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

Optimal. Leaf size=61 \[ \frac{(b c-a d)^3 \log (a+b x)}{a^2 b^2}-\frac{c^2 \log (x) (b c-3 a d)}{a^2}-\frac{c^3}{a x}+\frac{d^3 x}{b} \]

[Out]

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

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Rubi [A]  time = 0.112239, antiderivative size = 61, 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 \[ \frac{(b c-a d)^3 \log (a+b x)}{a^2 b^2}-\frac{c^2 \log (x) (b c-3 a d)}{a^2}-\frac{c^3}{a x}+\frac{d^3 x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{3} \int \frac{1}{b}\, dx - \frac{c^{3}}{a x} + \frac{c^{2} \left (3 a d - b c\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*Integral(1/b, x) - c**3/(a*x) + c**2*(3*a*d - b*c)*log(x)/a**2 - (a*d - b*c
)**3*log(a + b*x)/(a**2*b**2)

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Mathematica [A]  time = 0.0534791, size = 66, normalized size = 1.08 \[ \frac{b^2 c^2 x \log (x) (3 a d-b c)+a b \left (a d^3 x^2-b c^3\right )+x (b c-a d)^3 \log (a+b x)}{a^2 b^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 102, normalized size = 1.7 \[{\frac{{d}^{3}x}{b}}-{\frac{{c}^{3}}{ax}}+3\,{\frac{{c}^{2}\ln \left ( x \right ) d}{a}}-{\frac{{c}^{3}\ln \left ( x \right ) b}{{a}^{2}}}-{\frac{a\ln \left ( bx+a \right ){d}^{3}}{{b}^{2}}}+3\,{\frac{\ln \left ( bx+a \right ) c{d}^{2}}{b}}-3\,{\frac{\ln \left ( bx+a \right ){c}^{2}d}{a}}+{\frac{b\ln \left ( bx+a \right ){c}^{3}}{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/x^2/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.33753, size = 120, normalized size = 1.97 \[ \frac{d^{3} x}{b} - \frac{c^{3}}{a x} - \frac{{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (x\right )}{a^{2}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.217234, size = 132, normalized size = 2.16 \[ \frac{a^{2} b d^{3} x^{2} - a b^{2} c^{3} +{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x \log \left (b x + a\right ) -{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d\right )} x \log \left (x\right )}{a^{2} b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.19624, size = 196, normalized size = 3.21 \[ \frac{d^{3} x}{b} - \frac{c^{3}}{a x} + \frac{c^{2} \left (3 a d - b c\right ) \log{\left (x + \frac{3 a^{2} b c^{2} d - a b^{2} c^{3} - a b c^{2} \left (3 a d - b c\right )}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{2}} - \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{3 a^{2} b c^{2} d - a b^{2} c^{3} + \frac{a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.288914, size = 123, normalized size = 2.02 \[ \frac{d^{3} x}{b} - \frac{c^{3}}{a x} - \frac{{\left (b c^{3} - 3 \, a c^{2} d\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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