3.220 \(\int \frac{x^5}{(a+b x) (c+d x)^3} \, dx\)

Optimal. Leaf size=161 \[ -\frac{a^5 \log (a+b x)}{b^3 (b c-a d)^3}+\frac{c^3 \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^3}-\frac{x (a d+3 b c)}{b^2 d^4}-\frac{c^5}{2 d^5 (c+d x)^2 (b c-a d)}+\frac{c^4 (4 b c-5 a d)}{d^5 (c+d x) (b c-a d)^2}+\frac{x^2}{2 b d^3} \]

[Out]

-(((3*b*c + a*d)*x)/(b^2*d^4)) + x^2/(2*b*d^3) - c^5/(2*d^5*(b*c - a*d)*(c + d*x
)^2) + (c^4*(4*b*c - 5*a*d))/(d^5*(b*c - a*d)^2*(c + d*x)) - (a^5*Log[a + b*x])/
(b^3*(b*c - a*d)^3) + (c^3*(6*b^2*c^2 - 15*a*b*c*d + 10*a^2*d^2)*Log[c + d*x])/(
d^5*(b*c - a*d)^3)

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Rubi [A]  time = 0.398896, antiderivative size = 161, 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 \[ -\frac{a^5 \log (a+b x)}{b^3 (b c-a d)^3}+\frac{c^3 \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^3}-\frac{x (a d+3 b c)}{b^2 d^4}-\frac{c^5}{2 d^5 (c+d x)^2 (b c-a d)}+\frac{c^4 (4 b c-5 a d)}{d^5 (c+d x) (b c-a d)^2}+\frac{x^2}{2 b d^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((a + b*x)*(c + d*x)^3),x]

[Out]

-(((3*b*c + a*d)*x)/(b^2*d^4)) + x^2/(2*b*d^3) - c^5/(2*d^5*(b*c - a*d)*(c + d*x
)^2) + (c^4*(4*b*c - 5*a*d))/(d^5*(b*c - a*d)^2*(c + d*x)) - (a^5*Log[a + b*x])/
(b^3*(b*c - a*d)^3) + (c^3*(6*b^2*c^2 - 15*a*b*c*d + 10*a^2*d^2)*Log[c + d*x])/(
d^5*(b*c - a*d)^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{5} \log{\left (a + b x \right )}}{b^{3} \left (a d - b c\right )^{3}} + \frac{c^{5}}{2 d^{5} \left (c + d x\right )^{2} \left (a d - b c\right )} - \frac{c^{4} \left (5 a d - 4 b c\right )}{d^{5} \left (c + d x\right ) \left (a d - b c\right )^{2}} - \frac{c^{3} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right ) \log{\left (c + d x \right )}}{d^{5} \left (a d - b c\right )^{3}} - \frac{\left (a d + 3 b c\right ) \int \frac{1}{b^{2}}\, dx}{d^{4}} + \frac{\int x\, dx}{b d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x+a)/(d*x+c)**3,x)

[Out]

a**5*log(a + b*x)/(b**3*(a*d - b*c)**3) + c**5/(2*d**5*(c + d*x)**2*(a*d - b*c))
- c**4*(5*a*d - 4*b*c)/(d**5*(c + d*x)*(a*d - b*c)**2) - c**3*(10*a**2*d**2 - 1
5*a*b*c*d + 6*b**2*c**2)*log(c + d*x)/(d**5*(a*d - b*c)**3) - (a*d + 3*b*c)*Inte
gral(b**(-2), x)/d**4 + Integral(x, x)/(b*d**3)

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Mathematica [A]  time = 0.350869, size = 161, normalized size = 1. \[ \frac{1}{2} \left (-\frac{2 a^5 \log (a+b x)}{b^3 (b c-a d)^3}-\frac{2 c^3 \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right ) \log (c+d x)}{d^5 (a d-b c)^3}-\frac{2 x (a d+3 b c)}{b^2 d^4}+\frac{c^5}{d^5 (c+d x)^2 (a d-b c)}+\frac{2 c^4 (4 b c-5 a d)}{d^5 (c+d x) (b c-a d)^2}+\frac{x^2}{b d^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((a + b*x)*(c + d*x)^3),x]

[Out]

((-2*(3*b*c + a*d)*x)/(b^2*d^4) + x^2/(b*d^3) + c^5/(d^5*(-(b*c) + a*d)*(c + d*x
)^2) + (2*c^4*(4*b*c - 5*a*d))/(d^5*(b*c - a*d)^2*(c + d*x)) - (2*a^5*Log[a + b*
x])/(b^3*(b*c - a*d)^3) - (2*c^3*(6*b^2*c^2 - 15*a*b*c*d + 10*a^2*d^2)*Log[c + d
*x])/(d^5*(-(b*c) + a*d)^3))/2

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Maple [A]  time = 0.02, size = 213, normalized size = 1.3 \[{\frac{{x}^{2}}{2\,b{d}^{3}}}-{\frac{ax}{{d}^{3}{b}^{2}}}-3\,{\frac{cx}{b{d}^{4}}}-5\,{\frac{{c}^{4}a}{{d}^{4} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+4\,{\frac{b{c}^{5}}{ \left ( ad-bc \right ) ^{2}{d}^{5} \left ( dx+c \right ) }}+{\frac{{c}^{5}}{2\,{d}^{5} \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}-10\,{\frac{{c}^{3}\ln \left ( dx+c \right ){a}^{2}}{{d}^{3} \left ( ad-bc \right ) ^{3}}}+15\,{\frac{{c}^{4}\ln \left ( dx+c \right ) ab}{{d}^{4} \left ( ad-bc \right ) ^{3}}}-6\,{\frac{{c}^{5}\ln \left ( dx+c \right ){b}^{2}}{{d}^{5} \left ( ad-bc \right ) ^{3}}}+{\frac{{a}^{5}\ln \left ( bx+a \right ) }{{b}^{3} \left ( ad-bc \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x+a)/(d*x+c)^3,x)

[Out]

1/2*x^2/b/d^3-1/b^2/d^3*a*x-3/b/d^4*x*c-5/d^4*c^4/(a*d-b*c)^2/(d*x+c)*a+4/d^5*c^
5/(a*d-b*c)^2/(d*x+c)*b+1/2/d^5*c^5/(a*d-b*c)/(d*x+c)^2-10/d^3*c^3/(a*d-b*c)^3*l
n(d*x+c)*a^2+15/d^4*c^4/(a*d-b*c)^3*ln(d*x+c)*a*b-6/d^5*c^5/(a*d-b*c)^3*ln(d*x+c
)*b^2+1/b^3*a^5/(a*d-b*c)^3*ln(b*x+a)

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Maxima [A]  time = 1.37131, size = 392, normalized size = 2.43 \[ -\frac{a^{5} \log \left (b x + a\right )}{b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}} + \frac{{\left (6 \, b^{2} c^{5} - 15 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}} + \frac{7 \, b c^{6} - 9 \, a c^{5} d + 2 \,{\left (4 \, b c^{5} d - 5 \, a c^{4} d^{2}\right )} x}{2 \,{\left (b^{2} c^{4} d^{5} - 2 \, a b c^{3} d^{6} + a^{2} c^{2} d^{7} +{\left (b^{2} c^{2} d^{7} - 2 \, a b c d^{8} + a^{2} d^{9}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d^{6} - 2 \, a b c^{2} d^{7} + a^{2} c d^{8}\right )} x\right )}} + \frac{b d x^{2} - 2 \,{\left (3 \, b c + a d\right )} x}{2 \, b^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)*(d*x + c)^3),x, algorithm="maxima")

[Out]

-a^5*log(b*x + a)/(b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3) + (6
*b^2*c^5 - 15*a*b*c^4*d + 10*a^2*c^3*d^2)*log(d*x + c)/(b^3*c^3*d^5 - 3*a*b^2*c^
2*d^6 + 3*a^2*b*c*d^7 - a^3*d^8) + 1/2*(7*b*c^6 - 9*a*c^5*d + 2*(4*b*c^5*d - 5*a
*c^4*d^2)*x)/(b^2*c^4*d^5 - 2*a*b*c^3*d^6 + a^2*c^2*d^7 + (b^2*c^2*d^7 - 2*a*b*c
*d^8 + a^2*d^9)*x^2 + 2*(b^2*c^3*d^6 - 2*a*b*c^2*d^7 + a^2*c*d^8)*x) + 1/2*(b*d*
x^2 - 2*(3*b*c + a*d)*x)/(b^2*d^4)

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Fricas [A]  time = 0.256465, size = 782, normalized size = 4.86 \[ \frac{7 \, b^{5} c^{7} - 16 \, a b^{4} c^{6} d + 9 \, a^{2} b^{3} c^{5} d^{2} +{\left (b^{5} c^{3} d^{4} - 3 \, a b^{4} c^{2} d^{5} + 3 \, a^{2} b^{3} c d^{6} - a^{3} b^{2} d^{7}\right )} x^{4} - 2 \,{\left (2 \, b^{5} c^{4} d^{3} - 5 \, a b^{4} c^{3} d^{4} + 3 \, a^{2} b^{3} c^{2} d^{5} + a^{3} b^{2} c d^{6} - a^{4} b d^{7}\right )} x^{3} -{\left (11 \, b^{5} c^{5} d^{2} - 29 \, a b^{4} c^{4} d^{3} + 21 \, a^{2} b^{3} c^{3} d^{4} + a^{3} b^{2} c^{2} d^{5} - 4 \, a^{4} b c d^{6}\right )} x^{2} + 2 \,{\left (b^{5} c^{6} d - a b^{4} c^{5} d^{2} - a^{2} b^{3} c^{4} d^{3} + a^{4} b c^{2} d^{5}\right )} x - 2 \,{\left (a^{5} d^{7} x^{2} + 2 \, a^{5} c d^{6} x + a^{5} c^{2} d^{5}\right )} \log \left (b x + a\right ) + 2 \,{\left (6 \, b^{5} c^{7} - 15 \, a b^{4} c^{6} d + 10 \, a^{2} b^{3} c^{5} d^{2} +{\left (6 \, b^{5} c^{5} d^{2} - 15 \, a b^{4} c^{4} d^{3} + 10 \, a^{2} b^{3} c^{3} d^{4}\right )} x^{2} + 2 \,{\left (6 \, b^{5} c^{6} d - 15 \, a b^{4} c^{5} d^{2} + 10 \, a^{2} b^{3} c^{4} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (b^{6} c^{5} d^{5} - 3 \, a b^{5} c^{4} d^{6} + 3 \, a^{2} b^{4} c^{3} d^{7} - a^{3} b^{3} c^{2} d^{8} +{\left (b^{6} c^{3} d^{7} - 3 \, a b^{5} c^{2} d^{8} + 3 \, a^{2} b^{4} c d^{9} - a^{3} b^{3} d^{10}\right )} x^{2} + 2 \,{\left (b^{6} c^{4} d^{6} - 3 \, a b^{5} c^{3} d^{7} + 3 \, a^{2} b^{4} c^{2} d^{8} - a^{3} b^{3} c d^{9}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)*(d*x + c)^3),x, algorithm="fricas")

[Out]

1/2*(7*b^5*c^7 - 16*a*b^4*c^6*d + 9*a^2*b^3*c^5*d^2 + (b^5*c^3*d^4 - 3*a*b^4*c^2
*d^5 + 3*a^2*b^3*c*d^6 - a^3*b^2*d^7)*x^4 - 2*(2*b^5*c^4*d^3 - 5*a*b^4*c^3*d^4 +
3*a^2*b^3*c^2*d^5 + a^3*b^2*c*d^6 - a^4*b*d^7)*x^3 - (11*b^5*c^5*d^2 - 29*a*b^4
*c^4*d^3 + 21*a^2*b^3*c^3*d^4 + a^3*b^2*c^2*d^5 - 4*a^4*b*c*d^6)*x^2 + 2*(b^5*c^
6*d - a*b^4*c^5*d^2 - a^2*b^3*c^4*d^3 + a^4*b*c^2*d^5)*x - 2*(a^5*d^7*x^2 + 2*a^
5*c*d^6*x + a^5*c^2*d^5)*log(b*x + a) + 2*(6*b^5*c^7 - 15*a*b^4*c^6*d + 10*a^2*b
^3*c^5*d^2 + (6*b^5*c^5*d^2 - 15*a*b^4*c^4*d^3 + 10*a^2*b^3*c^3*d^4)*x^2 + 2*(6*
b^5*c^6*d - 15*a*b^4*c^5*d^2 + 10*a^2*b^3*c^4*d^3)*x)*log(d*x + c))/(b^6*c^5*d^5
- 3*a*b^5*c^4*d^6 + 3*a^2*b^4*c^3*d^7 - a^3*b^3*c^2*d^8 + (b^6*c^3*d^7 - 3*a*b^
5*c^2*d^8 + 3*a^2*b^4*c*d^9 - a^3*b^3*d^10)*x^2 + 2*(b^6*c^4*d^6 - 3*a*b^5*c^3*d
^7 + 3*a^2*b^4*c^2*d^8 - a^3*b^3*c*d^9)*x)

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Sympy [A]  time = 26.5566, size = 745, normalized size = 4.63 \[ \frac{a^{5} \log{\left (x + \frac{\frac{a^{9} d^{8}}{b \left (a d - b c\right )^{3}} - \frac{4 a^{8} c d^{7}}{\left (a d - b c\right )^{3}} + \frac{6 a^{7} b c^{2} d^{6}}{\left (a d - b c\right )^{3}} - \frac{4 a^{6} b^{2} c^{3} d^{5}}{\left (a d - b c\right )^{3}} + \frac{a^{5} b^{3} c^{4} d^{4}}{\left (a d - b c\right )^{3}} + a^{5} c d^{4} + 10 a^{3} b^{2} c^{3} d^{2} - 15 a^{2} b^{3} c^{4} d + 6 a b^{4} c^{5}}{a^{5} d^{5} + 10 a^{2} b^{3} c^{3} d^{2} - 15 a b^{4} c^{4} d + 6 b^{5} c^{5}} \right )}}{b^{3} \left (a d - b c\right )^{3}} - \frac{c^{3} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right ) \log{\left (x + \frac{a^{5} c d^{4} - \frac{a^{4} b^{2} c^{3} d^{3} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b^{3} c^{4} d^{2} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 10 a^{3} b^{2} c^{3} d^{2} - \frac{6 a^{2} b^{4} c^{5} d \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} - 15 a^{2} b^{3} c^{4} d + \frac{4 a b^{5} c^{6} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 6 a b^{4} c^{5} - \frac{b^{6} c^{7} \left (10 a^{2} d^{2} - 15 a b c d + 6 b^{2} c^{2}\right )}{d \left (a d - b c\right )^{3}}}{a^{5} d^{5} + 10 a^{2} b^{3} c^{3} d^{2} - 15 a b^{4} c^{4} d + 6 b^{5} c^{5}} \right )}}{d^{5} \left (a d - b c\right )^{3}} - \frac{9 a c^{5} d - 7 b c^{6} + x \left (10 a c^{4} d^{2} - 8 b c^{5} d\right )}{2 a^{2} c^{2} d^{7} - 4 a b c^{3} d^{6} + 2 b^{2} c^{4} d^{5} + x^{2} \left (2 a^{2} d^{9} - 4 a b c d^{8} + 2 b^{2} c^{2} d^{7}\right ) + x \left (4 a^{2} c d^{8} - 8 a b c^{2} d^{7} + 4 b^{2} c^{3} d^{6}\right )} + \frac{x^{2}}{2 b d^{3}} - \frac{x \left (a d + 3 b c\right )}{b^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x+a)/(d*x+c)**3,x)

[Out]

a**5*log(x + (a**9*d**8/(b*(a*d - b*c)**3) - 4*a**8*c*d**7/(a*d - b*c)**3 + 6*a*
*7*b*c**2*d**6/(a*d - b*c)**3 - 4*a**6*b**2*c**3*d**5/(a*d - b*c)**3 + a**5*b**3
*c**4*d**4/(a*d - b*c)**3 + a**5*c*d**4 + 10*a**3*b**2*c**3*d**2 - 15*a**2*b**3*
c**4*d + 6*a*b**4*c**5)/(a**5*d**5 + 10*a**2*b**3*c**3*d**2 - 15*a*b**4*c**4*d +
6*b**5*c**5))/(b**3*(a*d - b*c)**3) - c**3*(10*a**2*d**2 - 15*a*b*c*d + 6*b**2*
c**2)*log(x + (a**5*c*d**4 - a**4*b**2*c**3*d**3*(10*a**2*d**2 - 15*a*b*c*d + 6*
b**2*c**2)/(a*d - b*c)**3 + 4*a**3*b**3*c**4*d**2*(10*a**2*d**2 - 15*a*b*c*d + 6
*b**2*c**2)/(a*d - b*c)**3 + 10*a**3*b**2*c**3*d**2 - 6*a**2*b**4*c**5*d*(10*a**
2*d**2 - 15*a*b*c*d + 6*b**2*c**2)/(a*d - b*c)**3 - 15*a**2*b**3*c**4*d + 4*a*b*
*5*c**6*(10*a**2*d**2 - 15*a*b*c*d + 6*b**2*c**2)/(a*d - b*c)**3 + 6*a*b**4*c**5
- b**6*c**7*(10*a**2*d**2 - 15*a*b*c*d + 6*b**2*c**2)/(d*(a*d - b*c)**3))/(a**5
*d**5 + 10*a**2*b**3*c**3*d**2 - 15*a*b**4*c**4*d + 6*b**5*c**5))/(d**5*(a*d - b
*c)**3) - (9*a*c**5*d - 7*b*c**6 + x*(10*a*c**4*d**2 - 8*b*c**5*d))/(2*a**2*c**2
*d**7 - 4*a*b*c**3*d**6 + 2*b**2*c**4*d**5 + x**2*(2*a**2*d**9 - 4*a*b*c*d**8 +
2*b**2*c**2*d**7) + x*(4*a**2*c*d**8 - 8*a*b*c**2*d**7 + 4*b**2*c**3*d**6)) + x*
*2/(2*b*d**3) - x*(a*d + 3*b*c)/(b**2*d**4)

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GIAC/XCAS [A]  time = 0.297383, size = 339, normalized size = 2.11 \[ -\frac{a^{5}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}} + \frac{{\left (6 \, b^{2} c^{5} - 15 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}} + \frac{b d^{3} x^{2} - 6 \, b c d^{2} x - 2 \, a d^{3} x}{2 \, b^{2} d^{6}} + \frac{7 \, b^{2} c^{7} - 16 \, a b c^{6} d + 9 \, a^{2} c^{5} d^{2} + 2 \,{\left (4 \, b^{2} c^{6} d - 9 \, a b c^{5} d^{2} + 5 \, a^{2} c^{4} d^{3}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)*(d*x + c)^3),x, algorithm="giac")

[Out]

-a^5*ln(abs(b*x + a))/(b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)
+ (6*b^2*c^5 - 15*a*b*c^4*d + 10*a^2*c^3*d^2)*ln(abs(d*x + c))/(b^3*c^3*d^5 - 3*
a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a^3*d^8) + 1/2*(b*d^3*x^2 - 6*b*c*d^2*x - 2*a*d^
3*x)/(b^2*d^6) + 1/2*(7*b^2*c^7 - 16*a*b*c^6*d + 9*a^2*c^5*d^2 + 2*(4*b^2*c^6*d
- 9*a*b*c^5*d^2 + 5*a^2*c^4*d^3)*x)/((b*c - a*d)^3*(d*x + c)^2*d^5)