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

Optimal. Leaf size=87 \[ \frac{(b c-a d)^2 (a d+2 b c) \log (a+b x)}{a^3 b^2}-\frac{c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac{(b c-a d)^3}{a^2 b^2 (a+b x)}-\frac{c^3}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.166809, antiderivative size = 87, 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 \[ \frac{(b c-a d)^2 (a d+2 b c) \log (a+b x)}{a^3 b^2}-\frac{c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac{(b c-a d)^3}{a^2 b^2 (a+b x)}-\frac{c^3}{a^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.9994, size = 78, normalized size = 0.9 \[ - \frac{c^{3}}{a^{2} x} + \frac{\left (a d - b c\right )^{3}}{a^{2} b^{2} \left (a + b x\right )} + \frac{c^{2} \left (3 a d - 2 b c\right ) \log{\left (x \right )}}{a^{3}} + \frac{\left (a d - b c\right )^{2} \left (a d + 2 b c\right ) \log{\left (a + b x \right )}}{a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.125257, size = 79, normalized size = 0.91 \[ \frac{\frac{a (a d-b c)^3}{b^2 (a+b x)}+\frac{(b c-a d)^2 (a d+2 b c) \log (a+b x)}{b^2}+c^2 \log (x) (3 a d-2 b c)-\frac{a c^3}{x}}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 141, normalized size = 1.6 \[ -{\frac{{c}^{3}}{{a}^{2}x}}+3\,{\frac{{c}^{2}\ln \left ( x \right ) d}{{a}^{2}}}-2\,{\frac{{c}^{3}\ln \left ( x \right ) b}{{a}^{3}}}+{\frac{\ln \left ( bx+a \right ){d}^{3}}{{b}^{2}}}-3\,{\frac{\ln \left ( bx+a \right ){c}^{2}d}{{a}^{2}}}+2\,{\frac{b\ln \left ( bx+a \right ){c}^{3}}{{a}^{3}}}+{\frac{a{d}^{3}}{{b}^{2} \left ( bx+a \right ) }}-3\,{\frac{c{d}^{2}}{b \left ( bx+a \right ) }}+3\,{\frac{{c}^{2}d}{a \left ( bx+a \right ) }}-{\frac{{c}^{3}b}{{a}^{2} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/x^2/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.35293, size = 178, normalized size = 2.05 \[ -\frac{a b^{2} c^{3} +{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{a^{2} b^{3} x^{2} + a^{3} b^{2} x} - \frac{{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x\right )}{a^{3}} + \frac{{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.213567, size = 267, normalized size = 3.07 \[ -\frac{a^{2} b^{2} c^{3} +{\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x -{\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{2} +{\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}\right )} x\right )} \log \left (b x + a\right ) +{\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d\right )} x^{2} +{\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x\right )} \log \left (x\right )}{a^{3} b^{3} x^{2} + a^{4} b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.80823, size = 250, normalized size = 2.87 \[ \frac{- a b^{2} c^{3} + x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{a^{3} b^{2} x + a^{2} b^{3} x^{2}} + \frac{c^{2} \left (3 a d - 2 b c\right ) \log{\left (x + \frac{- 3 a^{2} b c^{2} d + 2 a b^{2} c^{3} + a b c^{2} \left (3 a d - 2 b c\right )}{a^{3} d^{3} - 6 a b^{2} c^{2} d + 4 b^{3} c^{3}} \right )}}{a^{3}} + \frac{\left (a d - b c\right )^{2} \left (a d + 2 b c\right ) \log{\left (x + \frac{- 3 a^{2} b c^{2} d + 2 a b^{2} c^{3} + \frac{a \left (a d - b c\right )^{2} \left (a d + 2 b c\right )}{b}}{a^{3} d^{3} - 6 a b^{2} c^{2} d + 4 b^{3} c^{3}} \right )}}{a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269251, size = 223, normalized size = 2.56 \[ -\frac{d^{3}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{2}} + \frac{b c^{3}}{a^{3}{\left (\frac{a}{b x + a} - 1\right )}} - \frac{{\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d\right )}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{3} b} - \frac{\frac{b^{5} c^{3}}{b x + a} - \frac{3 \, a b^{4} c^{2} d}{b x + a} + \frac{3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac{a^{3} b^{2} d^{3}}{b x + a}}{a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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