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

Optimal. Leaf size=167 \[ \frac{b^2 (b c-a d)^2}{a^5 (a+b x)}+\frac{b^2 \log (x) (5 b c-3 a d) (b c-a d)}{a^6}-\frac{b^2 (5 b c-3 a d) (b c-a d) \log (a+b x)}{a^6}-\frac{(b c-a d) (3 b c-a d)}{2 a^4 x^2}+\frac{2 c (b c-a d)}{3 a^3 x^3}+\frac{2 b (b c-a d) (2 b c-a d)}{a^5 x}-\frac{c^2}{4 a^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.15398, antiderivative size = 167, 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} \[ \frac{b^2 (b c-a d)^2}{a^5 (a+b x)}+\frac{b^2 \log (x) (5 b c-3 a d) (b c-a d)}{a^6}-\frac{b^2 (5 b c-3 a d) (b c-a d) \log (a+b x)}{a^6}-\frac{(b c-a d) (3 b c-a d)}{2 a^4 x^2}+\frac{2 c (b c-a d)}{3 a^3 x^3}+\frac{2 b (b c-a d) (2 b c-a d)}{a^5 x}-\frac{c^2}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x^5*(a + b*x)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.1385, size = 182, normalized size = 1.09 \[ \frac{-\frac{6 a^2 \left (a^2 d^2-4 a b c d+3 b^2 c^2\right )}{x^2}+\frac{24 a b \left (a^2 d^2-3 a b c d+2 b^2 c^2\right )}{x}+12 b^2 \log (x) \left (3 a^2 d^2-8 a b c d+5 b^2 c^2\right )-12 b^2 \left (3 a^2 d^2-8 a b c d+5 b^2 c^2\right ) \log (a+b x)-\frac{8 a^3 c (a d-b c)}{x^3}-\frac{3 a^4 c^2}{x^4}+\frac{12 a b^2 (b c-a d)^2}{a+b x}}{12 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*a^4*c^2)/x^4 - (8*a^3*c*(-(b*c) + a*d))/x^3 - (6*a^2*(3*b^2*c^2 - 4*a*b*c*d + a^2*d^2))/x^2 + (24*a*b*(2*
b^2*c^2 - 3*a*b*c*d + a^2*d^2))/x + (12*a*b^2*(b*c - a*d)^2)/(a + b*x) + 12*b^2*(5*b^2*c^2 - 8*a*b*c*d + 3*a^2
*d^2)*Log[x] - 12*b^2*(5*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*Log[a + b*x])/(12*a^6)

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Maple [A]  time = 0.013, size = 249, normalized size = 1.5 \begin{align*} -{\frac{{c}^{2}}{4\,{a}^{2}{x}^{4}}}-{\frac{{d}^{2}}{2\,{a}^{2}{x}^{2}}}+2\,{\frac{cdb}{{a}^{3}{x}^{2}}}-{\frac{3\,{b}^{2}{c}^{2}}{2\,{a}^{4}{x}^{2}}}+3\,{\frac{{b}^{2}\ln \left ( x \right ){d}^{2}}{{a}^{4}}}-8\,{\frac{{b}^{3}\ln \left ( x \right ) cd}{{a}^{5}}}+5\,{\frac{{b}^{4}\ln \left ( x \right ){c}^{2}}{{a}^{6}}}+2\,{\frac{{d}^{2}b}{{a}^{3}x}}-6\,{\frac{cd{b}^{2}}{{a}^{4}x}}+4\,{\frac{{c}^{2}{b}^{3}}{{a}^{5}x}}-{\frac{2\,cd}{3\,{a}^{2}{x}^{3}}}+{\frac{2\,{c}^{2}b}{3\,{a}^{3}{x}^{3}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ){d}^{2}}{{a}^{4}}}+8\,{\frac{{b}^{3}\ln \left ( bx+a \right ) cd}{{a}^{5}}}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ){c}^{2}}{{a}^{6}}}+{\frac{{d}^{2}{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-2\,{\frac{cd{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}+{\frac{{c}^{2}{b}^{4}}{{a}^{5} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c^2/a^2/x^4-1/2/a^2/x^2*d^2+2/a^3/x^2*b*c*d-3/2/a^4/x^2*b^2*c^2+3*b^2/a^4*ln(x)*d^2-8*b^3/a^5*ln(x)*c*d+5
*b^4/a^6*ln(x)*c^2+2*b/a^3/x*d^2-6*b^2/a^4/x*c*d+4*b^3/a^5/x*c^2-2/3*c/a^2/x^3*d+2/3*c^2/a^3/x^3*b-3*b^2/a^4*l
n(b*x+a)*d^2+8*b^3/a^5*ln(b*x+a)*c*d-5*b^4/a^6*ln(b*x+a)*c^2+b^2/a^3/(b*x+a)*d^2-2*b^3/a^4/(b*x+a)*c*d+b^4/a^5
/(b*x+a)*c^2

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Maxima [A]  time = 1.03769, size = 301, normalized size = 1.8 \begin{align*} -\frac{3 \, a^{4} c^{2} - 12 \,{\left (5 \, b^{4} c^{2} - 8 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} - 6 \,{\left (5 \, a b^{3} c^{2} - 8 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + 2 \,{\left (5 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} x^{2} -{\left (5 \, a^{3} b c^{2} - 8 \, a^{4} c d\right )} x}{12 \,{\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} - \frac{{\left (5 \, b^{4} c^{2} - 8 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{6}} + \frac{{\left (5 \, b^{4} c^{2} - 8 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} \log \left (x\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*a^4*c^2 - 12*(5*b^4*c^2 - 8*a*b^3*c*d + 3*a^2*b^2*d^2)*x^4 - 6*(5*a*b^3*c^2 - 8*a^2*b^2*c*d + 3*a^3*b
*d^2)*x^3 + 2*(5*a^2*b^2*c^2 - 8*a^3*b*c*d + 3*a^4*d^2)*x^2 - (5*a^3*b*c^2 - 8*a^4*c*d)*x)/(a^5*b*x^5 + a^6*x^
4) - (5*b^4*c^2 - 8*a*b^3*c*d + 3*a^2*b^2*d^2)*log(b*x + a)/a^6 + (5*b^4*c^2 - 8*a*b^3*c*d + 3*a^2*b^2*d^2)*lo
g(x)/a^6

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Fricas [A]  time = 2.32285, size = 621, normalized size = 3.72 \begin{align*} -\frac{3 \, a^{5} c^{2} - 12 \,{\left (5 \, a b^{4} c^{2} - 8 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{4} - 6 \,{\left (5 \, a^{2} b^{3} c^{2} - 8 \, a^{3} b^{2} c d + 3 \, a^{4} b d^{2}\right )} x^{3} + 2 \,{\left (5 \, a^{3} b^{2} c^{2} - 8 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} x^{2} -{\left (5 \, a^{4} b c^{2} - 8 \, a^{5} c d\right )} x + 12 \,{\left ({\left (5 \, b^{5} c^{2} - 8 \, a b^{4} c d + 3 \, a^{2} b^{3} d^{2}\right )} x^{5} +{\left (5 \, a b^{4} c^{2} - 8 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{4}\right )} \log \left (b x + a\right ) - 12 \,{\left ({\left (5 \, b^{5} c^{2} - 8 \, a b^{4} c d + 3 \, a^{2} b^{3} d^{2}\right )} x^{5} +{\left (5 \, a b^{4} c^{2} - 8 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{4}\right )} \log \left (x\right )}{12 \,{\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.72496, size = 377, normalized size = 2.26 \begin{align*} \frac{- 3 a^{4} c^{2} + x^{4} \left (36 a^{2} b^{2} d^{2} - 96 a b^{3} c d + 60 b^{4} c^{2}\right ) + x^{3} \left (18 a^{3} b d^{2} - 48 a^{2} b^{2} c d + 30 a b^{3} c^{2}\right ) + x^{2} \left (- 6 a^{4} d^{2} + 16 a^{3} b c d - 10 a^{2} b^{2} c^{2}\right ) + x \left (- 8 a^{4} c d + 5 a^{3} b c^{2}\right )}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} + \frac{b^{2} \left (a d - b c\right ) \left (3 a d - 5 b c\right ) \log{\left (x + \frac{3 a^{3} b^{2} d^{2} - 8 a^{2} b^{3} c d + 5 a b^{4} c^{2} - a b^{2} \left (a d - b c\right ) \left (3 a d - 5 b c\right )}{6 a^{2} b^{3} d^{2} - 16 a b^{4} c d + 10 b^{5} c^{2}} \right )}}{a^{6}} - \frac{b^{2} \left (a d - b c\right ) \left (3 a d - 5 b c\right ) \log{\left (x + \frac{3 a^{3} b^{2} d^{2} - 8 a^{2} b^{3} c d + 5 a b^{4} c^{2} + a b^{2} \left (a d - b c\right ) \left (3 a d - 5 b c\right )}{6 a^{2} b^{3} d^{2} - 16 a b^{4} c d + 10 b^{5} c^{2}} \right )}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*a**4*c**2 + x**4*(36*a**2*b**2*d**2 - 96*a*b**3*c*d + 60*b**4*c**2) + x**3*(18*a**3*b*d**2 - 48*a**2*b**2*
c*d + 30*a*b**3*c**2) + x**2*(-6*a**4*d**2 + 16*a**3*b*c*d - 10*a**2*b**2*c**2) + x*(-8*a**4*c*d + 5*a**3*b*c*
*2))/(12*a**6*x**4 + 12*a**5*b*x**5) + b**2*(a*d - b*c)*(3*a*d - 5*b*c)*log(x + (3*a**3*b**2*d**2 - 8*a**2*b**
3*c*d + 5*a*b**4*c**2 - a*b**2*(a*d - b*c)*(3*a*d - 5*b*c))/(6*a**2*b**3*d**2 - 16*a*b**4*c*d + 10*b**5*c**2))
/a**6 - b**2*(a*d - b*c)*(3*a*d - 5*b*c)*log(x + (3*a**3*b**2*d**2 - 8*a**2*b**3*c*d + 5*a*b**4*c**2 + a*b**2*
(a*d - b*c)*(3*a*d - 5*b*c))/(6*a**2*b**3*d**2 - 16*a*b**4*c*d + 10*b**5*c**2))/a**6

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Giac [A]  time = 1.20974, size = 382, normalized size = 2.29 \begin{align*} \frac{{\left (5 \, b^{5} c^{2} - 8 \, a b^{4} c d + 3 \, a^{2} b^{3} d^{2}\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{6} b} + \frac{\frac{b^{9} c^{2}}{b x + a} - \frac{2 \, a b^{8} c d}{b x + a} + \frac{a^{2} b^{7} d^{2}}{b x + a}}{a^{5} b^{5}} + \frac{77 \, b^{4} c^{2} - 104 \, a b^{3} c d + 30 \, a^{2} b^{2} d^{2} - \frac{4 \,{\left (65 \, a b^{5} c^{2} - 86 \, a^{2} b^{4} c d + 24 \, a^{3} b^{3} d^{2}\right )}}{{\left (b x + a\right )} b} + \frac{6 \,{\left (50 \, a^{2} b^{6} c^{2} - 64 \, a^{3} b^{5} c d + 17 \, a^{4} b^{4} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac{12 \,{\left (10 \, a^{3} b^{7} c^{2} - 12 \, a^{4} b^{6} c d + 3 \, a^{5} b^{5} d^{2}\right )}}{{\left (b x + a\right )}^{3} b^{3}}}{12 \, a^{6}{\left (\frac{a}{b x + a} - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*b^5*c^2 - 8*a*b^4*c*d + 3*a^2*b^3*d^2)*log(abs(-a/(b*x + a) + 1))/(a^6*b) + (b^9*c^2/(b*x + a) - 2*a*b^8*c*
d/(b*x + a) + a^2*b^7*d^2/(b*x + a))/(a^5*b^5) + 1/12*(77*b^4*c^2 - 104*a*b^3*c*d + 30*a^2*b^2*d^2 - 4*(65*a*b
^5*c^2 - 86*a^2*b^4*c*d + 24*a^3*b^3*d^2)/((b*x + a)*b) + 6*(50*a^2*b^6*c^2 - 64*a^3*b^5*c*d + 17*a^4*b^4*d^2)
/((b*x + a)^2*b^2) - 12*(10*a^3*b^7*c^2 - 12*a^4*b^6*c*d + 3*a^5*b^5*d^2)/((b*x + a)^3*b^3))/(a^6*(a/(b*x + a)
- 1)^4)