3.201 \(\int \frac{x^5}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=145 \[ -\frac{a^5 \log (a+b x)}{b^5 (b c-a d)}-\frac{x (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{b^4 d^4}+\frac{x^2 \left (a^2 d^2+a b c d+b^2 c^2\right )}{2 b^3 d^3}-\frac{x^3 (a d+b c)}{3 b^2 d^2}+\frac{c^5 \log (c+d x)}{d^5 (b c-a d)}+\frac{x^4}{4 b d} \]

[Out]

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

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Rubi [A]  time = 0.298371, antiderivative size = 145, 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 \[ -\frac{a^5 \log (a+b x)}{b^5 (b c-a d)}-\frac{x (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{b^4 d^4}+\frac{x^2 \left (a^2 d^2+a b c d+b^2 c^2\right )}{2 b^3 d^3}-\frac{x^3 (a d+b c)}{3 b^2 d^2}+\frac{c^5 \log (c+d x)}{d^5 (b c-a d)}+\frac{x^4}{4 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((a + b*x)*(c + d*x)),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{5} \log{\left (a + b x \right )}}{b^{5} \left (a d - b c\right )} - \frac{c^{5} \log{\left (c + d x \right )}}{d^{5} \left (a d - b c\right )} - \frac{\left (a d + b c\right ) \left (a^{2} d^{2} + b^{2} c^{2}\right ) \int \frac{1}{b^{4}}\, dx}{d^{4}} + \frac{x^{4}}{4 b d} - \frac{x^{3} \left (a d + b c\right )}{3 b^{2} d^{2}} + \frac{\left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right ) \int x\, dx}{b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x+a)/(d*x+c),x)

[Out]

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

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Mathematica [A]  time = 0.100766, size = 133, normalized size = 0.92 \[ \frac{-12 a^5 d^5 \log (a+b x)+b d x \left (12 a^4 d^4-6 a^3 b d^4 x+4 a^2 b^2 d^4 x^2-3 a b^3 d^4 x^3+b^4 c \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )\right )+12 b^5 c^5 \log (c+d x)}{12 b^5 d^5 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((a + b*x)*(c + d*x)),x]

[Out]

(b*d*x*(12*a^4*d^4 - 6*a^3*b*d^4*x + 4*a^2*b^2*d^4*x^2 - 3*a*b^3*d^4*x^3 + b^4*c
*(-12*c^3 + 6*c^2*d*x - 4*c*d^2*x^2 + 3*d^3*x^3)) - 12*a^5*d^5*Log[a + b*x] + 12
*b^5*c^5*Log[c + d*x])/(12*b^5*d^5*(b*c - a*d))

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Maple [A]  time = 0.012, size = 175, normalized size = 1.2 \[{\frac{{x}^{4}}{4\,bd}}-{\frac{{x}^{3}a}{3\,{b}^{2}d}}-{\frac{c{x}^{3}}{3\,b{d}^{2}}}+{\frac{{a}^{2}{x}^{2}}{2\,{b}^{3}d}}+{\frac{{x}^{2}ac}{2\,{b}^{2}{d}^{2}}}+{\frac{{x}^{2}{c}^{2}}{2\,b{d}^{3}}}-{\frac{{a}^{3}x}{{b}^{4}d}}-{\frac{{a}^{2}cx}{{b}^{3}{d}^{2}}}-{\frac{a{c}^{2}x}{{b}^{2}{d}^{3}}}-{\frac{{c}^{3}x}{b{d}^{4}}}-{\frac{{c}^{5}\ln \left ( dx+c \right ) }{{d}^{5} \left ( ad-bc \right ) }}+{\frac{{a}^{5}\ln \left ( bx+a \right ) }{{b}^{5} \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x+a)/(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.35363, size = 217, normalized size = 1.5 \[ -\frac{a^{5} \log \left (b x + a\right )}{b^{6} c - a b^{5} d} + \frac{c^{5} \log \left (d x + c\right )}{b c d^{5} - a d^{6}} + \frac{3 \, b^{3} d^{3} x^{4} - 4 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 6 \,{\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 12 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} x}{12 \, b^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)*(d*x + c)),x, algorithm="maxima")

[Out]

-a^5*log(b*x + a)/(b^6*c - a*b^5*d) + c^5*log(d*x + c)/(b*c*d^5 - a*d^6) + 1/12*
(3*b^3*d^3*x^4 - 4*(b^3*c*d^2 + a*b^2*d^3)*x^3 + 6*(b^3*c^2*d + a*b^2*c*d^2 + a^
2*b*d^3)*x^2 - 12*(b^3*c^3 + a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*x)/(b^4*d^4)

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Fricas [A]  time = 0.222746, size = 201, normalized size = 1.39 \[ -\frac{12 \, a^{5} d^{5} \log \left (b x + a\right ) - 12 \, b^{5} c^{5} \log \left (d x + c\right ) - 3 \,{\left (b^{5} c d^{4} - a b^{4} d^{5}\right )} x^{4} + 4 \,{\left (b^{5} c^{2} d^{3} - a^{2} b^{3} d^{5}\right )} x^{3} - 6 \,{\left (b^{5} c^{3} d^{2} - a^{3} b^{2} d^{5}\right )} x^{2} + 12 \,{\left (b^{5} c^{4} d - a^{4} b d^{5}\right )} x}{12 \,{\left (b^{6} c d^{5} - a b^{5} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)*(d*x + c)),x, algorithm="fricas")

[Out]

-1/12*(12*a^5*d^5*log(b*x + a) - 12*b^5*c^5*log(d*x + c) - 3*(b^5*c*d^4 - a*b^4*
d^5)*x^4 + 4*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^3 - 6*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^2
+ 12*(b^5*c^4*d - a^4*b*d^5)*x)/(b^6*c*d^5 - a*b^5*d^6)

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Sympy [A]  time = 11.1271, size = 298, normalized size = 2.06 \[ \frac{a^{5} \log{\left (x + \frac{\frac{a^{7} d^{6}}{b \left (a d - b c\right )} - \frac{2 a^{6} c d^{5}}{a d - b c} + \frac{a^{5} b c^{2} d^{4}}{a d - b c} + a^{5} c d^{4} + a b^{4} c^{5}}{a^{5} d^{5} + b^{5} c^{5}} \right )}}{b^{5} \left (a d - b c\right )} - \frac{c^{5} \log{\left (x + \frac{a^{5} c d^{4} - \frac{a^{2} b^{4} c^{5} d}{a d - b c} + \frac{2 a b^{5} c^{6}}{a d - b c} + a b^{4} c^{5} - \frac{b^{6} c^{7}}{d \left (a d - b c\right )}}{a^{5} d^{5} + b^{5} c^{5}} \right )}}{d^{5} \left (a d - b c\right )} + \frac{x^{4}}{4 b d} - \frac{x^{3} \left (a d + b c\right )}{3 b^{2} d^{2}} + \frac{x^{2} \left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right )}{2 b^{3} d^{3}} - \frac{x \left (a^{3} d^{3} + a^{2} b c d^{2} + a b^{2} c^{2} d + b^{3} c^{3}\right )}{b^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x+a)/(d*x+c),x)

[Out]

a**5*log(x + (a**7*d**6/(b*(a*d - b*c)) - 2*a**6*c*d**5/(a*d - b*c) + a**5*b*c**
2*d**4/(a*d - b*c) + a**5*c*d**4 + a*b**4*c**5)/(a**5*d**5 + b**5*c**5))/(b**5*(
a*d - b*c)) - c**5*log(x + (a**5*c*d**4 - a**2*b**4*c**5*d/(a*d - b*c) + 2*a*b**
5*c**6/(a*d - b*c) + a*b**4*c**5 - b**6*c**7/(d*(a*d - b*c)))/(a**5*d**5 + b**5*
c**5))/(d**5*(a*d - b*c)) + x**4/(4*b*d) - x**3*(a*d + b*c)/(3*b**2*d**2) + x**2
*(a**2*d**2 + a*b*c*d + b**2*c**2)/(2*b**3*d**3) - x*(a**3*d**3 + a**2*b*c*d**2
+ a*b**2*c**2*d + b**3*c**3)/(b**4*d**4)

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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^5/((b*x + a)*(d*x + c)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError