3.1902 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^2} \, dx\)

Optimal. Leaf size=75 \[ -\frac{b^2 x (2 b d-3 a e)}{e^3}+\frac{(b d-a e)^3}{e^4 (d+e x)}+\frac{3 b (b d-a e)^2 \log (d+e x)}{e^4}+\frac{b^3 x^2}{2 e^2} \]

[Out]

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

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Rubi [A]  time = 0.126457, antiderivative size = 75, 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 \[ -\frac{b^2 x (2 b d-3 a e)}{e^3}+\frac{(b d-a e)^3}{e^4 (d+e x)}+\frac{3 b (b d-a e)^2 \log (d+e x)}{e^4}+\frac{b^3 x^2}{2 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0670025, size = 114, normalized size = 1.52 \[ \frac{3 \left (a^2 b e^2-2 a b^2 d e+b^3 d^2\right ) \log (d+e x)}{e^4}+\frac{-a^3 e^3+3 a^2 b d e^2-3 a b^2 d^2 e+b^3 d^3}{e^4 (d+e x)}-\frac{b^2 x (2 b d-3 a e)}{e^3}+\frac{b^3 x^2}{2 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 149, normalized size = 2. \[{\frac{{x}^{2}{b}^{3}}{2\,{e}^{2}}}+3\,{\frac{{b}^{2}xa}{{e}^{2}}}-2\,{\frac{{b}^{3}xd}{{e}^{3}}}+3\,{\frac{b\ln \left ( ex+d \right ){a}^{2}}{{e}^{2}}}-6\,{\frac{{b}^{2}\ln \left ( ex+d \right ) ad}{{e}^{3}}}+3\,{\frac{{b}^{3}\ln \left ( ex+d \right ){d}^{2}}{{e}^{4}}}-{\frac{{a}^{3}}{e \left ( ex+d \right ) }}+3\,{\frac{{a}^{2}bd}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{a{d}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{{b}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }} \]

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)^2,x)

[Out]

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

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Maxima [A]  time = 0.715406, size = 158, normalized size = 2.11 \[ \frac{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}}{e^{5} x + d e^{4}} + \frac{b^{3} e x^{2} - 2 \,{\left (2 \, b^{3} d - 3 \, a b^{2} e\right )} x}{2 \, e^{3}} + \frac{3 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \]

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)^2,x, algorithm="maxima")

[Out]

(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)/(e^5*x + d*e^4) + 1/2*(b^3*e
*x^2 - 2*(2*b^3*d - 3*a*b^2*e)*x)/e^3 + 3*(b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*lo
g(e*x + d)/e^4

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Fricas [A]  time = 0.289618, size = 232, normalized size = 3.09 \[ \frac{b^{3} e^{3} x^{3} + 2 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} - 3 \,{\left (b^{3} d e^{2} - 2 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2}\right )} x + 6 \,{\left (b^{3} d^{3} - 2 \, a b^{2} d^{2} e + a^{2} b d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x + d e^{4}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.85244, size = 100, normalized size = 1.33 \[ \frac{b^{3} x^{2}}{2 e^{2}} + \frac{3 b \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{4}} - \frac{a^{3} e^{3} - 3 a^{2} b d e^{2} + 3 a b^{2} d^{2} e - b^{3} d^{3}}{d e^{4} + e^{5} x} + \frac{x \left (3 a b^{2} e - 2 b^{3} d\right )}{e^{3}} \]

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)**2,x)

[Out]

b**3*x**2/(2*e**2) + 3*b*(a*e - b*d)**2*log(d + e*x)/e**4 - (a**3*e**3 - 3*a**2*
b*d*e**2 + 3*a*b**2*d**2*e - b**3*d**3)/(d*e**4 + e**5*x) + x*(3*a*b**2*e - 2*b*
*3*d)/e**3

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GIAC/XCAS [A]  time = 0.287066, size = 221, normalized size = 2.95 \[ \frac{1}{2} \,{\left (b^{3} - \frac{6 \,{\left (b^{3} d e - a b^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-4\right )} - 3 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} e^{\left (-4\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{b^{3} d^{3} e^{2}}{x e + d} - \frac{3 \, a b^{2} d^{2} e^{3}}{x e + d} + \frac{3 \, a^{2} b d e^{4}}{x e + d} - \frac{a^{3} e^{5}}{x e + d}\right )} e^{\left (-6\right )} \]

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)^2,x, algorithm="giac")

[Out]

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