3.1024 \(\int \frac{(a+b x)^3 (A+B x)}{d+e x} \, dx\)

Optimal. Leaf size=124 \[ \frac{(b d-a e)^3 (B d-A e) \log (d+e x)}{e^5}-\frac{b x (b d-a e)^2 (B d-A e)}{e^4}+\frac{(a+b x)^2 (b d-a e) (B d-A e)}{2 e^3}-\frac{(a+b x)^3 (B d-A e)}{3 e^2}+\frac{B (a+b x)^4}{4 b e} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.194018, antiderivative size = 124, 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 \[ \frac{(b d-a e)^3 (B d-A e) \log (d+e x)}{e^5}-\frac{b x (b d-a e)^2 (B d-A e)}{e^4}+\frac{(a+b x)^2 (b d-a e) (B d-A e)}{2 e^3}-\frac{(a+b x)^3 (B d-A e)}{3 e^2}+\frac{B (a+b x)^4}{4 b e} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.154064, size = 169, normalized size = 1.36 \[ \frac{e x \left (12 a^3 B e^3+18 a^2 b e^2 (2 A e-2 B d+B e x)+6 a b^2 e \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+b^3 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )\right )+12 (b d-a e)^3 (B d-A e) \log (d+e x)}{12 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(e*x*(12*a^3*B*e^3 + 18*a^2*b*e^2*(-2*B*d + 2*A*e + B*e*x) + 6*a*b^2*e*(3*A*e*(-
2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + b^3*(2*A*e*(6*d^2 - 3*d*e*x + 2*
e^2*x^2) + B*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3))) + 12*(b*d - a*e)^
3*(B*d - A*e)*Log[d + e*x])/(12*e^5)

_______________________________________________________________________________________

Maple [B]  time = 0.007, size = 341, normalized size = 2.8 \[{\frac{B{b}^{3}{x}^{4}}{4\,e}}+{\frac{A{b}^{3}{x}^{3}}{3\,e}}+{\frac{B{x}^{3}a{b}^{2}}{e}}-{\frac{{b}^{3}B{x}^{3}d}{3\,{e}^{2}}}+{\frac{3\,aA{b}^{2}{x}^{2}}{2\,e}}-{\frac{A{x}^{2}{b}^{3}d}{2\,{e}^{2}}}+{\frac{3\,B{x}^{2}{a}^{2}b}{2\,e}}-{\frac{3\,B{x}^{2}a{b}^{2}d}{2\,{e}^{2}}}+{\frac{{b}^{3}B{x}^{2}{d}^{2}}{2\,{e}^{3}}}+3\,{\frac{{a}^{2}Abx}{e}}-3\,{\frac{a{b}^{2}Adx}{{e}^{2}}}+{\frac{{b}^{3}A{d}^{2}x}{{e}^{3}}}+{\frac{{a}^{3}Bx}{e}}-3\,{\frac{{a}^{2}bBdx}{{e}^{2}}}+3\,{\frac{a{b}^{2}B{d}^{2}x}{{e}^{3}}}-{\frac{{b}^{3}B{d}^{3}x}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ){a}^{3}A}{e}}-3\,{\frac{\ln \left ( ex+d \right ) A{a}^{2}bd}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ) Aa{b}^{2}{d}^{2}}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) A{b}^{3}{d}^{3}}{{e}^{4}}}-{\frac{\ln \left ( ex+d \right ) B{a}^{3}d}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ) B{a}^{2}b{d}^{2}}{{e}^{3}}}-3\,{\frac{\ln \left ( ex+d \right ) Ba{b}^{2}{d}^{3}}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ){b}^{3}B{d}^{4}}{{e}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.33862, size = 348, normalized size = 2.81 \[ \frac{3 \, B b^{3} e^{3} x^{4} - 4 \,{\left (B b^{3} d e^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{3} + 6 \,{\left (B b^{3} d^{2} e -{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{2} - 12 \,{\left (B b^{3} d^{3} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x}{12 \, e^{4}} + \frac{{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} \log \left (e x + d\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.210003, size = 351, normalized size = 2.83 \[ \frac{3 \, B b^{3} e^{4} x^{4} - 4 \,{\left (B b^{3} d e^{3} -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 6 \,{\left (B b^{3} d^{2} e^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 12 \,{\left (B b^{3} d^{3} e -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x + 12 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.65603, size = 214, normalized size = 1.73 \[ \frac{B b^{3} x^{4}}{4 e} + \frac{x^{3} \left (A b^{3} e + 3 B a b^{2} e - B b^{3} d\right )}{3 e^{2}} + \frac{x^{2} \left (3 A a b^{2} e^{2} - A b^{3} d e + 3 B a^{2} b e^{2} - 3 B a b^{2} d e + B b^{3} d^{2}\right )}{2 e^{3}} + \frac{x \left (3 A a^{2} b e^{3} - 3 A a b^{2} d e^{2} + A b^{3} d^{2} e + B a^{3} e^{3} - 3 B a^{2} b d e^{2} + 3 B a b^{2} d^{2} e - B b^{3} d^{3}\right )}{e^{4}} - \frac{\left (- A e + B d\right ) \left (a e - b d\right )^{3} \log{\left (d + e x \right )}}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225781, size = 383, normalized size = 3.09 \[{\left (B b^{3} d^{4} - 3 \, B a b^{2} d^{3} e - A b^{3} d^{3} e + 3 \, B a^{2} b d^{2} e^{2} + 3 \, A a b^{2} d^{2} e^{2} - B a^{3} d e^{3} - 3 \, A a^{2} b d e^{3} + A a^{3} e^{4}\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, B b^{3} x^{4} e^{3} - 4 \, B b^{3} d x^{3} e^{2} + 6 \, B b^{3} d^{2} x^{2} e - 12 \, B b^{3} d^{3} x + 12 \, B a b^{2} x^{3} e^{3} + 4 \, A b^{3} x^{3} e^{3} - 18 \, B a b^{2} d x^{2} e^{2} - 6 \, A b^{3} d x^{2} e^{2} + 36 \, B a b^{2} d^{2} x e + 12 \, A b^{3} d^{2} x e + 18 \, B a^{2} b x^{2} e^{3} + 18 \, A a b^{2} x^{2} e^{3} - 36 \, B a^{2} b d x e^{2} - 36 \, A a b^{2} d x e^{2} + 12 \, B a^{3} x e^{3} + 36 \, A a^{2} b x e^{3}\right )} e^{\left (-4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b^3*d^4 - 3*B*a*b^2*d^3*e - A*b^3*d^3*e + 3*B*a^2*b*d^2*e^2 + 3*A*a*b^2*d^2*e
^2 - B*a^3*d*e^3 - 3*A*a^2*b*d*e^3 + A*a^3*e^4)*e^(-5)*ln(abs(x*e + d)) + 1/12*(
3*B*b^3*x^4*e^3 - 4*B*b^3*d*x^3*e^2 + 6*B*b^3*d^2*x^2*e - 12*B*b^3*d^3*x + 12*B*
a*b^2*x^3*e^3 + 4*A*b^3*x^3*e^3 - 18*B*a*b^2*d*x^2*e^2 - 6*A*b^3*d*x^2*e^2 + 36*
B*a*b^2*d^2*x*e + 12*A*b^3*d^2*x*e + 18*B*a^2*b*x^2*e^3 + 18*A*a*b^2*x^2*e^3 - 3
6*B*a^2*b*d*x*e^2 - 36*A*a*b^2*d*x*e^2 + 12*B*a^3*x*e^3 + 36*A*a^2*b*x*e^3)*e^(-
4)