3.1445 \(\int \frac{a^2+2 a b x+b^2 x^2}{d+e x} \, dx\)

Optimal. Leaf size=50 \[ \frac{(b d-a e)^2 \log (d+e x)}{e^3}-\frac{b x (b d-a e)}{e^2}+\frac{(a+b x)^2}{2 e} \]

[Out]

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

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Rubi [A]  time = 0.0674319, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(b d-a e)^2 \log (d+e x)}{e^3}-\frac{b x (b d-a e)}{e^2}+\frac{(a+b x)^2}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.6695, size = 41, normalized size = 0.82 \[ \frac{b x \left (a e - b d\right )}{e^{2}} + \frac{\left (a + b x\right )^{2}}{2 e} + \frac{\left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0324213, size = 43, normalized size = 0.86 \[ \frac{b e x (4 a e-2 b d+b e x)+2 (b d-a e)^2 \log (d+e x)}{2 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 74, normalized size = 1.5 \[{\frac{{b}^{2}{x}^{2}}{2\,e}}+2\,{\frac{abx}{e}}-{\frac{{b}^{2}xd}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){a}^{2}}{e}}-2\,{\frac{\ln \left ( ex+d \right ) dab}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){b}^{2}{d}^{2}}{{e}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.685643, size = 81, normalized size = 1.62 \[ \frac{b^{2} e x^{2} - 2 \,{\left (b^{2} d - 2 \, a b e\right )} x}{2 \, e^{2}} + \frac{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.199191, size = 84, normalized size = 1.68 \[ \frac{b^{2} e^{2} x^{2} - 2 \,{\left (b^{2} d e - 2 \, a b e^{2}\right )} x + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.53329, size = 44, normalized size = 0.88 \[ \frac{b^{2} x^{2}}{2 e} + \frac{x \left (2 a b e - b^{2} d\right )}{e^{2}} + \frac{\left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211165, size = 82, normalized size = 1.64 \[{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{2} x^{2} e - 2 \, b^{2} d x + 4 \, a b x e\right )} e^{\left (-2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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