3.1901 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx\)

Optimal. Leaf size=74 \[ -\frac{(b d-a e)^3 \log (d+e x)}{e^4}+\frac{b x (b d-a e)^2}{e^3}-\frac{(a+b x)^2 (b d-a e)}{2 e^2}+\frac{(a+b x)^3}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.0786336, antiderivative size = 74, 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 \[ -\frac{(b d-a e)^3 \log (d+e x)}{e^4}+\frac{b x (b d-a e)^2}{e^3}-\frac{(a+b x)^2 (b d-a e)}{2 e^2}+\frac{(a+b x)^3}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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),x)

[Out]

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

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Mathematica [A]  time = 0.0532836, size = 74, normalized size = 1. \[ \frac{b e x \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (b d-a e)^3 \log (d+e x)}{6 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 133, normalized size = 1.8 \[{\frac{{b}^{3}{x}^{3}}{3\,e}}+{\frac{3\,{b}^{2}{x}^{2}a}{2\,e}}-{\frac{{b}^{3}{x}^{2}d}{2\,{e}^{2}}}+3\,{\frac{{a}^{2}bx}{e}}-3\,{\frac{a{b}^{2}dx}{{e}^{2}}}+{\frac{{b}^{3}{d}^{2}x}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ){a}^{3}}{e}}-3\,{\frac{\ln \left ( ex+d \right ){a}^{2}bd}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ){b}^{3}{d}^{3}}{{e}^{4}}} \]

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),x)

[Out]

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

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Maxima [A]  time = 0.709324, size = 154, normalized size = 2.08 \[ \frac{2 \, b^{3} e^{2} x^{3} - 3 \,{\left (b^{3} d e - 3 \, a b^{2} e^{2}\right )} x^{2} + 6 \,{\left (b^{3} d^{2} - 3 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x}{6 \, e^{3}} - \frac{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{4}} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.281198, size = 155, normalized size = 2.09 \[ \frac{2 \, b^{3} e^{3} x^{3} - 3 \,{\left (b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} + 6 \,{\left (b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.82274, size = 82, normalized size = 1.11 \[ \frac{b^{3} x^{3}}{3 e} + \frac{x^{2} \left (3 a b^{2} e - b^{3} d\right )}{2 e^{2}} + \frac{x \left (3 a^{2} b e^{2} - 3 a b^{2} d e + b^{3} d^{2}\right )}{e^{3}} + \frac{\left (a e - b d\right )^{3} \log{\left (d + e x \right )}}{e^{4}} \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.28099, size = 153, normalized size = 2.07 \[ -{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, b^{3} x^{3} e^{2} - 3 \, b^{3} d x^{2} e + 6 \, b^{3} d^{2} x + 9 \, a b^{2} x^{2} e^{2} - 18 \, a b^{2} d x e + 18 \, a^{2} b x e^{2}\right )} e^{\left (-3\right )} \]

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),x, algorithm="giac")

[Out]

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