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

Optimal. Leaf size=103 \[ -\frac{\log (x) (b c-a d)^3}{a^4}+\frac{(b c-a d)^3 \log (a+b x)}{a^4}+\frac{c^2 (b c-3 a d)}{2 a^2 x^2}-\frac{c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3 x}-\frac{c^3}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.160945, antiderivative size = 103, 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 \[ -\frac{\log (x) (b c-a d)^3}{a^4}+\frac{(b c-a d)^3 \log (a+b x)}{a^4}+\frac{c^2 (b c-3 a d)}{2 a^2 x^2}-\frac{c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3 x}-\frac{c^3}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.8179, size = 94, normalized size = 0.91 \[ - \frac{c^{3}}{3 a x^{3}} - \frac{c^{2} \left (3 a d - b c\right )}{2 a^{2} x^{2}} - \frac{c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{a^{3} x} + \frac{\left (a d - b c\right )^{3} \log{\left (x \right )}}{a^{4}} - \frac{\left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.107384, size = 93, normalized size = 0.9 \[ -\frac{\frac{a c \left (a^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )-3 a b c x (c+6 d x)+6 b^2 c^2 x^2\right )}{x^3}+6 \log (x) (b c-a d)^3-6 (b c-a d)^3 \log (a+b x)}{6 a^4} \]

Antiderivative was successfully verified.

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

[Out]

-((a*c*(6*b^2*c^2*x^2 - 3*a*b*c*x*(c + 6*d*x) + a^2*(2*c^2 + 9*c*d*x + 18*d^2*x^
2)))/x^3 + 6*(b*c - a*d)^3*Log[x] - 6*(b*c - a*d)^3*Log[a + b*x])/(6*a^4)

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Maple [A]  time = 0.013, size = 188, normalized size = 1.8 \[ -{\frac{{c}^{3}}{3\,a{x}^{3}}}+{\frac{\ln \left ( x \right ){d}^{3}}{a}}-3\,{\frac{\ln \left ( x \right ) cb{d}^{2}}{{a}^{2}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}{c}^{2}d}{{a}^{3}}}-{\frac{\ln \left ( x \right ){b}^{3}{c}^{3}}{{a}^{4}}}-3\,{\frac{c{d}^{2}}{ax}}+3\,{\frac{{c}^{2}bd}{{a}^{2}x}}-{\frac{{c}^{3}{b}^{2}}{{a}^{3}x}}-{\frac{3\,{c}^{2}d}{2\,a{x}^{2}}}+{\frac{{c}^{3}b}{2\,{a}^{2}{x}^{2}}}-{\frac{\ln \left ( bx+a \right ){d}^{3}}{a}}+3\,{\frac{\ln \left ( bx+a \right ) cb{d}^{2}}{{a}^{2}}}-3\,{\frac{\ln \left ( bx+a \right ){b}^{2}{c}^{2}d}{{a}^{3}}}+{\frac{\ln \left ( bx+a \right ){b}^{3}{c}^{3}}{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/x^4/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.35657, size = 211, normalized size = 2.05 \[ \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{4}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x\right )}{a^{4}} - \frac{2 \, a^{2} c^{3} + 6 \,{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{2} - 3 \,{\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x}{6 \, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.217698, size = 217, normalized size = 2.11 \[ -\frac{2 \, a^{3} c^{3} - 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (b x + a\right ) + 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (x\right ) + 6 \,{\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 3 \,{\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} d\right )} x}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.56108, size = 289, normalized size = 2.81 \[ - \frac{2 a^{2} c^{3} + x^{2} \left (18 a^{2} c d^{2} - 18 a b c^{2} d + 6 b^{2} c^{3}\right ) + x \left (9 a^{2} c^{2} d - 3 a b c^{3}\right )}{6 a^{3} x^{3}} + \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3} - a \left (a d - b c\right )^{3}}{2 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 6 a b^{3} c^{2} d - 2 b^{4} c^{3}} \right )}}{a^{4}} - \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3} + a \left (a d - b c\right )^{3}}{2 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 6 a b^{3} c^{2} d - 2 b^{4} c^{3}} \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270129, size = 228, normalized size = 2.21 \[ -\frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, a^{3} c^{3} + 6 \,{\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 3 \,{\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} d\right )} x}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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