3.1809 \(\int \frac{(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx\)

Optimal. Leaf size=78 \[ -\frac{3 b^2 (b c-a d) \log (c+d x)}{d^4}-\frac{3 b (b c-a d)^2}{d^4 (c+d x)}+\frac{(b c-a d)^3}{2 d^4 (c+d x)^2}+\frac{b^3 x}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.15878, antiderivative size = 78, 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 \[ -\frac{3 b^2 (b c-a d) \log (c+d x)}{d^4}-\frac{3 b (b c-a d)^2}{d^4 (c+d x)}+\frac{(b c-a d)^3}{2 d^4 (c+d x)^2}+\frac{b^3 x}{d^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0771361, size = 114, normalized size = 1.46 \[ \frac{-a^3 d^3-3 a^2 b d^2 (c+2 d x)+3 a b^2 c d (3 c+4 d x)-6 b^2 (c+d x)^2 (b c-a d) \log (c+d x)+b^3 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )}{2 d^4 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

(-(a^3*d^3) - 3*a^2*b*d^2*(c + 2*d*x) + 3*a*b^2*c*d*(3*c + 4*d*x) + b^3*(-5*c^3
- 4*c^2*d*x + 4*c*d^2*x^2 + 2*d^3*x^3) - 6*b^2*(b*c - a*d)*(c + d*x)^2*Log[c + d
*x])/(2*d^4*(c + d*x)^2)

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Maple [B]  time = 0.01, size = 160, normalized size = 2.1 \[{\frac{{b}^{3}x}{{d}^{3}}}-3\,{\frac{{a}^{2}b}{{d}^{2} \left ( dx+c \right ) }}+6\,{\frac{ac{b}^{2}}{{d}^{3} \left ( dx+c \right ) }}-3\,{\frac{{b}^{3}{c}^{2}}{{d}^{4} \left ( dx+c \right ) }}+3\,{\frac{{b}^{2}\ln \left ( dx+c \right ) a}{{d}^{3}}}-3\,{\frac{{b}^{3}\ln \left ( dx+c \right ) c}{{d}^{4}}}-{\frac{{a}^{3}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{3\,{a}^{2}bc}{2\,{d}^{2} \left ( dx+c \right ) ^{2}}}-{\frac{3\,a{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( dx+c \right ) ^{2}}}+{\frac{{c}^{3}{b}^{3}}{2\,{d}^{4} \left ( dx+c \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6/(a*c+(a*d+b*c)*x+x^2*b*d)^3,x)

[Out]

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

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Maxima [A]  time = 0.781899, size = 169, normalized size = 2.17 \[ \frac{b^{3} x}{d^{3}} - \frac{5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}} - \frac{3 \,{\left (b^{3} c - a b^{2} d\right )} \log \left (d x + c\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^6/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="maxima")

[Out]

b^3*x/d^3 - 1/2*(5*b^3*c^3 - 9*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3 + 6*(b^3*c^
2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x)/(d^6*x^2 + 2*c*d^5*x + c^2*d^4) - 3*(b^3*c -
a*b^2*d)*log(d*x + c)/d^4

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Fricas [A]  time = 0.20744, size = 254, normalized size = 3.26 \[ \frac{2 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c d^{2} x^{2} - 5 \, b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - a^{3} d^{3} - 2 \,{\left (2 \, b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x - 6 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d +{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (b^{3} c^{2} d - a b^{2} c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^6/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="fricas")

[Out]

1/2*(2*b^3*d^3*x^3 + 4*b^3*c*d^2*x^2 - 5*b^3*c^3 + 9*a*b^2*c^2*d - 3*a^2*b*c*d^2
- a^3*d^3 - 2*(2*b^3*c^2*d - 6*a*b^2*c*d^2 + 3*a^2*b*d^3)*x - 6*(b^3*c^3 - a*b^
2*c^2*d + (b^3*c*d^2 - a*b^2*d^3)*x^2 + 2*(b^3*c^2*d - a*b^2*c*d^2)*x)*log(d*x +
c))/(d^6*x^2 + 2*c*d^5*x + c^2*d^4)

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Sympy [A]  time = 4.37259, size = 128, normalized size = 1.64 \[ \frac{b^{3} x}{d^{3}} + \frac{3 b^{2} \left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{4}} - \frac{a^{3} d^{3} + 3 a^{2} b c d^{2} - 9 a b^{2} c^{2} d + 5 b^{3} c^{3} + x \left (6 a^{2} b d^{3} - 12 a b^{2} c d^{2} + 6 b^{3} c^{2} d\right )}{2 c^{2} d^{4} + 4 c d^{5} x + 2 d^{6} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**6/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

b**3*x/d**3 + 3*b**2*(a*d - b*c)*log(c + d*x)/d**4 - (a**3*d**3 + 3*a**2*b*c*d**
2 - 9*a*b**2*c**2*d + 5*b**3*c**3 + x*(6*a**2*b*d**3 - 12*a*b**2*c*d**2 + 6*b**3
*c**2*d))/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2)

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GIAC/XCAS [A]  time = 0.215568, size = 151, normalized size = 1.94 \[ \frac{b^{3} x}{d^{3}} - \frac{3 \,{\left (b^{3} c - a b^{2} d\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{4}} - \frac{5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \,{\left (d x + c\right )}^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^6/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="giac")

[Out]

b^3*x/d^3 - 3*(b^3*c - a*b^2*d)*ln(abs(d*x + c))/d^4 - 1/2*(5*b^3*c^3 - 9*a*b^2*
c^2*d + 3*a^2*b*c*d^2 + a^3*d^3 + 6*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x)/(
(d*x + c)^2*d^4)