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

Optimal. Leaf size=42 \[ -\frac{(b c-a d)^2 \log (a+b x)}{a b^2}+\frac{c^2 \log (x)}{a}+\frac{d^2 x}{b} \]

[Out]

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

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Rubi [A]  time = 0.0312313, antiderivative size = 42, 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 = {72} \[ -\frac{(b c-a d)^2 \log (a+b x)}{a b^2}+\frac{c^2 \log (x)}{a}+\frac{d^2 x}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x*(a + b*x)),x]

[Out]

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0175927, size = 42, normalized size = 1. \[ \frac{-(b c-a d)^2 \log (a+b x)+a b d^2 x+b^2 c^2 \log (x)}{a b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 61, normalized size = 1.5 \begin{align*}{\frac{{d}^{2}x}{b}}+{\frac{{c}^{2}\ln \left ( x \right ) }{a}}-{\frac{a\ln \left ( bx+a \right ){d}^{2}}{{b}^{2}}}+2\,{\frac{\ln \left ( bx+a \right ) cd}{b}}-{\frac{\ln \left ( bx+a \right ){c}^{2}}{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04671, size = 72, normalized size = 1.71 \begin{align*} \frac{d^{2} x}{b} + \frac{c^{2} \log \left (x\right )}{a} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.01297, size = 115, normalized size = 2.74 \begin{align*} \frac{a b d^{2} x + b^{2} c^{2} \log \left (x\right ) -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.10084, size = 73, normalized size = 1.74 \begin{align*} \frac{d^{2} x}{b} + \frac{c^{2} \log{\left (x \right )}}{a} - \frac{\left (a d - b c\right )^{2} \log{\left (x + \frac{a b c^{2} + \frac{a \left (a d - b c\right )^{2}}{b}}{a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}} \right )}}{a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.25439, size = 74, normalized size = 1.76 \begin{align*} \frac{d^{2} x}{b} + \frac{c^{2} \log \left ({\left | x \right |}\right )}{a} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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