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

Optimal. Leaf size=147 \[ \frac{x \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{b^3 d^4}-\frac{a^5 \log (a+b x)}{b^4 (b c-a d)^2}-\frac{x^2 (a d+2 b c)}{2 b^2 d^3}-\frac{c^5}{d^5 (c+d x) (b c-a d)}-\frac{c^4 (4 b c-5 a d) \log (c+d x)}{d^5 (b c-a d)^2}+\frac{x^3}{3 b d^2} \]

[Out]

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

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Rubi [A]  time = 0.162902, antiderivative size = 147, 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} \[ \frac{x \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{b^3 d^4}-\frac{a^5 \log (a+b x)}{b^4 (b c-a d)^2}-\frac{x^2 (a d+2 b c)}{2 b^2 d^3}-\frac{c^5}{d^5 (c+d x) (b c-a d)}-\frac{c^4 (4 b c-5 a d) \log (c+d x)}{d^5 (b c-a d)^2}+\frac{x^3}{3 b d^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x)*(c + d*x)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.149157, size = 147, normalized size = 1. \[ \frac{x \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{b^3 d^4}-\frac{a^5 \log (a+b x)}{b^4 (b c-a d)^2}-\frac{x^2 (a d+2 b c)}{2 b^2 d^3}+\frac{c^5}{d^5 (c+d x) (a d-b c)}+\frac{\left (5 a c^4 d-4 b c^5\right ) \log (c+d x)}{d^5 (b c-a d)^2}+\frac{x^3}{3 b d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 169, normalized size = 1.2 \begin{align*}{\frac{{x}^{3}}{3\,b{d}^{2}}}-{\frac{a{x}^{2}}{2\,{b}^{2}{d}^{2}}}-{\frac{c{x}^{2}}{b{d}^{3}}}+{\frac{{a}^{2}x}{{b}^{3}{d}^{2}}}+2\,{\frac{acx}{{b}^{2}{d}^{3}}}+3\,{\frac{{c}^{2}x}{b{d}^{4}}}+5\,{\frac{{c}^{4}\ln \left ( dx+c \right ) a}{{d}^{4} \left ( ad-bc \right ) ^{2}}}-4\,{\frac{{c}^{5}\ln \left ( dx+c \right ) b}{{d}^{5} \left ( ad-bc \right ) ^{2}}}+{\frac{{c}^{5}}{{d}^{5} \left ( ad-bc \right ) \left ( dx+c \right ) }}-{\frac{{a}^{5}\ln \left ( bx+a \right ) }{{b}^{4} \left ( ad-bc \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06269, size = 259, normalized size = 1.76 \begin{align*} -\frac{a^{5} \log \left (b x + a\right )}{b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}} - \frac{c^{5}}{b c^{2} d^{5} - a c d^{6} +{\left (b c d^{6} - a d^{7}\right )} x} - \frac{{\left (4 \, b c^{5} - 5 \, a c^{4} d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}} + \frac{2 \, b^{2} d^{2} x^{3} - 3 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x^{2} + 6 \,{\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x}{6 \, b^{3} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.41959, size = 678, normalized size = 4.61 \begin{align*} -\frac{6 \, b^{5} c^{6} - 6 \, a b^{4} c^{5} d - 2 \,{\left (b^{5} c^{2} d^{4} - 2 \, a b^{4} c d^{5} + a^{2} b^{3} d^{6}\right )} x^{4} +{\left (4 \, b^{5} c^{3} d^{3} - 5 \, a b^{4} c^{2} d^{4} - 2 \, a^{2} b^{3} c d^{5} + 3 \, a^{3} b^{2} d^{6}\right )} x^{3} - 3 \,{\left (4 \, b^{5} c^{4} d^{2} - 5 \, a b^{4} c^{3} d^{3} - a^{3} b^{2} c d^{5} + 2 \, a^{4} b d^{6}\right )} x^{2} - 6 \,{\left (3 \, b^{5} c^{5} d - 4 \, a b^{4} c^{4} d^{2} + a^{4} b c d^{5}\right )} x + 6 \,{\left (a^{5} d^{6} x + a^{5} c d^{5}\right )} \log \left (b x + a\right ) + 6 \,{\left (4 \, b^{5} c^{6} - 5 \, a b^{4} c^{5} d +{\left (4 \, b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2}\right )} x\right )} \log \left (d x + c\right )}{6 \,{\left (b^{6} c^{3} d^{5} - 2 \, a b^{5} c^{2} d^{6} + a^{2} b^{4} c d^{7} +{\left (b^{6} c^{2} d^{6} - 2 \, a b^{5} c d^{7} + a^{2} 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)/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.93202, size = 461, normalized size = 3.14 \begin{align*} - \frac{a^{5} \log{\left (x + \frac{\frac{a^{8} d^{7}}{b \left (a d - b c\right )^{2}} - \frac{3 a^{7} c d^{6}}{\left (a d - b c\right )^{2}} + \frac{3 a^{6} b c^{2} d^{5}}{\left (a d - b c\right )^{2}} - \frac{a^{5} b^{2} c^{3} d^{4}}{\left (a d - b c\right )^{2}} + a^{5} c d^{4} + 5 a^{2} b^{3} c^{4} d - 4 a b^{4} c^{5}}{a^{5} d^{5} + 5 a b^{4} c^{4} d - 4 b^{5} c^{5}} \right )}}{b^{4} \left (a d - b c\right )^{2}} + \frac{c^{5}}{a c d^{6} - b c^{2} d^{5} + x \left (a d^{7} - b c d^{6}\right )} + \frac{c^{4} \left (5 a d - 4 b c\right ) \log{\left (x + \frac{a^{5} c d^{4} - \frac{a^{3} b^{3} c^{4} d^{2} \left (5 a d - 4 b c\right )}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{4} c^{5} d \left (5 a d - 4 b c\right )}{\left (a d - b c\right )^{2}} + 5 a^{2} b^{3} c^{4} d - \frac{3 a b^{5} c^{6} \left (5 a d - 4 b c\right )}{\left (a d - b c\right )^{2}} - 4 a b^{4} c^{5} + \frac{b^{6} c^{7} \left (5 a d - 4 b c\right )}{d \left (a d - b c\right )^{2}}}{a^{5} d^{5} + 5 a b^{4} c^{4} d - 4 b^{5} c^{5}} \right )}}{d^{5} \left (a d - b c\right )^{2}} + \frac{x^{3}}{3 b d^{2}} - \frac{x^{2} \left (a d + 2 b c\right )}{2 b^{2} d^{3}} + \frac{x \left (a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right )}{b^{3} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19989, size = 332, normalized size = 2.26 \begin{align*} -\frac{c^{5} d^{4}}{{\left (b c d^{9} - a d^{10}\right )}{\left (d x + c\right )}} - \frac{a^{5} d \log \left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}} + \frac{{\left (2 \, b^{3} - \frac{3 \,{\left (4 \, b^{3} c d + a b^{2} d^{2}\right )}}{{\left (d x + c\right )} d} + \frac{6 \,{\left (6 \, b^{3} c^{2} d^{2} + 3 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )}}{{\left (d x + c\right )}^{2} d^{2}}\right )}{\left (d x + c\right )}^{3}}{6 \, b^{4} d^{5}} + \frac{{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{b^{4} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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