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

Optimal. Leaf size=137 \[ \frac{3 c \log (x) (b c-a d) (2 b c-a d)}{a^5}-\frac{3 c (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}+\frac{3 c^2 (b c-a d)}{a^4 x}+\frac{3 c (b c-a d)^2}{a^4 (a+b x)}+\frac{(b c-a d)^3}{2 a^3 b (a+b x)^2}-\frac{c^3}{2 a^3 x^2} \]

[Out]

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

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Rubi [A]  time = 0.285196, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{3 c \log (x) (b c-a d) (2 b c-a d)}{a^5}-\frac{3 c (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}+\frac{3 c^2 (b c-a d)}{a^4 x}+\frac{3 c (b c-a d)^2}{a^4 (a+b x)}+\frac{(b c-a d)^3}{2 a^3 b (a+b x)^2}-\frac{c^3}{2 a^3 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.1652, size = 124, normalized size = 0.91 \[ - \frac{c^{3}}{2 a^{3} x^{2}} - \frac{\left (a d - b c\right )^{3}}{2 a^{3} b \left (a + b x\right )^{2}} - \frac{3 c^{2} \left (a d - b c\right )}{a^{4} x} + \frac{3 c \left (a d - b c\right )^{2}}{a^{4} \left (a + b x\right )} + \frac{3 c \left (a d - 2 b c\right ) \left (a d - b c\right ) \log{\left (x \right )}}{a^{5}} - \frac{3 c \left (a d - 2 b c\right ) \left (a d - b c\right ) \log{\left (a + b x \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.28118, size = 138, normalized size = 1.01 \[ -\frac{-6 c \log (x) \left (a^2 d^2-3 a b c d+2 b^2 c^2\right )+6 c \left (a^2 d^2-3 a b c d+2 b^2 c^2\right ) \log (a+b x)+\frac{a^2 (a d-b c)^3}{b (a+b x)^2}+\frac{a^2 c^3}{x^2}+\frac{6 a c^2 (a d-b c)}{x}-\frac{6 a c (b c-a d)^2}{a+b x}}{2 a^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 238, normalized size = 1.7 \[ -{\frac{{c}^{3}}{2\,{a}^{3}{x}^{2}}}+3\,{\frac{c\ln \left ( x \right ){d}^{2}}{{a}^{3}}}-9\,{\frac{{c}^{2}\ln \left ( x \right ) bd}{{a}^{4}}}+6\,{\frac{{c}^{3}\ln \left ( x \right ){b}^{2}}{{a}^{5}}}-3\,{\frac{{c}^{2}d}{{a}^{3}x}}+3\,{\frac{{c}^{3}b}{{a}^{4}x}}-{\frac{{d}^{3}}{2\,b \left ( bx+a \right ) ^{2}}}+{\frac{3\,c{d}^{2}}{2\,a \left ( bx+a \right ) ^{2}}}-{\frac{3\,{c}^{2}db}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}}+{\frac{{c}^{3}{b}^{2}}{2\,{a}^{3} \left ( bx+a \right ) ^{2}}}-3\,{\frac{c\ln \left ( bx+a \right ){d}^{2}}{{a}^{3}}}+9\,{\frac{{c}^{2}\ln \left ( bx+a \right ) bd}{{a}^{4}}}-6\,{\frac{{c}^{3}\ln \left ( bx+a \right ){b}^{2}}{{a}^{5}}}+3\,{\frac{c{d}^{2}}{{a}^{2} \left ( bx+a \right ) }}-6\,{\frac{{c}^{2}db}{{a}^{3} \left ( bx+a \right ) }}+3\,{\frac{{c}^{3}{b}^{2}}{{a}^{4} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/x^3/(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.35966, size = 293, normalized size = 2.14 \[ -\frac{a^{3} b c^{3} - 6 \,{\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2}\right )} x^{3} -{\left (18 \, a b^{3} c^{3} - 27 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2} - 2 \,{\left (2 \, a^{2} b^{2} c^{3} - 3 \, a^{3} b c^{2} d\right )} x}{2 \,{\left (a^{4} b^{3} x^{4} + 2 \, a^{5} b^{2} x^{3} + a^{6} b x^{2}\right )}} - \frac{3 \,{\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (b x + a\right )}{a^{5}} + \frac{3 \,{\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (x\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(a^3*b*c^3 - 6*(2*b^4*c^3 - 3*a*b^3*c^2*d + a^2*b^2*c*d^2)*x^3 - (18*a*b^3*
c^3 - 27*a^2*b^2*c^2*d + 9*a^3*b*c*d^2 - a^4*d^3)*x^2 - 2*(2*a^2*b^2*c^3 - 3*a^3
*b*c^2*d)*x)/(a^4*b^3*x^4 + 2*a^5*b^2*x^3 + a^6*b*x^2) - 3*(2*b^2*c^3 - 3*a*b*c^
2*d + a^2*c*d^2)*log(b*x + a)/a^5 + 3*(2*b^2*c^3 - 3*a*b*c^2*d + a^2*c*d^2)*log(
x)/a^5

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Fricas [A]  time = 0.220369, size = 520, normalized size = 3.8 \[ -\frac{a^{4} b c^{3} - 6 \,{\left (2 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2}\right )} x^{3} -{\left (18 \, a^{2} b^{3} c^{3} - 27 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{2} - 2 \,{\left (2 \, a^{3} b^{2} c^{3} - 3 \, a^{4} b c^{2} d\right )} x + 6 \,{\left ({\left (2 \, b^{5} c^{3} - 3 \, a b^{4} c^{2} d + a^{2} b^{3} c d^{2}\right )} x^{4} + 2 \,{\left (2 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2}\right )} x^{3} +{\left (2 \, a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) - 6 \,{\left ({\left (2 \, b^{5} c^{3} - 3 \, a b^{4} c^{2} d + a^{2} b^{3} c d^{2}\right )} x^{4} + 2 \,{\left (2 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2}\right )} x^{3} +{\left (2 \, a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{5} b^{3} x^{4} + 2 \, a^{6} b^{2} x^{3} + a^{7} b x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(a^4*b*c^3 - 6*(2*a*b^4*c^3 - 3*a^2*b^3*c^2*d + a^3*b^2*c*d^2)*x^3 - (18*a^
2*b^3*c^3 - 27*a^3*b^2*c^2*d + 9*a^4*b*c*d^2 - a^5*d^3)*x^2 - 2*(2*a^3*b^2*c^3 -
3*a^4*b*c^2*d)*x + 6*((2*b^5*c^3 - 3*a*b^4*c^2*d + a^2*b^3*c*d^2)*x^4 + 2*(2*a*
b^4*c^3 - 3*a^2*b^3*c^2*d + a^3*b^2*c*d^2)*x^3 + (2*a^2*b^3*c^3 - 3*a^3*b^2*c^2*
d + a^4*b*c*d^2)*x^2)*log(b*x + a) - 6*((2*b^5*c^3 - 3*a*b^4*c^2*d + a^2*b^3*c*d
^2)*x^4 + 2*(2*a*b^4*c^3 - 3*a^2*b^3*c^2*d + a^3*b^2*c*d^2)*x^3 + (2*a^2*b^3*c^3
- 3*a^3*b^2*c^2*d + a^4*b*c*d^2)*x^2)*log(x))/(a^5*b^3*x^4 + 2*a^6*b^2*x^3 + a^
7*b*x^2)

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Sympy [A]  time = 11.5944, size = 371, normalized size = 2.71 \[ \frac{- a^{3} b c^{3} + x^{3} \left (6 a^{2} b^{2} c d^{2} - 18 a b^{3} c^{2} d + 12 b^{4} c^{3}\right ) + x^{2} \left (- a^{4} d^{3} + 9 a^{3} b c d^{2} - 27 a^{2} b^{2} c^{2} d + 18 a b^{3} c^{3}\right ) + x \left (- 6 a^{3} b c^{2} d + 4 a^{2} b^{2} c^{3}\right )}{2 a^{6} b x^{2} + 4 a^{5} b^{2} x^{3} + 2 a^{4} b^{3} x^{4}} + \frac{3 c \left (a d - 2 b c\right ) \left (a d - b c\right ) \log{\left (x + \frac{3 a^{3} c d^{2} - 9 a^{2} b c^{2} d + 6 a b^{2} c^{3} - 3 a c \left (a d - 2 b c\right ) \left (a d - b c\right )}{6 a^{2} b c d^{2} - 18 a b^{2} c^{2} d + 12 b^{3} c^{3}} \right )}}{a^{5}} - \frac{3 c \left (a d - 2 b c\right ) \left (a d - b c\right ) \log{\left (x + \frac{3 a^{3} c d^{2} - 9 a^{2} b c^{2} d + 6 a b^{2} c^{3} + 3 a c \left (a d - 2 b c\right ) \left (a d - b c\right )}{6 a^{2} b c d^{2} - 18 a b^{2} c^{2} d + 12 b^{3} c^{3}} \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-a**3*b*c**3 + x**3*(6*a**2*b**2*c*d**2 - 18*a*b**3*c**2*d + 12*b**4*c**3) + x*
*2*(-a**4*d**3 + 9*a**3*b*c*d**2 - 27*a**2*b**2*c**2*d + 18*a*b**3*c**3) + x*(-6
*a**3*b*c**2*d + 4*a**2*b**2*c**3))/(2*a**6*b*x**2 + 4*a**5*b**2*x**3 + 2*a**4*b
**3*x**4) + 3*c*(a*d - 2*b*c)*(a*d - b*c)*log(x + (3*a**3*c*d**2 - 9*a**2*b*c**2
*d + 6*a*b**2*c**3 - 3*a*c*(a*d - 2*b*c)*(a*d - b*c))/(6*a**2*b*c*d**2 - 18*a*b*
*2*c**2*d + 12*b**3*c**3))/a**5 - 3*c*(a*d - 2*b*c)*(a*d - b*c)*log(x + (3*a**3*
c*d**2 - 9*a**2*b*c**2*d + 6*a*b**2*c**3 + 3*a*c*(a*d - 2*b*c)*(a*d - b*c))/(6*a
**2*b*c*d**2 - 18*a*b**2*c**2*d + 12*b**3*c**3))/a**5

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GIAC/XCAS [A]  time = 0.252893, size = 296, normalized size = 2.16 \[ \frac{3 \,{\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} - \frac{3 \,{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{2} b c d^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{5} b} + \frac{12 \, b^{4} c^{3} x^{3} - 18 \, a b^{3} c^{2} d x^{3} + 6 \, a^{2} b^{2} c d^{2} x^{3} + 18 \, a b^{3} c^{3} x^{2} - 27 \, a^{2} b^{2} c^{2} d x^{2} + 9 \, a^{3} b c d^{2} x^{2} - a^{4} d^{3} x^{2} + 4 \, a^{2} b^{2} c^{3} x - 6 \, a^{3} b c^{2} d x - a^{3} b c^{3}}{2 \,{\left (b x^{2} + a x\right )}^{2} a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(2*b^2*c^3 - 3*a*b*c^2*d + a^2*c*d^2)*ln(abs(x))/a^5 - 3*(2*b^3*c^3 - 3*a*b^2*
c^2*d + a^2*b*c*d^2)*ln(abs(b*x + a))/(a^5*b) + 1/2*(12*b^4*c^3*x^3 - 18*a*b^3*c
^2*d*x^3 + 6*a^2*b^2*c*d^2*x^3 + 18*a*b^3*c^3*x^2 - 27*a^2*b^2*c^2*d*x^2 + 9*a^3
*b*c*d^2*x^2 - a^4*d^3*x^2 + 4*a^2*b^2*c^3*x - 6*a^3*b*c^2*d*x - a^3*b*c^3)/((b*
x^2 + a*x)^2*a^4*b)