3.1118 \(\int \frac{(A+B x) (d+e x)^3}{(a+b x)^3} \, dx\)

Optimal. Leaf size=141 \[ -\frac{(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^5 (a+b x)}-\frac{(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}+\frac{3 e (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{b^5}+\frac{e^2 x (-3 a B e+A b e+3 b B d)}{b^4}+\frac{B e^3 x^2}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.337095, antiderivative size = 141, 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 \[ -\frac{(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^5 (a+b x)}-\frac{(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}+\frac{3 e (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{b^5}+\frac{e^2 x (-3 a B e+A b e+3 b B d)}{b^4}+\frac{B e^3 x^2}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B e^{3} \int x\, dx}{b^{3}} + e^{2} \left (A b e - 3 B a e + 3 B b d\right ) \int \frac{1}{b^{4}}\, dx - \frac{3 e \left (a e - b d\right ) \left (A b e - 2 B a e + B b d\right ) \log{\left (a + b x \right )}}{b^{5}} - \frac{\left (a e - b d\right )^{2} \left (3 A b e - 4 B a e + B b d\right )}{b^{5} \left (a + b x\right )} + \frac{\left (A b - B a\right ) \left (a e - b d\right )^{3}}{2 b^{5} \left (a + b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.244658, size = 245, normalized size = 1.74 \[ \frac{-A b \left (5 a^3 e^3+a^2 b e^2 (4 e x-9 d)+a b^2 e \left (3 d^2-12 d e x-4 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x-2 e^3 x^3\right )\right )+B \left (7 a^4 e^3+a^3 b e^2 (2 e x-15 d)+a^2 b^2 e \left (9 d^2-12 d e x-11 e^2 x^2\right )-a b^3 \left (d^3-12 d^2 e x-12 d e^2 x^2+4 e^3 x^3\right )+b^4 x \left (-2 d^3+6 d e^2 x^2+e^3 x^3\right )\right )+6 e (a+b x)^2 (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{2 b^5 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-(A*b*(5*a^3*e^3 + a^2*b*e^2*(-9*d + 4*e*x) + a*b^2*e*(3*d^2 - 12*d*e*x - 4*e^2
*x^2) + b^3*(d^3 + 6*d^2*e*x - 2*e^3*x^3))) + B*(7*a^4*e^3 + a^3*b*e^2*(-15*d +
2*e*x) + a^2*b^2*e*(9*d^2 - 12*d*e*x - 11*e^2*x^2) + b^4*x*(-2*d^3 + 6*d*e^2*x^2
+ e^3*x^3) - a*b^3*(d^3 - 12*d^2*e*x - 12*d*e^2*x^2 + 4*e^3*x^3)) + 6*e*(b*d -
a*e)*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^2*Log[a + b*x])/(2*b^5*(a + b*x)^2)

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Maple [B]  time = 0.016, size = 404, normalized size = 2.9 \[{\frac{B{e}^{3}{x}^{2}}{2\,{b}^{3}}}+{\frac{{e}^{3}Ax}{{b}^{3}}}-3\,{\frac{Ba{e}^{3}x}{{b}^{4}}}+3\,{\frac{{e}^{2}Bdx}{{b}^{3}}}-3\,{\frac{{e}^{3}\ln \left ( bx+a \right ) Aa}{{b}^{4}}}+3\,{\frac{{e}^{2}\ln \left ( bx+a \right ) Ad}{{b}^{3}}}+6\,{\frac{{e}^{3}\ln \left ( bx+a \right ) B{a}^{2}}{{b}^{5}}}-9\,{\frac{{e}^{2}\ln \left ( bx+a \right ) Bad}{{b}^{4}}}+3\,{\frac{e\ln \left ( bx+a \right ) B{d}^{2}}{{b}^{3}}}-3\,{\frac{A{a}^{2}{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}+6\,{\frac{aAd{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}-3\,{\frac{A{d}^{2}e}{{b}^{2} \left ( bx+a \right ) }}+4\,{\frac{B{a}^{3}{e}^{3}}{{b}^{5} \left ( bx+a \right ) }}-9\,{\frac{B{a}^{2}d{e}^{2}}{{b}^{4} \left ( bx+a \right ) }}+6\,{\frac{Ba{d}^{2}e}{{b}^{3} \left ( bx+a \right ) }}-{\frac{B{d}^{3}}{{b}^{2} \left ( bx+a \right ) }}+{\frac{{a}^{3}A{e}^{3}}{2\, \left ( bx+a \right ) ^{2}{b}^{4}}}-{\frac{3\,A{a}^{2}d{e}^{2}}{2\, \left ( bx+a \right ) ^{2}{b}^{3}}}+{\frac{3\,aA{d}^{2}e}{2\, \left ( bx+a \right ) ^{2}{b}^{2}}}-{\frac{A{d}^{3}}{2\, \left ( bx+a \right ) ^{2}b}}-{\frac{B{a}^{4}{e}^{3}}{2\, \left ( bx+a \right ) ^{2}{b}^{5}}}+{\frac{3\,B{a}^{3}d{e}^{2}}{2\, \left ( bx+a \right ) ^{2}{b}^{4}}}-{\frac{3\,B{a}^{2}{d}^{2}e}{2\, \left ( bx+a \right ) ^{2}{b}^{3}}}+{\frac{Ba{d}^{3}}{2\, \left ( bx+a \right ) ^{2}{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37778, size = 381, normalized size = 2.7 \[ -\frac{{\left (B a b^{3} + A b^{4}\right )} d^{3} - 3 \,{\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \,{\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{2} -{\left (7 \, B a^{4} - 5 \, A a^{3} b\right )} e^{3} + 2 \,{\left (B b^{4} d^{3} - 3 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{2} -{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac{B b e^{3} x^{2} + 2 \,{\left (3 \, B b d e^{2} -{\left (3 \, B a - A b\right )} e^{3}\right )} x}{2 \, b^{4}} + \frac{3 \,{\left (B b^{2} d^{2} e -{\left (3 \, B a b - A b^{2}\right )} d e^{2} +{\left (2 \, B a^{2} - A a b\right )} e^{3}\right )} \log \left (b x + a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*((B*a*b^3 + A*b^4)*d^3 - 3*(3*B*a^2*b^2 - A*a*b^3)*d^2*e + 3*(5*B*a^3*b - 3
*A*a^2*b^2)*d*e^2 - (7*B*a^4 - 5*A*a^3*b)*e^3 + 2*(B*b^4*d^3 - 3*(2*B*a*b^3 - A*
b^4)*d^2*e + 3*(3*B*a^2*b^2 - 2*A*a*b^3)*d*e^2 - (4*B*a^3*b - 3*A*a^2*b^2)*e^3)*
x)/(b^7*x^2 + 2*a*b^6*x + a^2*b^5) + 1/2*(B*b*e^3*x^2 + 2*(3*B*b*d*e^2 - (3*B*a
- A*b)*e^3)*x)/b^4 + 3*(B*b^2*d^2*e - (3*B*a*b - A*b^2)*d*e^2 + (2*B*a^2 - A*a*b
)*e^3)*log(b*x + a)/b^5

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Fricas [A]  time = 0.217784, size = 597, normalized size = 4.23 \[ \frac{B b^{4} e^{3} x^{4} -{\left (B a b^{3} + A b^{4}\right )} d^{3} + 3 \,{\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e - 3 \,{\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{2} +{\left (7 \, B a^{4} - 5 \, A a^{3} b\right )} e^{3} + 2 \,{\left (3 \, B b^{4} d e^{2} -{\left (2 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} +{\left (12 \, B a b^{3} d e^{2} -{\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{3}\right )} x^{2} - 2 \,{\left (B b^{4} d^{3} - 3 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 6 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{2} -{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x + 6 \,{\left (B a^{2} b^{2} d^{2} e -{\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} d e^{2} +{\left (2 \, B a^{4} - A a^{3} b\right )} e^{3} +{\left (B b^{4} d^{2} e -{\left (3 \, B a b^{3} - A b^{4}\right )} d e^{2} +{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \,{\left (B a b^{3} d^{2} e -{\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d e^{2} +{\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(B*b^4*e^3*x^4 - (B*a*b^3 + A*b^4)*d^3 + 3*(3*B*a^2*b^2 - A*a*b^3)*d^2*e - 3
*(5*B*a^3*b - 3*A*a^2*b^2)*d*e^2 + (7*B*a^4 - 5*A*a^3*b)*e^3 + 2*(3*B*b^4*d*e^2
- (2*B*a*b^3 - A*b^4)*e^3)*x^3 + (12*B*a*b^3*d*e^2 - (11*B*a^2*b^2 - 4*A*a*b^3)*
e^3)*x^2 - 2*(B*b^4*d^3 - 3*(2*B*a*b^3 - A*b^4)*d^2*e + 6*(B*a^2*b^2 - A*a*b^3)*
d*e^2 - (B*a^3*b - 2*A*a^2*b^2)*e^3)*x + 6*(B*a^2*b^2*d^2*e - (3*B*a^3*b - A*a^2
*b^2)*d*e^2 + (2*B*a^4 - A*a^3*b)*e^3 + (B*b^4*d^2*e - (3*B*a*b^3 - A*b^4)*d*e^2
+ (2*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 + 2*(B*a*b^3*d^2*e - (3*B*a^2*b^2 - A*a*b^3)
*d*e^2 + (2*B*a^3*b - A*a^2*b^2)*e^3)*x)*log(b*x + a))/(b^7*x^2 + 2*a*b^6*x + a^
2*b^5)

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Sympy [A]  time = 19.9014, size = 296, normalized size = 2.1 \[ \frac{B e^{3} x^{2}}{2 b^{3}} + \frac{- 5 A a^{3} b e^{3} + 9 A a^{2} b^{2} d e^{2} - 3 A a b^{3} d^{2} e - A b^{4} d^{3} + 7 B a^{4} e^{3} - 15 B a^{3} b d e^{2} + 9 B a^{2} b^{2} d^{2} e - B a b^{3} d^{3} + x \left (- 6 A a^{2} b^{2} e^{3} + 12 A a b^{3} d e^{2} - 6 A b^{4} d^{2} e + 8 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 12 B a b^{3} d^{2} e - 2 B b^{4} d^{3}\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} - \frac{x \left (- A b e^{3} + 3 B a e^{3} - 3 B b d e^{2}\right )}{b^{4}} + \frac{3 e \left (a e - b d\right ) \left (- A b e + 2 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*e**3*x**2/(2*b**3) + (-5*A*a**3*b*e**3 + 9*A*a**2*b**2*d*e**2 - 3*A*a*b**3*d**
2*e - A*b**4*d**3 + 7*B*a**4*e**3 - 15*B*a**3*b*d*e**2 + 9*B*a**2*b**2*d**2*e -
B*a*b**3*d**3 + x*(-6*A*a**2*b**2*e**3 + 12*A*a*b**3*d*e**2 - 6*A*b**4*d**2*e +
8*B*a**3*b*e**3 - 18*B*a**2*b**2*d*e**2 + 12*B*a*b**3*d**2*e - 2*B*b**4*d**3))/(
2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - x*(-A*b*e**3 + 3*B*a*e**3 - 3*B*b*d*e*
*2)/b**4 + 3*e*(a*e - b*d)*(-A*b*e + 2*B*a*e - B*b*d)*log(a + b*x)/b**5

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GIAC/XCAS [A]  time = 0.231069, size = 367, normalized size = 2.6 \[ \frac{3 \,{\left (B b^{2} d^{2} e - 3 \, B a b d e^{2} + A b^{2} d e^{2} + 2 \, B a^{2} e^{3} - A a b e^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5}} + \frac{B b^{3} x^{2} e^{3} + 6 \, B b^{3} d x e^{2} - 6 \, B a b^{2} x e^{3} + 2 \, A b^{3} x e^{3}}{2 \, b^{6}} - \frac{B a b^{3} d^{3} + A b^{4} d^{3} - 9 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 15 \, B a^{3} b d e^{2} - 9 \, A a^{2} b^{2} d e^{2} - 7 \, B a^{4} e^{3} + 5 \, A a^{3} b e^{3} + 2 \,{\left (B b^{4} d^{3} - 6 \, B a b^{3} d^{2} e + 3 \, A b^{4} d^{2} e + 9 \, B a^{2} b^{2} d e^{2} - 6 \, A a b^{3} d e^{2} - 4 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(B*b^2*d^2*e - 3*B*a*b*d*e^2 + A*b^2*d*e^2 + 2*B*a^2*e^3 - A*a*b*e^3)*ln(abs(b
*x + a))/b^5 + 1/2*(B*b^3*x^2*e^3 + 6*B*b^3*d*x*e^2 - 6*B*a*b^2*x*e^3 + 2*A*b^3*
x*e^3)/b^6 - 1/2*(B*a*b^3*d^3 + A*b^4*d^3 - 9*B*a^2*b^2*d^2*e + 3*A*a*b^3*d^2*e
+ 15*B*a^3*b*d*e^2 - 9*A*a^2*b^2*d*e^2 - 7*B*a^4*e^3 + 5*A*a^3*b*e^3 + 2*(B*b^4*
d^3 - 6*B*a*b^3*d^2*e + 3*A*b^4*d^2*e + 9*B*a^2*b^2*d*e^2 - 6*A*a*b^3*d*e^2 - 4*
B*a^3*b*e^3 + 3*A*a^2*b^2*e^3)*x)/((b*x + a)^2*b^5)