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

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

[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 = 0.187244, 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, Rules used = {88} \[ \frac{c^3 \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^3}-\frac{a^5 \log (a+b x)}{b^3 (b c-a d)^3}-\frac{x (a d+3 b c)}{b^2 d^4}+\frac{c^4 (4 b c-5 a d)}{d^5 (c+d x) (b c-a d)^2}-\frac{c^5}{2 d^5 (c+d x)^2 (b c-a d)}+\frac{x^2}{2 b d^3} \]

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)

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) (c+d x)^3} \, dx &=\int \left (\frac{-3 b c-a d}{b^2 d^4}+\frac{x}{b d^3}-\frac{a^5}{b^2 (b c-a d)^3 (a+b x)}-\frac{c^5}{d^4 (-b c+a d) (c+d x)^3}-\frac{c^4 (4 b c-5 a d)}{d^4 (-b c+a d)^2 (c+d x)^2}-\frac{c^3 \left (6 b^2 c^2-15 a b c d+10 a^2 d^2\right )}{d^4 (-b c+a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac{(3 b c+a d) x}{b^2 d^4}+\frac{x^2}{2 b d^3}-\frac{c^5}{2 d^5 (b c-a d) (c+d x)^2}+\frac{c^4 (4 b c-5 a d)}{d^5 (b c-a d)^2 (c+d x)}-\frac{a^5 \log (a+b x)}{b^3 (b c-a d)^3}+\frac{c^3 \left (6 b^2 c^2-15 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.204326, size = 161, normalized size = 1. \[ \frac{1}{2} \left (-\frac{2 c^3 \left (10 a^2 d^2-15 a b c d+6 b^2 c^2\right ) \log (c+d x)}{d^5 (a d-b c)^3}-\frac{2 a^5 \log (a+b x)}{b^3 (b c-a d)^3}-\frac{2 x (a d+3 b c)}{b^2 d^4}+\frac{2 c^4 (4 b c-5 a d)}{d^5 (c+d x) (b c-a d)^2}+\frac{c^5}{d^5 (c+d x)^2 (a d-b c)}+\frac{x^2}{b d^3}\right ) \]

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 + 1
0*a^2*d^2)*Log[c + d*x])/(d^5*(-(b*c) + a*d)^3))/2

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Maple [A]  time = 0.011, size = 213, normalized size = 1.3 \begin{align*}{\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{{c}^{5}b}{ \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}}} \end{align*}

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*ln(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 = 1.27882, size = 392, normalized size = 2.43 \begin{align*} -\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}} \end{align*}

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 [B]  time = 2.84422, size = 1139, normalized size = 7.07 \begin{align*} \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 )}} \end{align*}

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 [B]  time = 6.25795, size = 745, normalized size = 4.63 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.15285, size = 339, normalized size = 2.11 \begin{align*} -\frac{a^{5} \log \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 )} \log \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}} \end{align*}

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*log(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)*log(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)