3.246 \(\int \frac{(c+d x)^3}{x^4 (a+b x)^2} \, dx\)

Optimal. Leaf size=132 \[ -\frac{\log (x) (b c-a d)^2 (4 b c-a d)}{a^5}+\frac{(b c-a d)^2 (4 b c-a d) \log (a+b x)}{a^5}-\frac{3 c (b c-a d)^2}{a^4 x}-\frac{(b c-a d)^3}{a^4 (a+b x)}+\frac{c^2 (2 b c-3 a d)}{2 a^3 x^2}-\frac{c^3}{3 a^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.244194, antiderivative size = 132, 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 \[ -\frac{\log (x) (b c-a d)^2 (4 b c-a d)}{a^5}+\frac{(b c-a d)^2 (4 b c-a d) \log (a+b x)}{a^5}-\frac{3 c (b c-a d)^2}{a^4 x}-\frac{(b c-a d)^3}{a^4 (a+b x)}+\frac{c^2 (2 b c-3 a d)}{2 a^3 x^2}-\frac{c^3}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^3/(x^4*(a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 32.6664, size = 117, normalized size = 0.89 \[ - \frac{c^{3}}{3 a^{2} x^{3}} - \frac{c^{2} \left (3 a d - 2 b c\right )}{2 a^{3} x^{2}} - \frac{3 c \left (a d - b c\right )^{2}}{a^{4} x} + \frac{\left (a d - b c\right )^{3}}{a^{4} \left (a + b x\right )} + \frac{\left (a d - 4 b c\right ) \left (a d - b c\right )^{2} \log{\left (x \right )}}{a^{5}} - \frac{\left (a d - 4 b c\right ) \left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**3/x**4/(b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.105146, size = 126, normalized size = 0.95 \[ -\frac{\frac{2 a^3 c^3}{x^3}+\frac{3 a^2 c^2 (3 a d-2 b c)}{x^2}+\frac{18 a c (b c-a d)^2}{x}-\frac{6 a (a d-b c)^3}{a+b x}+6 \log (x) (b c-a d)^2 (4 b c-a d)-6 (b c-a d)^2 (4 b c-a d) \log (a+b x)}{6 a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^3/(x^4*(a + b*x)^2),x]

[Out]

-((2*a^3*c^3)/x^3 + (3*a^2*c^2*(-2*b*c + 3*a*d))/x^2 + (18*a*c*(b*c - a*d)^2)/x
- (6*a*(-(b*c) + a*d)^3)/(a + b*x) + 6*(b*c - a*d)^2*(4*b*c - a*d)*Log[x] - 6*(b
*c - a*d)^2*(4*b*c - a*d)*Log[a + b*x])/(6*a^5)

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Maple [A]  time = 0.02, size = 256, normalized size = 1.9 \[ -{\frac{{c}^{3}}{3\,{a}^{2}{x}^{3}}}+{\frac{\ln \left ( x \right ){d}^{3}}{{a}^{2}}}-6\,{\frac{\ln \left ( x \right ) cb{d}^{2}}{{a}^{3}}}+9\,{\frac{\ln \left ( x \right ){b}^{2}{c}^{2}d}{{a}^{4}}}-4\,{\frac{\ln \left ( x \right ){b}^{3}{c}^{3}}{{a}^{5}}}-{\frac{3\,{c}^{2}d}{2\,{a}^{2}{x}^{2}}}+{\frac{{c}^{3}b}{{a}^{3}{x}^{2}}}-3\,{\frac{c{d}^{2}}{{a}^{2}x}}+6\,{\frac{{c}^{2}db}{{a}^{3}x}}-3\,{\frac{{c}^{3}{b}^{2}}{{a}^{4}x}}-{\frac{\ln \left ( bx+a \right ){d}^{3}}{{a}^{2}}}+6\,{\frac{\ln \left ( bx+a \right ) cb{d}^{2}}{{a}^{3}}}-9\,{\frac{\ln \left ( bx+a \right ){b}^{2}{c}^{2}d}{{a}^{4}}}+4\,{\frac{\ln \left ( bx+a \right ){b}^{3}{c}^{3}}{{a}^{5}}}+{\frac{{d}^{3}}{a \left ( bx+a \right ) }}-3\,{\frac{c{d}^{2}b}{{a}^{2} \left ( bx+a \right ) }}+3\,{\frac{{c}^{2}d{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-{\frac{{b}^{3}{c}^{3}}{{a}^{4} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/x^4/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.34353, size = 296, normalized size = 2.24 \[ -\frac{2 \, a^{3} c^{3} + 6 \,{\left (4 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} + 3 \,{\left (4 \, a b^{2} c^{3} - 9 \, a^{2} b c^{2} d + 6 \, a^{3} c d^{2}\right )} x^{2} -{\left (4 \, a^{2} b c^{3} - 9 \, a^{3} c^{2} d\right )} x}{6 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac{{\left (4 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{5}} - \frac{{\left (4 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((b*x + a)^2*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.216087, size = 435, normalized size = 3.3 \[ -\frac{2 \, a^{4} c^{3} + 6 \,{\left (4 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 3 \,{\left (4 \, a^{2} b^{2} c^{3} - 9 \, a^{3} b c^{2} d + 6 \, a^{4} c d^{2}\right )} x^{2} -{\left (4 \, a^{3} b c^{3} - 9 \, a^{4} c^{2} d\right )} x - 6 \,{\left ({\left (4 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} +{\left (4 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) + 6 \,{\left ({\left (4 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} +{\left (4 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((b*x + a)^2*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 10.1899, size = 386, normalized size = 2.92 \[ \frac{- 2 a^{3} c^{3} + x^{3} \left (6 a^{3} d^{3} - 36 a^{2} b c d^{2} + 54 a b^{2} c^{2} d - 24 b^{3} c^{3}\right ) + x^{2} \left (- 18 a^{3} c d^{2} + 27 a^{2} b c^{2} d - 12 a b^{2} c^{3}\right ) + x \left (- 9 a^{3} c^{2} d + 4 a^{2} b c^{3}\right )}{6 a^{5} x^{3} + 6 a^{4} b x^{4}} + \frac{\left (a d - 4 b c\right ) \left (a d - b c\right )^{2} \log{\left (x + \frac{a^{4} d^{3} - 6 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - 4 a b^{3} c^{3} - a \left (a d - 4 b c\right ) \left (a d - b c\right )^{2}}{2 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 18 a b^{3} c^{2} d - 8 b^{4} c^{3}} \right )}}{a^{5}} - \frac{\left (a d - 4 b c\right ) \left (a d - b c\right )^{2} \log{\left (x + \frac{a^{4} d^{3} - 6 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - 4 a b^{3} c^{3} + a \left (a d - 4 b c\right ) \left (a d - b c\right )^{2}}{2 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 18 a b^{3} c^{2} d - 8 b^{4} c^{3}} \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**3/x**4/(b*x+a)**2,x)

[Out]

(-2*a**3*c**3 + x**3*(6*a**3*d**3 - 36*a**2*b*c*d**2 + 54*a*b**2*c**2*d - 24*b**
3*c**3) + x**2*(-18*a**3*c*d**2 + 27*a**2*b*c**2*d - 12*a*b**2*c**3) + x*(-9*a**
3*c**2*d + 4*a**2*b*c**3))/(6*a**5*x**3 + 6*a**4*b*x**4) + (a*d - 4*b*c)*(a*d -
b*c)**2*log(x + (a**4*d**3 - 6*a**3*b*c*d**2 + 9*a**2*b**2*c**2*d - 4*a*b**3*c**
3 - a*(a*d - 4*b*c)*(a*d - b*c)**2)/(2*a**3*b*d**3 - 12*a**2*b**2*c*d**2 + 18*a*
b**3*c**2*d - 8*b**4*c**3))/a**5 - (a*d - 4*b*c)*(a*d - b*c)**2*log(x + (a**4*d*
*3 - 6*a**3*b*c*d**2 + 9*a**2*b**2*c**2*d - 4*a*b**3*c**3 + a*(a*d - 4*b*c)*(a*d
- b*c)**2)/(2*a**3*b*d**3 - 12*a**2*b**2*c*d**2 + 18*a*b**3*c**2*d - 8*b**4*c**
3))/a**5

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GIAC/XCAS [A]  time = 0.268756, size = 378, normalized size = 2.86 \[ -\frac{{\left (4 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{5} b} - \frac{\frac{b^{7} c^{3}}{b x + a} - \frac{3 \, a b^{6} c^{2} d}{b x + a} + \frac{3 \, a^{2} b^{5} c d^{2}}{b x + a} - \frac{a^{3} b^{4} d^{3}}{b x + a}}{a^{4} b^{4}} + \frac{26 \, b^{3} c^{3} - 45 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - \frac{3 \,{\left (20 \, a b^{4} c^{3} - 33 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2}\right )}}{{\left (b x + a\right )} b} + \frac{18 \,{\left (2 \, a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + a^{4} b^{3} c d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{6 \, a^{5}{\left (\frac{a}{b x + a} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((b*x + a)^2*x^4),x, algorithm="giac")

[Out]

-(4*b^4*c^3 - 9*a*b^3*c^2*d + 6*a^2*b^2*c*d^2 - a^3*b*d^3)*ln(abs(-a/(b*x + a) +
1))/(a^5*b) - (b^7*c^3/(b*x + a) - 3*a*b^6*c^2*d/(b*x + a) + 3*a^2*b^5*c*d^2/(b
*x + a) - a^3*b^4*d^3/(b*x + a))/(a^4*b^4) + 1/6*(26*b^3*c^3 - 45*a*b^2*c^2*d +
18*a^2*b*c*d^2 - 3*(20*a*b^4*c^3 - 33*a^2*b^3*c^2*d + 12*a^3*b^2*c*d^2)/((b*x +
a)*b) + 18*(2*a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + a^4*b^3*c*d^2)/((b*x + a)^2*b^2))/
(a^5*(a/(b*x + a) - 1)^3)