3.266 \(\int \frac{(c+d x)^2}{x^3 (a+b x)^2} \, dx\)

Optimal. Leaf size=103 \[ \frac{2 c (b c-a d)}{a^3 x}+\frac{(b c-a d)^2}{a^3 (a+b x)}+\frac{\log (x) (b c-a d) (3 b c-a d)}{a^4}-\frac{(b c-a d) (3 b c-a d) \log (a+b x)}{a^4}-\frac{c^2}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0794943, antiderivative size = 103, 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} \[ \frac{2 c (b c-a d)}{a^3 x}+\frac{(b c-a d)^2}{a^3 (a+b x)}+\frac{\log (x) (b c-a d) (3 b c-a d)}{a^4}-\frac{(b c-a d) (3 b c-a d) \log (a+b x)}{a^4}-\frac{c^2}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x^3*(a + b*x)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.098269, size = 109, normalized size = 1.06 \[ -\frac{-2 \log (x) \left (a^2 d^2-4 a b c d+3 b^2 c^2\right )+2 \left (a^2 d^2-4 a b c d+3 b^2 c^2\right ) \log (a+b x)+\frac{a^2 c^2}{x^2}+\frac{4 a c (a d-b c)}{x}-\frac{2 a (b c-a d)^2}{a+b x}}{2 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(x^3*(a + b*x)^2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 158, normalized size = 1.5 \begin{align*} -{\frac{{c}^{2}}{2\,{a}^{2}{x}^{2}}}+{\frac{\ln \left ( x \right ){d}^{2}}{{a}^{2}}}-4\,{\frac{b\ln \left ( x \right ) cd}{{a}^{3}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}{c}^{2}}{{a}^{4}}}-2\,{\frac{cd}{{a}^{2}x}}+2\,{\frac{{c}^{2}b}{{a}^{3}x}}-{\frac{\ln \left ( bx+a \right ){d}^{2}}{{a}^{2}}}+4\,{\frac{\ln \left ( bx+a \right ) bcd}{{a}^{3}}}-3\,{\frac{\ln \left ( bx+a \right ){b}^{2}{c}^{2}}{{a}^{4}}}+{\frac{{d}^{2}}{a \left ( bx+a \right ) }}-2\,{\frac{cdb}{{a}^{2} \left ( bx+a \right ) }}+{\frac{{b}^{2}{c}^{2}}{{a}^{3} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/x^3/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.04243, size = 182, normalized size = 1.77 \begin{align*} -\frac{a^{2} c^{2} - 2 \,{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} x^{2} -{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x}{2 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac{{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{4}} + \frac{{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \log \left (x\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/x^3/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.4063, size = 429, normalized size = 4.17 \begin{align*} -\frac{a^{3} c^{2} - 2 \,{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} -{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d\right )} x + 2 \,{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} +{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} +{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/x^3/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.60715, size = 262, normalized size = 2.54 \begin{align*} \frac{- a^{2} c^{2} + x^{2} \left (2 a^{2} d^{2} - 8 a b c d + 6 b^{2} c^{2}\right ) + x \left (- 4 a^{2} c d + 3 a b c^{2}\right )}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} + \frac{\left (a d - 3 b c\right ) \left (a d - b c\right ) \log{\left (x + \frac{a^{3} d^{2} - 4 a^{2} b c d + 3 a b^{2} c^{2} - a \left (a d - 3 b c\right ) \left (a d - b c\right )}{2 a^{2} b d^{2} - 8 a b^{2} c d + 6 b^{3} c^{2}} \right )}}{a^{4}} - \frac{\left (a d - 3 b c\right ) \left (a d - b c\right ) \log{\left (x + \frac{a^{3} d^{2} - 4 a^{2} b c d + 3 a b^{2} c^{2} + a \left (a d - 3 b c\right ) \left (a d - b c\right )}{2 a^{2} b d^{2} - 8 a b^{2} c d + 6 b^{3} c^{2}} \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/x**3/(b*x+a)**2,x)

[Out]

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

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Giac [A]  time = 1.27804, size = 224, normalized size = 2.17 \begin{align*} \frac{{\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{4} b} + \frac{\frac{b^{5} c^{2}}{b x + a} - \frac{2 \, a b^{4} c d}{b x + a} + \frac{a^{2} b^{3} d^{2}}{b x + a}}{a^{3} b^{3}} + \frac{5 \, b^{2} c^{2} - 4 \, a b c d - \frac{2 \,{\left (3 \, a b^{3} c^{2} - 2 \, a^{2} b^{2} c d\right )}}{{\left (b x + a\right )} b}}{2 \, a^{4}{\left (\frac{a}{b x + a} - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/x^3/(b*x+a)^2,x, algorithm="giac")

[Out]

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