3.1929 \(\int \frac{(a+b x) (d+e x)^2}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{(b d-a e)^2 \log (a+b x)}{b^3}+\frac{e x (b d-a e)}{b^2}+\frac{(d+e x)^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0519659, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{(b d-a e)^2 \log (a+b x)}{b^3}+\frac{e x (b d-a e)}{b^2}+\frac{(d+e x)^2}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0293552, size = 43, normalized size = 0.88 \[ \frac{b e x (-2 a e+4 b d+b e x)+2 (b d-a e)^2 \log (a+b x)}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 74, normalized size = 1.5 \[{\frac{{e}^{2}{x}^{2}}{2\,b}}-{\frac{{e}^{2}xa}{{b}^{2}}}+2\,{\frac{dex}{b}}+{\frac{\ln \left ( bx+a \right ){a}^{2}{e}^{2}}{{b}^{3}}}-2\,{\frac{\ln \left ( bx+a \right ) ade}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{2}}{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.70926, size = 82, normalized size = 1.67 \[ \frac{b e^{2} x^{2} + 2 \,{\left (2 \, b d e - a e^{2}\right )} x}{2 \, b^{2}} + \frac{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (b x + a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.269893, size = 85, normalized size = 1.73 \[ \frac{b^{2} e^{2} x^{2} + 2 \,{\left (2 \, b^{2} d e - a b e^{2}\right )} x + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.55446, size = 44, normalized size = 0.9 \[ \frac{e^{2} x^{2}}{2 b} - \frac{x \left (a e^{2} - 2 b d e\right )}{b^{2}} + \frac{\left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.27723, size = 80, normalized size = 1.63 \[ \frac{b x^{2} e^{2} + 4 \, b d x e - 2 \, a x e^{2}}{2 \, b^{2}} + \frac{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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