3.1345 \(\int \frac{(a+b x)^3}{(c+d x)^2} \, dx\)

Optimal. Leaf size=75 \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(b c-a d)^3}{d^4 (c+d x)}+\frac{3 b (b c-a d)^2 \log (c+d x)}{d^4}+\frac{b^3 x^2}{2 d^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.13265, antiderivative size = 75, 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 \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(b c-a d)^3}{d^4 (c+d x)}+\frac{3 b (b c-a d)^2 \log (c+d x)}{d^4}+\frac{b^3 x^2}{2 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/(c + d*x)^2,x]

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.074438, size = 114, normalized size = 1.52 \[ \frac{3 \left (a^2 b d^2-2 a b^2 c d+b^3 c^2\right ) \log (c+d x)}{d^4}+\frac{-a^3 d^3+3 a^2 b c d^2-3 a b^2 c^2 d+b^3 c^3}{d^4 (c+d x)}-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{b^3 x^2}{2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.01, size = 149, normalized size = 2. \[{\frac{{b}^{3}{x}^{2}}{2\,{d}^{2}}}+3\,{\frac{a{b}^{2}x}{{d}^{2}}}-2\,{\frac{{b}^{3}xc}{{d}^{3}}}+3\,{\frac{b\ln \left ( dx+c \right ){a}^{2}}{{d}^{2}}}-6\,{\frac{{b}^{2}\ln \left ( dx+c \right ) ac}{{d}^{3}}}+3\,{\frac{{b}^{3}\ln \left ( dx+c \right ){c}^{2}}{{d}^{4}}}-{\frac{{a}^{3}}{d \left ( dx+c \right ) }}+3\,{\frac{{a}^{2}bc}{{d}^{2} \left ( dx+c \right ) }}-3\,{\frac{a{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }}+{\frac{{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(d*x+c)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.36981, size = 158, normalized size = 2.11 \[ \frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{d^{5} x + c d^{4}} + \frac{b^{3} d x^{2} - 2 \,{\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} x}{2 \, d^{3}} + \frac{3 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (d x + c\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(d^5*x + c*d^4) + 1/2*(b^3*d
*x^2 - 2*(2*b^3*c - 3*a*b^2*d)*x)/d^3 + 3*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*lo
g(d*x + c)/d^4

_______________________________________________________________________________________

Fricas [A]  time = 0.196995, size = 232, normalized size = 3.09 \[ \frac{b^{3} d^{3} x^{3} + 2 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3} - 3 \,{\left (b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} x + 6 \,{\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (d^{5} x + c d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.29762, size = 100, normalized size = 1.33 \[ \frac{b^{3} x^{2}}{2 d^{2}} + \frac{3 b \left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{4}} - \frac{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{c d^{4} + d^{5} x} + \frac{x \left (3 a b^{2} d - 2 b^{3} c\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**3*x**2/(2*d**2) + 3*b*(a*d - b*c)**2*log(c + d*x)/d**4 - (a**3*d**3 - 3*a**2*
b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3)/(c*d**4 + d**5*x) + x*(3*a*b**2*d - 2*b*
*3*c)/d**3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217943, size = 224, normalized size = 2.99 \[ \frac{{\left (b^{3} - \frac{6 \,{\left (b^{3} c d - a b^{2} d^{2}\right )}}{{\left (d x + c\right )} d}\right )}{\left (d x + c\right )}^{2}}{2 \, d^{4}} - \frac{3 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}{\rm ln}\left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{d^{4}} + \frac{\frac{b^{3} c^{3} d^{2}}{d x + c} - \frac{3 \, a b^{2} c^{2} d^{3}}{d x + c} + \frac{3 \, a^{2} b c d^{4}}{d x + c} - \frac{a^{3} d^{5}}{d x + c}}{d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^3 - 6*(b^3*c*d - a*b^2*d^2)/((d*x + c)*d))*(d*x + c)^2/d^4 - 3*(b^3*c^2 -
2*a*b^2*c*d + a^2*b*d^2)*ln(abs(d*x + c)/((d*x + c)^2*abs(d)))/d^4 + (b^3*c^3*d
^2/(d*x + c) - 3*a*b^2*c^2*d^3/(d*x + c) + 3*a^2*b*c*d^4/(d*x + c) - a^3*d^5/(d*
x + c))/d^6