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

Optimal. Leaf size=62 \[ \frac{b c-a d}{a^2 x}+\frac{b \log (x) (b c-a d)}{a^3}-\frac{b (b c-a d) \log (a+b x)}{a^3}-\frac{c}{2 a x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.039749, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{b c-a d}{a^2 x}+\frac{b \log (x) (b c-a d)}{a^3}-\frac{b (b c-a d) \log (a+b x)}{a^3}-\frac{c}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(x^3*(a + b*x)),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0316975, size = 58, normalized size = 0.94 \[ \frac{-\frac{a (a c+2 a d x-2 b c x)}{x^2}+2 b \log (x) (b c-a d)+2 b (a d-b c) \log (a+b x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 75, normalized size = 1.2 \begin{align*} -{\frac{c}{2\,a{x}^{2}}}-{\frac{d}{ax}}+{\frac{bc}{{a}^{2}x}}-{\frac{b\ln \left ( x \right ) d}{{a}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) c}{{a}^{3}}}+{\frac{b\ln \left ( bx+a \right ) d}{{a}^{2}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) c}{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.169, size = 85, normalized size = 1.37 \begin{align*} -\frac{{\left (b^{2} c - a b d\right )} \log \left (b x + a\right )}{a^{3}} + \frac{{\left (b^{2} c - a b d\right )} \log \left (x\right )}{a^{3}} - \frac{a c - 2 \,{\left (b c - a d\right )} x}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.98901, size = 154, normalized size = 2.48 \begin{align*} -\frac{2 \,{\left (b^{2} c - a b d\right )} x^{2} \log \left (b x + a\right ) - 2 \,{\left (b^{2} c - a b d\right )} x^{2} \log \left (x\right ) + a^{2} c - 2 \,{\left (a b c - a^{2} d\right )} x}{2 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.718709, size = 131, normalized size = 2.11 \begin{align*} - \frac{a c + x \left (2 a d - 2 b c\right )}{2 a^{2} x^{2}} - \frac{b \left (a d - b c\right ) \log{\left (x + \frac{a^{2} b d - a b^{2} c - a b \left (a d - b c\right )}{2 a b^{2} d - 2 b^{3} c} \right )}}{a^{3}} + \frac{b \left (a d - b c\right ) \log{\left (x + \frac{a^{2} b d - a b^{2} c + a b \left (a d - b c\right )}{2 a b^{2} d - 2 b^{3} c} \right )}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17258, size = 101, normalized size = 1.63 \begin{align*} \frac{{\left (b^{2} c - a b d\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} c - a b^{2} d\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac{a^{2} c - 2 \,{\left (a b c - a^{2} d\right )} x}{2 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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