3.2312 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^3} \, dx\)

Optimal. Leaf size=119 \[ \frac{A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{e^4 (d+e x)}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{\log (d+e x) (-A c e-b B e+3 B c d)}{e^4}+\frac{B c x}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.27428, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{e^4 (d+e x)}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{\log (d+e x) (-A c e-b B e+3 B c d)}{e^4}+\frac{B c x}{e^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2))/(d + e*x)^3,x]

[Out]

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

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

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.141475, size = 135, normalized size = 1.13 \[ \frac{-a B e^2-A b e^2+2 A c d e+2 b B d e-3 B c d^2}{e^4 (d+e x)}+\frac{-a A e^3+a B d e^2+A b d e^2-A c d^2 e-b B d^2 e+B c d^3}{2 e^4 (d+e x)^2}+\frac{\log (d+e x) (A c e+b B e-3 B c d)}{e^4}+\frac{B c x}{e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2))/(d + e*x)^3,x]

[Out]

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

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Maple [A]  time = 0.011, size = 217, normalized size = 1.8 \[{\frac{Bcx}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) Ac}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) bB}{{e}^{3}}}-3\,{\frac{\ln \left ( ex+d \right ) Bcd}{{e}^{4}}}-{\frac{Ab}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{Acd}{{e}^{3} \left ( ex+d \right ) }}-{\frac{aB}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{bBd}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{Bc{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{aA}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{Abd}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{Ac{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{aBd}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{B{d}^{2}b}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{Bc{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)/(e*x+d)^3,x)

[Out]

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

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Maxima [A]  time = 0.710295, size = 184, normalized size = 1.55 \[ -\frac{5 \, B c d^{3} + A a e^{3} - 3 \,{\left (B b + A c\right )} d^{2} e +{\left (B a + A b\right )} d e^{2} + 2 \,{\left (3 \, B c d^{2} e - 2 \,{\left (B b + A c\right )} d e^{2} +{\left (B a + A b\right )} e^{3}\right )} x}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac{B c x}{e^{3}} - \frac{{\left (3 \, B c d -{\left (B b + A c\right )} e\right )} \log \left (e x + d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/(e*x + d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.258117, size = 274, normalized size = 2.3 \[ \frac{2 \, B c e^{3} x^{3} + 4 \, B c d e^{2} x^{2} - 5 \, B c d^{3} - A a e^{3} + 3 \,{\left (B b + A c\right )} d^{2} e -{\left (B a + A b\right )} d e^{2} - 2 \,{\left (2 \, B c d^{2} e - 2 \,{\left (B b + A c\right )} d e^{2} +{\left (B a + A b\right )} e^{3}\right )} x - 2 \,{\left (3 \, B c d^{3} -{\left (B b + A c\right )} d^{2} e +{\left (3 \, B c d e^{2} -{\left (B b + A c\right )} e^{3}\right )} x^{2} + 2 \,{\left (3 \, B c d^{2} e -{\left (B b + A c\right )} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/(e*x + d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 18.9413, size = 162, normalized size = 1.36 \[ \frac{B c x}{e^{3}} - \frac{A a e^{3} + A b d e^{2} - 3 A c d^{2} e + B a d e^{2} - 3 B b d^{2} e + 5 B c d^{3} + x \left (2 A b e^{3} - 4 A c d e^{2} + 2 B a e^{3} - 4 B b d e^{2} + 6 B c d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} + \frac{\left (A c e + B b e - 3 B c d\right ) \log{\left (d + e x \right )}}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)/(e*x+d)**3,x)

[Out]

B*c*x/e**3 - (A*a*e**3 + A*b*d*e**2 - 3*A*c*d**2*e + B*a*d*e**2 - 3*B*b*d**2*e +
5*B*c*d**3 + x*(2*A*b*e**3 - 4*A*c*d*e**2 + 2*B*a*e**3 - 4*B*b*d*e**2 + 6*B*c*d
**2*e))/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + (A*c*e + B*b*e - 3*B*c*d)*log
(d + e*x)/e**4

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GIAC/XCAS [A]  time = 0.270404, size = 174, normalized size = 1.46 \[ B c x e^{\left (-3\right )} -{\left (3 \, B c d - B b e - A c e\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (5 \, B c d^{3} - 3 \, B b d^{2} e - 3 \, A c d^{2} e + B a d e^{2} + A b d e^{2} + A a e^{3} + 2 \,{\left (3 \, B c d^{2} e - 2 \, B b d e^{2} - 2 \, A c d e^{2} + B a e^{3} + A b e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/(e*x + d)^3,x, algorithm="giac")

[Out]

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