3.1025 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=150 \[ -\frac{b^2 (d+e x)^2 (-3 a B e-A b e+4 b B d)}{2 e^5}-\frac{(b d-a e)^3 (B d-A e)}{e^5 (d+e x)}-\frac{(b d-a e)^2 \log (d+e x) (-a B e-3 A b e+4 b B d)}{e^5}+\frac{3 b x (b d-a e) (-a B e-A b e+2 b B d)}{e^4}+\frac{b^3 B (d+e x)^3}{3 e^5} \]

[Out]

(3*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*x)/e^4 - ((b*d - a*e)^3*(B*d - A*e))/
(e^5*(d + e*x)) - (b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^2)/(2*e^5) + (b^3*B
*(d + e*x)^3)/(3*e^5) - ((b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*Log[d + e*x])
/e^5

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Rubi [A]  time = 0.414918, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{b^2 (d+e x)^2 (-3 a B e-A b e+4 b B d)}{2 e^5}-\frac{(b d-a e)^3 (B d-A e)}{e^5 (d+e x)}-\frac{(b d-a e)^2 \log (d+e x) (-a B e-3 A b e+4 b B d)}{e^5}+\frac{3 b x (b d-a e) (-a B e-A b e+2 b B d)}{e^4}+\frac{b^3 B (d+e x)^3}{3 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(3*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*x)/e^4 - ((b*d - a*e)^3*(B*d - A*e))/
(e^5*(d + e*x)) - (b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^2)/(2*e^5) + (b^3*B
*(d + e*x)^3)/(3*e^5) - ((b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*Log[d + e*x])
/e^5

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Rubi in Sympy [A]  time = 51.1853, size = 143, normalized size = 0.95 \[ \frac{B b^{3} \left (d + e x\right )^{3}}{3 e^{5}} + \frac{b^{2} \left (d + e x\right )^{2} \left (A b e + 3 B a e - 4 B b d\right )}{2 e^{5}} + \frac{3 b x \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{e^{4}} + \frac{\left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{3}}{e^{5} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.181714, size = 244, normalized size = 1.63 \[ \frac{6 a^3 e^3 (B d-A e)+18 a^2 b e^2 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+9 a b^2 e \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )-6 (d+e x) (b d-a e)^2 \log (d+e x) (-a B e-3 A b e+4 b B d)+b^3 \left (3 A e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+2 B \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )}{6 e^5 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

(6*a^3*e^3*(B*d - A*e) + 18*a^2*b*e^2*(A*d*e + B*(-d^2 + d*e*x + e^2*x^2)) + 9*a
*b^2*e*(2*A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^
3*x^3)) + b^3*(3*A*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 +
9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)) - 6*(b*d - a*e)^2*(4*b*B*d
- 3*A*b*e - a*B*e)*(d + e*x)*Log[d + e*x])/(6*e^5*(d + e*x))

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Maple [B]  time = 0.013, size = 376, normalized size = 2.5 \[{\frac{{b}^{3}B{x}^{3}}{3\,{e}^{2}}}+{\frac{{b}^{3}A{x}^{2}}{2\,{e}^{2}}}+{\frac{3\,{b}^{2}B{x}^{2}a}{2\,{e}^{2}}}-{\frac{{b}^{3}B{x}^{2}d}{{e}^{3}}}+3\,{\frac{a{b}^{2}Ax}{{e}^{2}}}-2\,{\frac{{b}^{3}Adx}{{e}^{3}}}+3\,{\frac{{a}^{2}bBx}{{e}^{2}}}-6\,{\frac{a{b}^{2}Bdx}{{e}^{3}}}+3\,{\frac{{b}^{3}B{d}^{2}x}{{e}^{4}}}+3\,{\frac{\ln \left ( ex+d \right ) A{a}^{2}b}{{e}^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) Aa{b}^{2}d}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) A{b}^{3}{d}^{2}}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ) B{a}^{3}}{{e}^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) B{a}^{2}bd}{{e}^{3}}}+9\,{\frac{\ln \left ( ex+d \right ) Ba{b}^{2}{d}^{2}}{{e}^{4}}}-4\,{\frac{\ln \left ( ex+d \right ){b}^{3}B{d}^{3}}{{e}^{5}}}-{\frac{{a}^{3}A}{e \left ( ex+d \right ) }}+3\,{\frac{Ad{a}^{2}b}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{a{b}^{2}A{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{{b}^{3}A{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}+{\frac{Bd{a}^{3}}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{b{a}^{2}{d}^{2}B}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{{b}^{2}a{d}^{3}B}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{b}^{3}B{d}^{4}}{{e}^{5} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b^3/e^2*B*x^3+1/2*b^3/e^2*A*x^2+3/2*b^2/e^2*B*x^2*a-b^3/e^3*B*x^2*d+3*b^2/e^
2*A*a*x-2*b^3/e^3*A*d*x+3*b/e^2*B*a^2*x-6*b^2/e^3*B*a*d*x+3*b^3/e^4*B*d^2*x+3/e^
2*ln(e*x+d)*A*a^2*b-6/e^3*ln(e*x+d)*A*a*b^2*d+3/e^4*ln(e*x+d)*A*b^3*d^2+1/e^2*ln
(e*x+d)*B*a^3-6/e^3*ln(e*x+d)*B*a^2*b*d+9/e^4*ln(e*x+d)*B*a*b^2*d^2-4/e^5*ln(e*x
+d)*b^3*B*d^3-1/e/(e*x+d)*a^3*A+3/e^2/(e*x+d)*A*d*a^2*b-3/e^3/(e*x+d)*A*a*b^2*d^
2+1/e^4/(e*x+d)*A*b^3*d^3+1/e^2/(e*x+d)*B*d*a^3-3/e^3/(e*x+d)*B*a^2*b*d^2+3/e^4/
(e*x+d)*B*a*b^2*d^3-1/e^5/(e*x+d)*b^3*B*d^4

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Maxima [A]  time = 1.36813, size = 360, normalized size = 2.4 \[ -\frac{B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}}{e^{6} x + d e^{5}} + \frac{2 \, B b^{3} e^{2} x^{3} - 3 \,{\left (2 \, B b^{3} d e -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 6 \,{\left (3 \, B b^{3} d^{2} - 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} x}{6 \, e^{4}} - \frac{{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.215078, size = 535, normalized size = 3.57 \[ \frac{2 \, B b^{3} e^{4} x^{4} - 6 \, B b^{3} d^{4} - 6 \, A a^{3} e^{4} + 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 18 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 6 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} -{\left (4 \, B b^{3} d e^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \,{\left (4 \, B b^{3} d^{2} e^{2} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \,{\left (3 \, B b^{3} d^{3} e - 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3}\right )} x - 6 \,{\left (4 \, B b^{3} d^{4} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} +{\left (4 \, B b^{3} d^{3} e - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{6} x + d e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.55926, size = 250, normalized size = 1.67 \[ \frac{B b^{3} x^{3}}{3 e^{2}} + \frac{- A a^{3} e^{4} + 3 A a^{2} b d e^{3} - 3 A a b^{2} d^{2} e^{2} + A b^{3} d^{3} e + B a^{3} d e^{3} - 3 B a^{2} b d^{2} e^{2} + 3 B a b^{2} d^{3} e - B b^{3} d^{4}}{d e^{5} + e^{6} x} + \frac{x^{2} \left (A b^{3} e + 3 B a b^{2} e - 2 B b^{3} d\right )}{2 e^{3}} + \frac{x \left (3 A a b^{2} e^{2} - 2 A b^{3} d e + 3 B a^{2} b e^{2} - 6 B a b^{2} d e + 3 B b^{3} d^{2}\right )}{e^{4}} + \frac{\left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right ) \log{\left (d + e x \right )}}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.231515, size = 489, normalized size = 3.26 \[ \frac{1}{6} \,{\left (2 \, B b^{3} - \frac{3 \,{\left (4 \, B b^{3} d e - 3 \, B a b^{2} e^{2} - A b^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{18 \,{\left (2 \, B b^{3} d^{2} e^{2} - 3 \, B a b^{2} d e^{3} - A b^{3} d e^{3} + B a^{2} b e^{4} + A a b^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-5\right )} +{\left (4 \, B b^{3} d^{3} - 9 \, B a b^{2} d^{2} e - 3 \, A b^{3} d^{2} e + 6 \, B a^{2} b d e^{2} + 6 \, A a b^{2} d e^{2} - B a^{3} e^{3} - 3 \, A a^{2} b e^{3}\right )} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{B b^{3} d^{4} e^{3}}{x e + d} - \frac{3 \, B a b^{2} d^{3} e^{4}}{x e + d} - \frac{A b^{3} d^{3} e^{4}}{x e + d} + \frac{3 \, B a^{2} b d^{2} e^{5}}{x e + d} + \frac{3 \, A a b^{2} d^{2} e^{5}}{x e + d} - \frac{B a^{3} d e^{6}}{x e + d} - \frac{3 \, A a^{2} b d e^{6}}{x e + d} + \frac{A a^{3} e^{7}}{x e + d}\right )} e^{\left (-8\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*B*b^3 - 3*(4*B*b^3*d*e - 3*B*a*b^2*e^2 - A*b^3*e^2)*e^(-1)/(x*e + d) + 18
*(2*B*b^3*d^2*e^2 - 3*B*a*b^2*d*e^3 - A*b^3*d*e^3 + B*a^2*b*e^4 + A*a*b^2*e^4)*e
^(-2)/(x*e + d)^2)*(x*e + d)^3*e^(-5) + (4*B*b^3*d^3 - 9*B*a*b^2*d^2*e - 3*A*b^3
*d^2*e + 6*B*a^2*b*d*e^2 + 6*A*a*b^2*d*e^2 - B*a^3*e^3 - 3*A*a^2*b*e^3)*e^(-5)*l
n(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (B*b^3*d^4*e^3/(x*e + d) - 3*B*a*b^2*d^3*e^
4/(x*e + d) - A*b^3*d^3*e^4/(x*e + d) + 3*B*a^2*b*d^2*e^5/(x*e + d) + 3*A*a*b^2*
d^2*e^5/(x*e + d) - B*a^3*d*e^6/(x*e + d) - 3*A*a^2*b*d*e^6/(x*e + d) + A*a^3*e^
7/(x*e + d))*e^(-8)