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

Optimal. Leaf size=58 \[ -\left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \log (a+b x)+\frac{c^2 \log (x)}{a^2}+\frac{(b c-a d)^2}{a b^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0416725, antiderivative size = 58, 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} \[ -\left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \log (a+b x)+\frac{c^2 \log (x)}{a^2}+\frac{(b c-a d)^2}{a b^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x*(a + b*x)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0487028, size = 60, normalized size = 1.03 \[ \frac{\frac{(a d-b c) ((a+b x) (a d+b c) \log (a+b x)+a (a d-b c))}{b^2 (a+b x)}+c^2 \log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 81, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}\ln \left ( x \right ) }{{a}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{2}}{{b}^{2}}}-{\frac{\ln \left ( bx+a \right ){c}^{2}}{{a}^{2}}}+{\frac{a{d}^{2}}{{b}^{2} \left ( bx+a \right ) }}-2\,{\frac{cd}{b \left ( bx+a \right ) }}+{\frac{{c}^{2}}{a \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05438, size = 105, normalized size = 1.81 \begin{align*} \frac{c^{2} \log \left (x\right )}{a^{2}} + \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{a b^{3} x + a^{2} b^{2}} - \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.27635, size = 209, normalized size = 3.6 \begin{align*} \frac{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} -{\left (a b^{2} c^{2} - a^{3} d^{2} +{\left (b^{3} c^{2} - a^{2} b d^{2}\right )} x\right )} \log \left (b x + a\right ) +{\left (b^{3} c^{2} x + a b^{2} c^{2}\right )} \log \left (x\right )}{a^{2} b^{3} x + a^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.11712, size = 107, normalized size = 1.84 \begin{align*} \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{a^{2} b^{2} + a b^{3} x} + \frac{c^{2} \log{\left (x \right )}}{a^{2}} + \frac{\left (a d - b c\right ) \left (a d + b c\right ) \log{\left (x + \frac{- a b c^{2} + \frac{a \left (a d - b c\right ) \left (a d + b c\right )}{b}}{a^{2} d^{2} - 2 b^{2} c^{2}} \right )}}{a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16944, size = 146, normalized size = 2.52 \begin{align*} -b{\left (\frac{d^{2} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} - \frac{c^{2} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{2} b} - \frac{\frac{b^{3} c^{2}}{b x + a} - \frac{2 \, a b^{2} c d}{b x + a} + \frac{a^{2} b d^{2}}{b x + a}}{a b^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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