3.1810 \(\int \frac{(a+b x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx\)

Optimal. Leaf size=59 \[ \frac{2 b (b c-a d)}{d^3 (c+d x)}-\frac{(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac{b^2 \log (c+d x)}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.11379, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{2 b (b c-a d)}{d^3 (c+d x)}-\frac{(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac{b^2 \log (c+d x)}{d^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 22.7649, size = 51, normalized size = 0.86 \[ \frac{b^{2} \log{\left (c + d x \right )}}{d^{3}} - \frac{2 b \left (a d - b c\right )}{d^{3} \left (c + d x\right )} - \frac{\left (a d - b c\right )^{2}}{2 d^{3} \left (c + d x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.04459, size = 48, normalized size = 0.81 \[ \frac{\frac{(b c-a d) (a d+3 b c+4 b d x)}{(c+d x)^2}+2 b^2 \log (c+d x)}{2 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 92, normalized size = 1.6 \[ -2\,{\frac{ab}{{d}^{2} \left ( dx+c \right ) }}+2\,{\frac{{b}^{2}c}{{d}^{3} \left ( dx+c \right ) }}-{\frac{{a}^{2}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{abc}{{d}^{2} \left ( dx+c \right ) ^{2}}}-{\frac{{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{2}\ln \left ( dx+c \right ) }{{d}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(a*c+(a*d+b*c)*x+x^2*b*d)^3,x)

[Out]

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

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Maxima [A]  time = 0.765027, size = 108, normalized size = 1.83 \[ \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} + \frac{b^{2} \log \left (d x + c\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.203066, size = 135, normalized size = 2.29 \[ \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x + 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right )}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.68478, size = 80, normalized size = 1.36 \[ \frac{b^{2} \log{\left (c + d x \right )}}{d^{3}} - \frac{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2} + x \left (4 a b d^{2} - 4 b^{2} c d\right )}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217349, size = 93, normalized size = 1.58 \[ \frac{b^{2}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{3}} + \frac{4 \,{\left (b^{2} c - a b d\right )} x + \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}}{d}}{2 \,{\left (d x + c\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="giac")

[Out]

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