(d + e*x)^3*Log[d + e*x])/(3*e^5*(d + e*x)^3)

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Maple [A]  time = 0.012, size = 189, normalized size = 1.6 \[{\frac{{c}^{2}x}{{e}^{4}}}+{\frac{{b}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{d}^{2}bc}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+2\,{\frac{c\ln \left ( ex+d \right ) b}{{e}^{4}}}-4\,{\frac{d{c}^{2}\ln \left ( ex+d \right ) }{{e}^{5}}}-{\frac{{b}^{2}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{d}^{3}bc}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+6\,{\frac{bcd}{{e}^{4} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^2/(e*x+d)^4,x)

[Out]

c^2*x/e^4+d/e^3/(e*x+d)^2*b^2-3*d^2/e^4/(e*x+d)^2*b*c+2*d^3/e^5/(e*x+d)^2*c^2+2*
c/e^4*ln(e*x+d)*b-4*c^2*d*ln(e*x+d)/e^5-1/3*d^2/e^3/(e*x+d)^3*b^2+2/3*d^3/e^4/(e
*x+d)^3*b*c-1/3*d^4/e^5/(e*x+d)^3*c^2-1/e^3/(e*x+d)*b^2+6/e^4/(e*x+d)*b*c*d-6/e^
5/(e*x+d)*c^2*d^2

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Maxima [A]  time = 0.709749, size = 215, normalized size = 1.79 \[ -\frac{13 \, c^{2} d^{4} - 11 \, b c d^{3} e + b^{2} d^{2} e^{2} + 3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 3 \,{\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac{c^{2} x}{e^{4}} - \frac{2 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (e x + d\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(13*c^2*d^4 - 11*b*c*d^3*e + b^2*d^2*e^2 + 3*(6*c^2*d^2*e^2 - 6*b*c*d*e^3 +
b^2*e^4)*x^2 + 3*(10*c^2*d^3*e - 9*b*c*d^2*e^2 + b^2*d*e^3)*x)/(e^8*x^3 + 3*d*e
^7*x^2 + 3*d^2*e^6*x + d^3*e^5) + c^2*x/e^4 - 2*(2*c^2*d - b*c*e)*log(e*x + d)/e
^5

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Fricas [A]  time = 0.222997, size = 331, normalized size = 2.76 \[ \frac{3 \, c^{2} e^{4} x^{4} + 9 \, c^{2} d e^{3} x^{3} - 13 \, c^{2} d^{4} + 11 \, b c d^{3} e - b^{2} d^{2} e^{2} - 3 \,{\left (3 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} - 3 \,{\left (9 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{4} - b c d^{3} e +{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 3 \,{\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} x^{2} + 3 \,{\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*c^2*e^4*x^4 + 9*c^2*d*e^3*x^3 - 13*c^2*d^4 + 11*b*c*d^3*e - b^2*d^2*e^2 -
3*(3*c^2*d^2*e^2 - 6*b*c*d*e^3 + b^2*e^4)*x^2 - 3*(9*c^2*d^3*e - 9*b*c*d^2*e^2
+ b^2*d*e^3)*x - 6*(2*c^2*d^4 - b*c*d^3*e + (2*c^2*d*e^3 - b*c*e^4)*x^3 + 3*(2*c
^2*d^2*e^2 - b*c*d*e^3)*x^2 + 3*(2*c^2*d^3*e - b*c*d^2*e^2)*x)*log(e*x + d))/(e^
8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

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Sympy [A]  time = 6.13564, size = 163, normalized size = 1.36 \[ \frac{c^{2} x}{e^{4}} + \frac{2 c \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{b^{2} d^{2} e^{2} - 11 b c d^{3} e + 13 c^{2} d^{4} + x^{2} \left (3 b^{2} e^{4} - 18 b c d e^{3} + 18 c^{2} d^{2} e^{2}\right ) + x \left (3 b^{2} d e^{3} - 27 b c d^{2} e^{2} + 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**2/(e*x+d)**4,x)

[Out]

c**2*x/e**4 + 2*c*(b*e - 2*c*d)*log(d + e*x)/e**5 - (b**2*d**2*e**2 - 11*b*c*d**
3*e + 13*c**2*d**4 + x**2*(3*b**2*e**4 - 18*b*c*d*e**3 + 18*c**2*d**2*e**2) + x*
(3*b**2*d*e**3 - 27*b*c*d**2*e**2 + 30*c**2*d**3*e))/(3*d**3*e**5 + 9*d**2*e**6*
x + 9*d*e**7*x**2 + 3*e**8*x**3)

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GIAC/XCAS [A]  time = 0.207942, size = 177, normalized size = 1.48 \[ c^{2} x e^{\left (-4\right )} - 2 \,{\left (2 \, c^{2} d - b c e\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (13 \, c^{2} d^{4} - 11 \, b c d^{3} e + b^{2} d^{2} e^{2} + 3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 3 \,{\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^2*x*e^(-4) - 2*(2*c^2*d - b*c*e)*e^(-5)*ln(abs(x*e + d)) - 1/3*(13*c^2*d^4 - 1
1*b*c*d^3*e + b^2*d^2*e^2 + 3*(6*c^2*d^2*e^2 - 6*b*c*d*e^3 + b^2*e^4)*x^2 + 3*(1
0*c^2*d^3*e - 9*b*c*d^2*e^2 + b^2*d*e^3)*x)*e^(-5)/(x*e + d)^3