3.2310 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )}{d+e x} \, dx\)

Optimal. Leaf size=111 \[ -\frac{x \left (A e (c d-b e)-B \left (c d^2-e (b d-a e)\right )\right )}{e^3}-\frac{(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{x^2 (-A c e-b B e+B c d)}{2 e^2}+\frac{B c x^3}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.29301, antiderivative size = 109, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{x \left (-B e (b d-a e)-A e (c d-b e)+B c d^2\right )}{e^3}-\frac{(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{x^2 (-A c e-b B e+B c d)}{2 e^2}+\frac{B c x^3}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c x^{3}}{3 e} + \left (A b e^{2} - A c d e + B a e^{2} - B b d e + B c d^{2}\right ) \int \frac{1}{e^{3}}\, dx + \frac{\left (A c e + B b e - B c d\right ) \int x\, dx}{e^{2}} + \frac{\left (A e - B d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.10312, size = 100, normalized size = 0.9 \[ \frac{e x \left (3 B e (2 a e-2 b d+b e x)+3 A e (2 b e-2 c d+c e x)+B c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (B d-A e) \log (d+e x) \left (e (a e-b d)+c d^2\right )}{6 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(e*x*(3*B*e*(-2*b*d + 2*a*e + b*e*x) + 3*A*e*(-2*c*d + 2*b*e + c*e*x) + B*c*(6*d
^2 - 3*d*e*x + 2*e^2*x^2)) - 6*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))*Log[d + e*
x])/(6*e^4)

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Maple [A]  time = 0.005, size = 171, normalized size = 1.5 \[{\frac{Bc{x}^{3}}{3\,e}}+{\frac{Ac{x}^{2}}{2\,e}}+{\frac{B{x}^{2}b}{2\,e}}-{\frac{Bc{x}^{2}d}{2\,{e}^{2}}}+{\frac{Abx}{e}}-{\frac{Acdx}{{e}^{2}}}+{\frac{aBx}{e}}-{\frac{bBdx}{{e}^{2}}}+{\frac{Bc{d}^{2}x}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) aA}{e}}-{\frac{\ln \left ( ex+d \right ) Abd}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) Ac{d}^{2}}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) aBd}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) Bb{d}^{2}}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) Bc{d}^{3}}{{e}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.695924, size = 165, normalized size = 1.49 \[ \frac{2 \, B c e^{2} x^{3} - 3 \,{\left (B c d e -{\left (B b + A c\right )} e^{2}\right )} x^{2} + 6 \,{\left (B c d^{2} -{\left (B b + A c\right )} d e +{\left (B a + A b\right )} e^{2}\right )} x}{6 \, e^{3}} - \frac{{\left (B c d^{3} - A a e^{3} -{\left (B b + A c\right )} d^{2} e +{\left (B a + A b\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.260872, size = 166, normalized size = 1.5 \[ \frac{2 \, B c e^{3} x^{3} - 3 \,{\left (B c d e^{2} -{\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e -{\left (B b + A c\right )} d e^{2} +{\left (B a + A b\right )} e^{3}\right )} x - 6 \,{\left (B c d^{3} - A a e^{3} -{\left (B b + A c\right )} d^{2} e +{\left (B a + A b\right )} d e^{2}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.52806, size = 104, normalized size = 0.94 \[ \frac{B c x^{3}}{3 e} + \frac{x^{2} \left (A c e + B b e - B c d\right )}{2 e^{2}} + \frac{x \left (A b e^{2} - A c d e + B a e^{2} - B b d e + B c d^{2}\right )}{e^{3}} - \frac{\left (- A e + B d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*c*x**3/(3*e) + x**2*(A*c*e + B*b*e - B*c*d)/(2*e**2) + x*(A*b*e**2 - A*c*d*e +
B*a*e**2 - B*b*d*e + B*c*d**2)/e**3 - (-A*e + B*d)*(a*e**2 - b*d*e + c*d**2)*lo
g(d + e*x)/e**4

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GIAC/XCAS [A]  time = 0.281029, size = 184, normalized size = 1.66 \[ -{\left (B c d^{3} - B b d^{2} e - A c d^{2} e + B a d e^{2} + A b d e^{2} - A a e^{3}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B c x^{3} e^{2} - 3 \, B c d x^{2} e + 6 \, B c d^{2} x + 3 \, B b x^{2} e^{2} + 3 \, A c x^{2} e^{2} - 6 \, B b d x e - 6 \, A c d x e + 6 \, B a x e^{2} + 6 \, A b x e^{2}\right )} e^{\left (-3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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