3.1138 \(\int \frac{(A+B x) (d+e x)^3}{b x+c x^2} \, dx\)

Optimal. Leaf size=128 \[ \frac{e x \left (A c e (3 c d-b e)+B \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{c^3}+\frac{(b B-A c) (c d-b e)^3 \log (b+c x)}{b c^4}+\frac{e^2 x^2 (A c e-b B e+3 B c d)}{2 c^2}+\frac{A d^3 \log (x)}{b}+\frac{B e^3 x^3}{3 c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.333259, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{e x \left (A c e (3 c d-b e)+B \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{c^3}+\frac{(b B-A c) (c d-b e)^3 \log (b+c x)}{b c^4}+\frac{e^2 x^2 (A c e-b B e+3 B c d)}{2 c^2}+\frac{A d^3 \log (x)}{b}+\frac{B e^3 x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.138421, size = 118, normalized size = 0.92 \[ \frac{b c e x \left (3 A c e (-2 b e+6 c d+c e x)+B \left (6 b^2 e^2-3 b c e (6 d+e x)+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right )-6 (b B-A c) (b e-c d)^3 \log (b+c x)+6 A c^4 d^3 \log (x)}{6 b c^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.012, size = 252, normalized size = 2. \[{\frac{B{e}^{3}{x}^{3}}{3\,c}}+{\frac{A{e}^{3}{x}^{2}}{2\,c}}-{\frac{B{e}^{3}{x}^{2}b}{2\,{c}^{2}}}+{\frac{3\,{e}^{2}B{x}^{2}d}{2\,c}}-{\frac{A{e}^{3}bx}{{c}^{2}}}+3\,{\frac{{e}^{2}Adx}{c}}+{\frac{B{e}^{3}{b}^{2}x}{{c}^{3}}}-3\,{\frac{B{e}^{2}bdx}{{c}^{2}}}+3\,{\frac{B{d}^{2}ex}{c}}+{\frac{A{d}^{3}\ln \left ( x \right ) }{b}}+{\frac{{b}^{2}\ln \left ( cx+b \right ) A{e}^{3}}{{c}^{3}}}-3\,{\frac{b\ln \left ( cx+b \right ) Ad{e}^{2}}{{c}^{2}}}+3\,{\frac{\ln \left ( cx+b \right ) A{d}^{2}e}{c}}-{\frac{\ln \left ( cx+b \right ) A{d}^{3}}{b}}-{\frac{{b}^{3}\ln \left ( cx+b \right ) B{e}^{3}}{{c}^{4}}}+3\,{\frac{{b}^{2}\ln \left ( cx+b \right ) Bd{e}^{2}}{{c}^{3}}}-3\,{\frac{b\ln \left ( cx+b \right ) B{d}^{2}e}{{c}^{2}}}+{\frac{\ln \left ( cx+b \right ) B{d}^{3}}{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.724177, size = 270, normalized size = 2.11 \[ \frac{A d^{3} \log \left (x\right )}{b} + \frac{2 \, B c^{2} e^{3} x^{3} + 3 \,{\left (3 \, B c^{2} d e^{2} -{\left (B b c - A c^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (3 \, B c^{2} d^{2} e - 3 \,{\left (B b c - A c^{2}\right )} d e^{2} +{\left (B b^{2} - A b c\right )} e^{3}\right )} x}{6 \, c^{3}} + \frac{{\left ({\left (B b c^{3} - A c^{4}\right )} d^{3} - 3 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} -{\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} \log \left (c x + b\right )}{b c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.304237, size = 292, normalized size = 2.28 \[ \frac{2 \, B b c^{3} e^{3} x^{3} + 6 \, A c^{4} d^{3} \log \left (x\right ) + 3 \,{\left (3 \, B b c^{3} d e^{2} -{\left (B b^{2} c^{2} - A b c^{3}\right )} e^{3}\right )} x^{2} + 6 \,{\left (3 \, B b c^{3} d^{2} e - 3 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} d e^{2} +{\left (B b^{3} c - A b^{2} c^{2}\right )} e^{3}\right )} x + 6 \,{\left ({\left (B b c^{3} - A c^{4}\right )} d^{3} - 3 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} -{\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} \log \left (c x + b\right )}{6 \, b c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 17.7382, size = 260, normalized size = 2.03 \[ \frac{A d^{3} \log{\left (x \right )}}{b} + \frac{B e^{3} x^{3}}{3 c} - \frac{x^{2} \left (- A c e^{3} + B b e^{3} - 3 B c d e^{2}\right )}{2 c^{2}} + \frac{x \left (- A b c e^{3} + 3 A c^{2} d e^{2} + B b^{2} e^{3} - 3 B b c d e^{2} + 3 B c^{2} d^{2} e\right )}{c^{3}} - \frac{\left (- A c + B b\right ) \left (b e - c d\right )^{3} \log{\left (x + \frac{A b c^{3} d^{3} + \frac{b \left (- A c + B b\right ) \left (b e - c d\right )^{3}}{c}}{- A b^{3} c e^{3} + 3 A b^{2} c^{2} d e^{2} - 3 A b c^{3} d^{2} e + 2 A c^{4} d^{3} + B b^{4} e^{3} - 3 B b^{3} c d e^{2} + 3 B b^{2} c^{2} d^{2} e - B b c^{3} d^{3}} \right )}}{b c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.279347, size = 279, normalized size = 2.18 \[ \frac{A d^{3}{\rm ln}\left ({\left | x \right |}\right )}{b} + \frac{2 \, B c^{2} x^{3} e^{3} + 9 \, B c^{2} d x^{2} e^{2} + 18 \, B c^{2} d^{2} x e - 3 \, B b c x^{2} e^{3} + 3 \, A c^{2} x^{2} e^{3} - 18 \, B b c d x e^{2} + 18 \, A c^{2} d x e^{2} + 6 \, B b^{2} x e^{3} - 6 \, A b c x e^{3}}{6 \, c^{3}} + \frac{{\left (B b c^{3} d^{3} - A c^{4} d^{3} - 3 \, B b^{2} c^{2} d^{2} e + 3 \, A b c^{3} d^{2} e + 3 \, B b^{3} c d e^{2} - 3 \, A b^{2} c^{2} d e^{2} - B b^{4} e^{3} + A b^{3} c e^{3}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*d^3*ln(abs(x))/b + 1/6*(2*B*c^2*x^3*e^3 + 9*B*c^2*d*x^2*e^2 + 18*B*c^2*d^2*x*e
- 3*B*b*c*x^2*e^3 + 3*A*c^2*x^2*e^3 - 18*B*b*c*d*x*e^2 + 18*A*c^2*d*x*e^2 + 6*B
*b^2*x*e^3 - 6*A*b*c*x*e^3)/c^3 + (B*b*c^3*d^3 - A*c^4*d^3 - 3*B*b^2*c^2*d^2*e +
3*A*b*c^3*d^2*e + 3*B*b^3*c*d*e^2 - 3*A*b^2*c^2*d*e^2 - B*b^4*e^3 + A*b^3*c*e^3
)*ln(abs(c*x + b))/(b*c^4)