3.194 \(\int \frac{(c+d x)^3}{a+b x} \, dx\)

Optimal. Leaf size=73 \[ \frac{(b c-a d)^3 \log (a+b x)}{b^4}+\frac{d x (b c-a d)^2}{b^3}+\frac{(c+d x)^2 (b c-a d)}{2 b^2}+\frac{(c+d x)^3}{3 b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0748069, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{(b c-a d)^3 \log (a+b x)}{b^4}+\frac{d x (b c-a d)^2}{b^3}+\frac{(c+d x)^2 (b c-a d)}{2 b^2}+\frac{(c+d x)^3}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^3/(a + b*x),x]

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**3/(b*x+a),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0460686, size = 74, normalized size = 1.01 \[ \frac{b d x \left (6 a^2 d^2-3 a b d (6 c+d x)+b^2 \left (18 c^2+9 c d x+2 d^2 x^2\right )\right )+6 (b c-a d)^3 \log (a+b x)}{6 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 133, normalized size = 1.8 \[{\frac{{d}^{3}{x}^{3}}{3\,b}}-{\frac{{d}^{3}{x}^{2}a}{2\,{b}^{2}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,b}}+{\frac{{d}^{3}{a}^{2}x}{{b}^{3}}}-3\,{\frac{{d}^{2}acx}{{b}^{2}}}+3\,{\frac{d{c}^{2}x}{b}}-{\frac{\ln \left ( bx+a \right ){a}^{3}{d}^{3}}{{b}^{4}}}+3\,{\frac{\ln \left ( bx+a \right ){a}^{2}c{d}^{2}}{{b}^{3}}}-3\,{\frac{\ln \left ( bx+a \right ) a{c}^{2}d}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{3}}{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/(b*x+a),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.3601, size = 154, normalized size = 2.11 \[ \frac{2 \, b^{2} d^{3} x^{3} + 3 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.202082, size = 157, normalized size = 2.15 \[ \frac{2 \, b^{3} d^{3} x^{3} + 3 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x + 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.89055, size = 82, normalized size = 1.12 \[ \frac{d^{3} x^{3}}{3 b} - \frac{x^{2} \left (a d^{3} - 3 b c d^{2}\right )}{2 b^{2}} + \frac{x \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{3}} - \frac{\left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**3/(b*x+a),x)

[Out]

d**3*x**3/(3*b) - x**2*(a*d**3 - 3*b*c*d**2)/(2*b**2) + x*(a**2*d**3 - 3*a*b*c*d
**2 + 3*b**2*c**2*d)/b**3 - (a*d - b*c)**3*log(a + b*x)/b**4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.357816, size = 155, normalized size = 2.12 \[ \frac{2 \, b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x^{2} - 3 \, a b d^{3} x^{2} + 18 \, b^{2} c^{2} d x - 18 \, a b c d^{2} x + 6 \, a^{2} d^{3} x}{6 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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