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

Optimal. Leaf size=77 \[ -\frac{a^3 \log (a+b x)}{b^3 (b c-a d)}-\frac{x (a d+b c)}{b^2 d^2}+\frac{c^3 \log (c+d x)}{d^3 (b c-a d)}+\frac{x^2}{2 b d} \]

[Out]

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

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Rubi [A]  time = 0.130673, antiderivative size = 77, 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 \[ -\frac{a^3 \log (a+b x)}{b^3 (b c-a d)}-\frac{x (a d+b c)}{b^2 d^2}+\frac{c^3 \log (c+d x)}{d^3 (b c-a d)}+\frac{x^2}{2 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0588202, size = 74, normalized size = 0.96 \[ \frac{-2 a^3 d^3 \log (a+b x)+b d x (b c-a d) (-2 a d-2 b c+b d x)+2 b^3 c^3 \log (c+d x)}{2 b^3 d^3 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 80, normalized size = 1. \[{\frac{{x}^{2}}{2\,bd}}-{\frac{ax}{{b}^{2}d}}-{\frac{cx}{b{d}^{2}}}-{\frac{{c}^{3}\ln \left ( dx+c \right ) }{{d}^{3} \left ( ad-bc \right ) }}+{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{3} \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)/(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.36947, size = 104, normalized size = 1.35 \[ -\frac{a^{3} \log \left (b x + a\right )}{b^{4} c - a b^{3} d} + \frac{c^{3} \log \left (d x + c\right )}{b c d^{3} - a d^{4}} + \frac{b d x^{2} - 2 \,{\left (b c + a d\right )} x}{2 \, b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.221891, size = 128, normalized size = 1.66 \[ -\frac{2 \, a^{3} d^{3} \log \left (b x + a\right ) - 2 \, b^{3} c^{3} \log \left (d x + c\right ) -{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x}{2 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.82358, size = 219, normalized size = 2.84 \[ \frac{a^{3} \log{\left (x + \frac{\frac{a^{5} d^{4}}{b \left (a d - b c\right )} - \frac{2 a^{4} c d^{3}}{a d - b c} + \frac{a^{3} b c^{2} d^{2}}{a d - b c} + a^{3} c d^{2} + a b^{2} c^{3}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{b^{3} \left (a d - b c\right )} - \frac{c^{3} \log{\left (x + \frac{a^{3} c d^{2} - \frac{a^{2} b^{2} c^{3} d}{a d - b c} + \frac{2 a b^{3} c^{4}}{a d - b c} + a b^{2} c^{3} - \frac{b^{4} c^{5}}{d \left (a d - b c\right )}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )} + \frac{x^{2}}{2 b d} - \frac{x \left (a d + b c\right )}{b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError