3.281 \(\int \frac{x^5}{(a+b x)^2 (c+d x)^2} \, dx\)

Optimal. Leaf size=142 \[ \frac{a^5}{b^4 (a+b x) (b c-a d)^2}+\frac{a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}-\frac{2 x (a d+b c)}{b^3 d^3}+\frac{c^5}{d^4 (c+d x) (b c-a d)^2}+\frac{c^4 (3 b c-5 a d) \log (c+d x)}{d^4 (b c-a d)^3}+\frac{x^2}{2 b^2 d^2} \]

[Out]

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

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Rubi [A]  time = 0.166142, antiderivative size = 142, 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, Rules used = {88} \[ \frac{a^5}{b^4 (a+b x) (b c-a d)^2}+\frac{a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}-\frac{2 x (a d+b c)}{b^3 d^3}+\frac{c^5}{d^4 (c+d x) (b c-a d)^2}+\frac{c^4 (3 b c-5 a d) \log (c+d x)}{d^4 (b c-a d)^3}+\frac{x^2}{2 b^2 d^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x)^2*(c + d*x)^2),x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{x^5}{(a+b x)^2 (c+d x)^2} \, dx &=\int \left (-\frac{2 (b c+a d)}{b^3 d^3}+\frac{x}{b^2 d^2}-\frac{a^5}{b^3 (b c-a d)^2 (a+b x)^2}-\frac{a^4 (-5 b c+3 a d)}{b^3 (b c-a d)^3 (a+b x)}-\frac{c^5}{d^3 (-b c+a d)^2 (c+d x)^2}-\frac{c^4 (3 b c-5 a d)}{d^3 (-b c+a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac{2 (b c+a d) x}{b^3 d^3}+\frac{x^2}{2 b^2 d^2}+\frac{a^5}{b^4 (b c-a d)^2 (a+b x)}+\frac{c^5}{d^4 (b c-a d)^2 (c+d x)}+\frac{a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}+\frac{c^4 (3 b c-5 a d) \log (c+d x)}{d^4 (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.15974, size = 142, normalized size = 1. \[ \frac{a^5}{b^4 (a+b x) (b c-a d)^2}+\frac{a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}-\frac{2 x (a d+b c)}{b^3 d^3}+\frac{c^5}{d^4 (c+d x) (b c-a d)^2}+\frac{c^4 (5 a d-3 b c) \log (c+d x)}{d^4 (a d-b c)^3}+\frac{x^2}{2 b^2 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/((a + b*x)^2*(c + d*x)^2),x]

[Out]

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

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Maple [A]  time = 0.013, size = 181, normalized size = 1.3 \begin{align*}{\frac{{x}^{2}}{2\,{b}^{2}{d}^{2}}}-2\,{\frac{ax}{{d}^{2}{b}^{3}}}-2\,{\frac{cx}{{d}^{3}{b}^{2}}}+5\,{\frac{{c}^{4}\ln \left ( dx+c \right ) a}{{d}^{3} \left ( ad-bc \right ) ^{3}}}-3\,{\frac{{c}^{5}\ln \left ( dx+c \right ) b}{{d}^{4} \left ( ad-bc \right ) ^{3}}}+{\frac{{c}^{5}}{{d}^{4} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+3\,{\frac{{a}^{5}\ln \left ( bx+a \right ) d}{{b}^{4} \left ( ad-bc \right ) ^{3}}}-5\,{\frac{{a}^{4}\ln \left ( bx+a \right ) c}{{b}^{3} \left ( ad-bc \right ) ^{3}}}+{\frac{{a}^{5}}{{b}^{4} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)^2/(d*x+c)^2,x)

[Out]

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

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Maxima [B]  time = 1.10152, size = 419, normalized size = 2.95 \begin{align*} \frac{{\left (5 \, a^{4} b c - 3 \, a^{5} d\right )} \log \left (b x + a\right )}{b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}} + \frac{{\left (3 \, b c^{5} - 5 \, a c^{4} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}} + \frac{a b^{4} c^{5} + a^{5} c d^{4} +{\left (b^{5} c^{5} + a^{5} d^{5}\right )} x}{a b^{6} c^{3} d^{4} - 2 \, a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} +{\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{2} +{\left (b^{7} c^{3} d^{4} - a b^{6} c^{2} d^{5} - a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x} + \frac{b d x^{2} - 4 \,{\left (b c + a d\right )} x}{2 \, b^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.40163, size = 1224, normalized size = 8.62 \begin{align*} \frac{2 \, a b^{5} c^{6} - 2 \, a^{2} b^{4} c^{5} d + 2 \, a^{5} b c^{2} d^{4} - 2 \, a^{6} c d^{5} +{\left (b^{6} c^{3} d^{3} - 3 \, a b^{5} c^{2} d^{4} + 3 \, a^{2} b^{4} c d^{5} - a^{3} b^{3} d^{6}\right )} x^{4} - 3 \,{\left (b^{6} c^{4} d^{2} - 2 \, a b^{5} c^{3} d^{3} + 2 \, a^{3} b^{3} c d^{5} - a^{4} b^{2} d^{6}\right )} x^{3} -{\left (4 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} - 5 \, a^{2} b^{4} c^{3} d^{3} + 5 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - 4 \, a^{5} b d^{6}\right )} x^{2} + 2 \,{\left (b^{6} c^{6} - 3 \, a b^{5} c^{5} d + 4 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{4} b^{2} c^{2} d^{4} + 3 \, a^{5} b c d^{5} - a^{6} d^{6}\right )} x + 2 \,{\left (5 \, a^{5} b c^{2} d^{4} - 3 \, a^{6} c d^{5} +{\left (5 \, a^{4} b^{2} c d^{5} - 3 \, a^{5} b d^{6}\right )} x^{2} +{\left (5 \, a^{4} b^{2} c^{2} d^{4} + 2 \, a^{5} b c d^{5} - 3 \, a^{6} d^{6}\right )} x\right )} \log \left (b x + a\right ) + 2 \,{\left (3 \, a b^{5} c^{6} - 5 \, a^{2} b^{4} c^{5} d +{\left (3 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2}\right )} x^{2} +{\left (3 \, b^{6} c^{6} - 2 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{7} c^{4} d^{4} - 3 \, a^{2} b^{6} c^{3} d^{5} + 3 \, a^{3} b^{5} c^{2} d^{6} - a^{4} b^{4} c d^{7} +{\left (b^{8} c^{3} d^{5} - 3 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} - a^{3} b^{5} d^{8}\right )} x^{2} +{\left (b^{8} c^{4} d^{4} - 2 \, a b^{7} c^{3} d^{5} + 2 \, a^{3} b^{5} c d^{7} - a^{4} b^{4} d^{8}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 6.10004, size = 726, normalized size = 5.11 \begin{align*} \frac{a^{4} \left (3 a d - 5 b c\right ) \log{\left (x + \frac{\frac{a^{8} d^{7} \left (3 a d - 5 b c\right )}{b \left (a d - b c\right )^{3}} - \frac{4 a^{7} c d^{6} \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} + \frac{6 a^{6} b c^{2} d^{5} \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} - \frac{4 a^{5} b^{2} c^{3} d^{4} \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} + 3 a^{5} c d^{4} + \frac{a^{4} b^{3} c^{4} d^{3} \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} - 5 a^{4} b c^{2} d^{3} - 5 a^{2} b^{3} c^{4} d + 3 a b^{4} c^{5}}{3 a^{5} d^{5} - 5 a^{4} b c d^{4} - 5 a b^{4} c^{4} d + 3 b^{5} c^{5}} \right )}}{b^{4} \left (a d - b c\right )^{3}} + \frac{c^{4} \left (5 a d - 3 b c\right ) \log{\left (x + \frac{3 a^{5} c d^{4} + \frac{a^{4} b^{3} c^{4} d^{3} \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 5 a^{4} b c^{2} d^{3} - \frac{4 a^{3} b^{4} c^{5} d^{2} \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{5} c^{6} d \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 5 a^{2} b^{3} c^{4} d - \frac{4 a b^{6} c^{7} \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + 3 a b^{4} c^{5} + \frac{b^{7} c^{8} \left (5 a d - 3 b c\right )}{d \left (a d - b c\right )^{3}}}{3 a^{5} d^{5} - 5 a^{4} b c d^{4} - 5 a b^{4} c^{4} d + 3 b^{5} c^{5}} \right )}}{d^{4} \left (a d - b c\right )^{3}} + \frac{a^{5} c d^{4} + a b^{4} c^{5} + x \left (a^{5} d^{5} + b^{5} c^{5}\right )}{a^{3} b^{4} c d^{6} - 2 a^{2} b^{5} c^{2} d^{5} + a b^{6} c^{3} d^{4} + x^{2} \left (a^{2} b^{5} d^{7} - 2 a b^{6} c d^{6} + b^{7} c^{2} d^{5}\right ) + x \left (a^{3} b^{4} d^{7} - a^{2} b^{5} c d^{6} - a b^{6} c^{2} d^{5} + b^{7} c^{3} d^{4}\right )} + \frac{x^{2}}{2 b^{2} d^{2}} - \frac{x \left (2 a d + 2 b c\right )}{b^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x+a)**2/(d*x+c)**2,x)

[Out]

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

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Giac [B]  time = 1.23702, size = 560, normalized size = 3.94 \begin{align*} \frac{a^{5} b^{4}}{{\left (b^{10} c^{2} - 2 \, a b^{9} c d + a^{2} b^{8} d^{2}\right )}{\left (b x + a\right )}} + \frac{{\left (3 \, b^{2} c^{5} - 5 \, a b c^{4} d\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{4} - 3 \, a b^{3} c^{2} d^{5} + 3 \, a^{2} b^{2} c d^{6} - a^{3} b d^{7}} - \frac{{\left (3 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{4} d^{4}} + \frac{{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6} - \frac{3 \, b^{5} c^{4} d^{2} - 2 \, a b^{4} c^{3} d^{3} - 12 \, a^{2} b^{3} c^{2} d^{4} + 18 \, a^{3} b^{2} c d^{5} - 7 \, a^{4} b d^{6}}{{\left (b x + a\right )} b} - \frac{2 \,{\left (3 \, b^{7} c^{5} d - 5 \, a b^{6} c^{4} d^{2} + 10 \, a^{3} b^{4} c^{2} d^{4} - 10 \, a^{4} b^{3} c d^{5} + 3 \, a^{5} b^{2} d^{6}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )}{\left (b x + a\right )}^{2}}{2 \,{\left (b c - a d\right )}^{3} b^{4}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(d*x+c)^2,x, algorithm="giac")

[Out]

a^5*b^4/((b^10*c^2 - 2*a*b^9*c*d + a^2*b^8*d^2)*(b*x + a)) + (3*b^2*c^5 - 5*a*b*c^4*d)*log(abs(b*c/(b*x + a) -
a*d/(b*x + a) + d))/(b^4*c^3*d^4 - 3*a*b^3*c^2*d^5 + 3*a^2*b^2*c*d^6 - a^3*b*d^7) - (3*b^2*c^2 + 4*a*b*c*d +
3*a^2*d^2)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/(b^4*d^4) + 1/2*(b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d
^5 - a^3*d^6 - (3*b^5*c^4*d^2 - 2*a*b^4*c^3*d^3 - 12*a^2*b^3*c^2*d^4 + 18*a^3*b^2*c*d^5 - 7*a^4*b*d^6)/((b*x +
a)*b) - 2*(3*b^7*c^5*d - 5*a*b^6*c^4*d^2 + 10*a^3*b^4*c^2*d^4 - 10*a^4*b^3*c*d^5 + 3*a^5*b^2*d^6)/((b*x + a)^
2*b^2))*(b*x + a)^2/((b*c - a*d)^3*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^4)