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

Optimal. Leaf size=114 \[ \frac{b^2 \log (x) (b c-a d)^2}{a^5}-\frac{b^2 (b c-a d)^2 \log (a+b x)}{a^5}+\frac{c (b c-2 a d)}{3 a^2 x^3}-\frac{(b c-a d)^2}{2 a^3 x^2}+\frac{b (b c-a d)^2}{a^4 x}-\frac{c^2}{4 a x^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0764039, antiderivative size = 114, 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} \[ \frac{b^2 \log (x) (b c-a d)^2}{a^5}-\frac{b^2 (b c-a d)^2 \log (a+b x)}{a^5}+\frac{c (b c-2 a d)}{3 a^2 x^3}-\frac{(b c-a d)^2}{2 a^3 x^2}+\frac{b (b c-a d)^2}{a^4 x}-\frac{c^2}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x^5*(a + b*x)),x]

[Out]

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

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

Mathematica [A]  time = 0.0844867, size = 127, normalized size = 1.11 \[ \frac{\frac{a \left (4 a^2 b x \left (c^2+3 c d x+3 d^2 x^2\right )+a^3 \left (-\left (3 c^2+8 c d x+6 d^2 x^2\right )\right )-6 a b^2 c x^2 (c+4 d x)+12 b^3 c^2 x^3\right )}{x^4}+12 b^2 \log (x) (b c-a d)^2-12 b^2 (b c-a d)^2 \log (a+b x)}{12 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 193, normalized size = 1.7 \begin{align*} -{\frac{{c}^{2}}{4\,a{x}^{4}}}-{\frac{{d}^{2}}{2\,a{x}^{2}}}+{\frac{bcd}{{a}^{2}{x}^{2}}}-{\frac{{b}^{2}{c}^{2}}{2\,{a}^{3}{x}^{2}}}-{\frac{2\,cd}{3\,a{x}^{3}}}+{\frac{{c}^{2}b}{3\,{a}^{2}{x}^{3}}}+{\frac{{b}^{2}\ln \left ( x \right ){d}^{2}}{{a}^{3}}}-2\,{\frac{{b}^{3}\ln \left ( x \right ) cd}{{a}^{4}}}+{\frac{{b}^{4}\ln \left ( x \right ){c}^{2}}{{a}^{5}}}+{\frac{b{d}^{2}}{{a}^{2}x}}-2\,{\frac{{b}^{2}cd}{{a}^{3}x}}+{\frac{{b}^{3}{c}^{2}}{{a}^{4}x}}-{\frac{{b}^{2}\ln \left ( bx+a \right ){d}^{2}}{{a}^{3}}}+2\,{\frac{{b}^{3}\ln \left ( bx+a \right ) cd}{{a}^{4}}}-{\frac{{b}^{4}\ln \left ( bx+a \right ){c}^{2}}{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.15541, size = 221, normalized size = 1.94 \begin{align*} -\frac{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{5}} + \frac{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left (x\right )}{a^{5}} - \frac{3 \, a^{3} c^{2} - 12 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 4 \,{\left (a^{2} b c^{2} - 2 \, a^{3} c d\right )} x}{12 \, a^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.21711, size = 362, normalized size = 3.18 \begin{align*} -\frac{3 \, a^{4} c^{2} + 12 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} \log \left (b x + a\right ) - 12 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} \log \left (x\right ) - 12 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3} + 6 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} - 4 \,{\left (a^{3} b c^{2} - 2 \, a^{4} c d\right )} x}{12 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 1.36932, size = 287, normalized size = 2.52 \begin{align*} \frac{- 3 a^{3} c^{2} + x^{3} \left (12 a^{2} b d^{2} - 24 a b^{2} c d + 12 b^{3} c^{2}\right ) + x^{2} \left (- 6 a^{3} d^{2} + 12 a^{2} b c d - 6 a b^{2} c^{2}\right ) + x \left (- 8 a^{3} c d + 4 a^{2} b c^{2}\right )}{12 a^{4} x^{4}} + \frac{b^{2} \left (a d - b c\right )^{2} \log{\left (x + \frac{a^{3} b^{2} d^{2} - 2 a^{2} b^{3} c d + a b^{4} c^{2} - a b^{2} \left (a d - b c\right )^{2}}{2 a^{2} b^{3} d^{2} - 4 a b^{4} c d + 2 b^{5} c^{2}} \right )}}{a^{5}} - \frac{b^{2} \left (a d - b c\right )^{2} \log{\left (x + \frac{a^{3} b^{2} d^{2} - 2 a^{2} b^{3} c d + a b^{4} c^{2} + a b^{2} \left (a d - b c\right )^{2}}{2 a^{2} b^{3} d^{2} - 4 a b^{4} c d + 2 b^{5} c^{2}} \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*a**3*c**2 + x**3*(12*a**2*b*d**2 - 24*a*b**2*c*d + 12*b**3*c**2) + x**2*(-6*a**3*d**2 + 12*a**2*b*c*d - 6*
a*b**2*c**2) + x*(-8*a**3*c*d + 4*a**2*b*c**2))/(12*a**4*x**4) + b**2*(a*d - b*c)**2*log(x + (a**3*b**2*d**2 -
2*a**2*b**3*c*d + a*b**4*c**2 - a*b**2*(a*d - b*c)**2)/(2*a**2*b**3*d**2 - 4*a*b**4*c*d + 2*b**5*c**2))/a**5
- b**2*(a*d - b*c)**2*log(x + (a**3*b**2*d**2 - 2*a**2*b**3*c*d + a*b**4*c**2 + a*b**2*(a*d - b*c)**2)/(2*a**2
*b**3*d**2 - 4*a*b**4*c*d + 2*b**5*c**2))/a**5

________________________________________________________________________________________

Giac [A]  time = 1.26041, size = 235, normalized size = 2.06 \begin{align*} \frac{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac{{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{3 \, a^{4} c^{2} - 12 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3} + 6 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} - 4 \,{\left (a^{3} b c^{2} - 2 \, a^{4} c d\right )} x}{12 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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