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

Optimal. Leaf size=97 \[ \frac{3 c \log (x) (b c-a d)^2}{a^4}-\frac{3 c (b c-a d)^2 \log (a+b x)}{a^4}+\frac{c^2 (2 b c-3 a d)}{a^3 x}+\frac{(b c-a d)^3}{a^3 b (a+b x)}-\frac{c^3}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.176806, antiderivative size = 97, 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 \[ \frac{3 c \log (x) (b c-a d)^2}{a^4}-\frac{3 c (b c-a d)^2 \log (a+b x)}{a^4}+\frac{c^2 (2 b c-3 a d)}{a^3 x}+\frac{(b c-a d)^3}{a^3 b (a+b x)}-\frac{c^3}{2 a^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.5652, size = 88, normalized size = 0.91 \[ - \frac{c^{3}}{2 a^{2} x^{2}} - \frac{c^{2} \left (3 a d - 2 b c\right )}{a^{3} x} - \frac{\left (a d - b c\right )^{3}}{a^{3} b \left (a + b x\right )} + \frac{3 c \left (a d - b c\right )^{2} \log{\left (x \right )}}{a^{4}} - \frac{3 c \left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.186132, size = 93, normalized size = 0.96 \[ -\frac{\frac{a^2 c^3}{x^2}+\frac{2 a c^2 (3 a d-2 b c)}{x}+\frac{2 a (a d-b c)^3}{b (a+b x)}-6 c \log (x) (b c-a d)^2+6 c (b c-a d)^2 \log (a+b x)}{2 a^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 186, normalized size = 1.9 \[ -{\frac{{c}^{3}}{2\,{a}^{2}{x}^{2}}}-3\,{\frac{{c}^{2}d}{{a}^{2}x}}+2\,{\frac{{c}^{3}b}{{a}^{3}x}}+3\,{\frac{c\ln \left ( x \right ){d}^{2}}{{a}^{2}}}-6\,{\frac{{c}^{2}\ln \left ( x \right ) bd}{{a}^{3}}}+3\,{\frac{{c}^{3}\ln \left ( x \right ){b}^{2}}{{a}^{4}}}-{\frac{{d}^{3}}{b \left ( bx+a \right ) }}+3\,{\frac{c{d}^{2}}{a \left ( bx+a \right ) }}-3\,{\frac{{c}^{2}db}{{a}^{2} \left ( bx+a \right ) }}+{\frac{{c}^{3}{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-3\,{\frac{c\ln \left ( bx+a \right ){d}^{2}}{{a}^{2}}}+6\,{\frac{{c}^{2}\ln \left ( bx+a \right ) bd}{{a}^{3}}}-3\,{\frac{{c}^{3}\ln \left ( bx+a \right ){b}^{2}}{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/x^3/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.35202, size = 220, normalized size = 2.27 \[ -\frac{a^{2} b c^{3} - 2 \,{\left (3 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2} - 3 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d\right )} x}{2 \,{\left (a^{3} b^{2} x^{3} + a^{4} b x^{2}\right )}} - \frac{3 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (b x + a\right )}{a^{4}} + \frac{3 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (x\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21256, size = 336, normalized size = 3.46 \[ -\frac{a^{3} b c^{3} - 2 \,{\left (3 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2} - 3 \,{\left (a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d\right )} x + 6 \,{\left ({\left (b^{4} c^{3} - 2 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2}\right )} x^{3} +{\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) - 6 \,{\left ({\left (b^{4} c^{3} - 2 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2}\right )} x^{3} +{\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{4} b^{2} x^{3} + a^{5} b x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.94706, size = 291, normalized size = 3. \[ - \frac{a^{2} b c^{3} + x^{2} \left (2 a^{3} d^{3} - 6 a^{2} b c d^{2} + 12 a b^{2} c^{2} d - 6 b^{3} c^{3}\right ) + x \left (6 a^{2} b c^{2} d - 3 a b^{2} c^{3}\right )}{2 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac{3 c \left (a d - b c\right )^{2} \log{\left (x + \frac{3 a^{3} c d^{2} - 6 a^{2} b c^{2} d + 3 a b^{2} c^{3} - 3 a c \left (a d - b c\right )^{2}}{6 a^{2} b c d^{2} - 12 a b^{2} c^{2} d + 6 b^{3} c^{3}} \right )}}{a^{4}} - \frac{3 c \left (a d - b c\right )^{2} \log{\left (x + \frac{3 a^{3} c d^{2} - 6 a^{2} b c^{2} d + 3 a b^{2} c^{3} + 3 a c \left (a d - b c\right )^{2}}{6 a^{2} b c d^{2} - 12 a b^{2} c^{2} d + 6 b^{3} c^{3}} \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269634, size = 262, normalized size = 2.7 \[ \frac{3 \,{\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2}\right )}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{4} b} + \frac{\frac{b^{5} c^{3}}{b x + a} - \frac{3 \, a b^{4} c^{2} d}{b x + a} + \frac{3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac{a^{3} b^{2} d^{3}}{b x + a}}{a^{3} b^{3}} + \frac{5 \, b^{2} c^{3} - 6 \, a b c^{2} d - \frac{6 \,{\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )}}{{\left (b x + a\right )} b}}{2 \, a^{4}{\left (\frac{a}{b x + a} - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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