3.1904 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^4} \, dx\)

Optimal. Leaf size=86 \[ \frac{3 b^2 (b d-a e)}{e^4 (d+e x)}-\frac{3 b (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac{(b d-a e)^3}{3 e^4 (d+e x)^3}+\frac{b^3 \log (d+e x)}{e^4} \]

[Out]

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

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Rubi [A]  time = 0.124046, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{3 b^2 (b d-a e)}{e^4 (d+e x)}-\frac{3 b (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac{(b d-a e)^3}{3 e^4 (d+e x)^3}+\frac{b^3 \log (d+e x)}{e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.6482, size = 76, normalized size = 0.88 \[ \frac{b^{3} \log{\left (d + e x \right )}}{e^{4}} - \frac{3 b^{2} \left (a e - b d\right )}{e^{4} \left (d + e x\right )} - \frac{3 b \left (a e - b d\right )^{2}}{2 e^{4} \left (d + e x\right )^{2}} - \frac{\left (a e - b d\right )^{3}}{3 e^{4} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**3*log(d + e*x)/e**4 - 3*b**2*(a*e - b*d)/(e**4*(d + e*x)) - 3*b*(a*e - b*d)**
2/(2*e**4*(d + e*x)**2) - (a*e - b*d)**3/(3*e**4*(d + e*x)**3)

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Mathematica [A]  time = 0.0769684, size = 79, normalized size = 0.92 \[ \frac{\frac{(b d-a e) \left (2 a^2 e^2+a b e (5 d+9 e x)+b^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )}{(d+e x)^3}+6 b^3 \log (d+e x)}{6 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 166, normalized size = 1.9 \[ -{\frac{{a}^{3}}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{{a}^{2}bd}{{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{a{d}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{3}{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{3}\ln \left ( ex+d \right ) }{{e}^{4}}}-{\frac{3\,{a}^{2}b}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{a{b}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{b}^{3}{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-3\,{\frac{a{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{d{b}^{3}}{{e}^{4} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.726018, size = 193, normalized size = 2.24 \[ \frac{11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac{b^{3} \log \left (e x + d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(11*b^3*d^3 - 6*a*b^2*d^2*e - 3*a^2*b*d*e^2 - 2*a^3*e^3 + 18*(b^3*d*e^2 - a*
b^2*e^3)*x^2 + 9*(3*b^3*d^2*e - 2*a*b^2*d*e^2 - a^2*b*e^3)*x)/(e^7*x^3 + 3*d*e^6
*x^2 + 3*d^2*e^5*x + d^3*e^4) + b^3*log(e*x + d)/e^4

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Fricas [A]  time = 0.277366, size = 239, normalized size = 2.78 \[ \frac{11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(11*b^3*d^3 - 6*a*b^2*d^2*e - 3*a^2*b*d*e^2 - 2*a^3*e^3 + 18*(b^3*d*e^2 - a*
b^2*e^3)*x^2 + 9*(3*b^3*d^2*e - 2*a*b^2*d*e^2 - a^2*b*e^3)*x + 6*(b^3*e^3*x^3 +
3*b^3*d*e^2*x^2 + 3*b^3*d^2*e*x + b^3*d^3)*log(e*x + d))/(e^7*x^3 + 3*d*e^6*x^2
+ 3*d^2*e^5*x + d^3*e^4)

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Sympy [A]  time = 4.9962, size = 148, normalized size = 1.72 \[ \frac{b^{3} \log{\left (d + e x \right )}}{e^{4}} - \frac{2 a^{3} e^{3} + 3 a^{2} b d e^{2} + 6 a b^{2} d^{2} e - 11 b^{3} d^{3} + x^{2} \left (18 a b^{2} e^{3} - 18 b^{3} d e^{2}\right ) + x \left (9 a^{2} b e^{3} + 18 a b^{2} d e^{2} - 27 b^{3} d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.280349, size = 158, normalized size = 1.84 \[ b^{3} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (18 \,{\left (b^{3} d e - a b^{2} e^{2}\right )} x^{2} + 9 \,{\left (3 \, b^{3} d^{2} - 2 \, a b^{2} d e - a^{2} b e^{2}\right )} x +{\left (11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^3*e^(-4)*ln(abs(x*e + d)) + 1/6*(18*(b^3*d*e - a*b^2*e^2)*x^2 + 9*(3*b^3*d^2 -
2*a*b^2*d*e - a^2*b*e^2)*x + (11*b^3*d^3 - 6*a*b^2*d^2*e - 3*a^2*b*d*e^2 - 2*a^
3*e^3)*e^(-1))*e^(-3)/(x*e + d)^3