3.212 \(\int \frac{c+d x}{x^4 (a+b x)} \, dx\)

Optimal. Leaf size=86 \[ -\frac{b^2 \log (x) (b c-a d)}{a^4}+\frac{b^2 (b c-a d) \log (a+b x)}{a^4}+\frac{b c-a d}{2 a^2 x^2}-\frac{b (b c-a d)}{a^3 x}-\frac{c}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0538269, antiderivative size = 86, 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} \[ -\frac{b^2 \log (x) (b c-a d)}{a^4}+\frac{b^2 (b c-a d) \log (a+b x)}{a^4}+\frac{b c-a d}{2 a^2 x^2}-\frac{b (b c-a d)}{a^3 x}-\frac{c}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(x^4*(a + b*x)),x]

[Out]

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

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

Mathematica [A]  time = 0.048604, size = 81, normalized size = 0.94 \[ \frac{\frac{a \left (a^2 (-(2 c+3 d x))+3 a b x (c+2 d x)-6 b^2 c x^2\right )}{x^3}+6 b^2 \log (x) (a d-b c)+6 b^2 (b c-a d) \log (a+b x)}{6 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(x^4*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.008, size = 101, normalized size = 1.2 \begin{align*} -{\frac{c}{3\,a{x}^{3}}}-{\frac{d}{2\,a{x}^{2}}}+{\frac{bc}{2\,{a}^{2}{x}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) d}{{a}^{3}}}-{\frac{{b}^{3}\ln \left ( x \right ) c}{{a}^{4}}}+{\frac{bd}{{a}^{2}x}}-{\frac{{b}^{2}c}{{a}^{3}x}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) d}{{a}^{3}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) c}{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)/x^4/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.05152, size = 120, normalized size = 1.4 \begin{align*} \frac{{\left (b^{3} c - a b^{2} d\right )} \log \left (b x + a\right )}{a^{4}} - \frac{{\left (b^{3} c - a b^{2} d\right )} \log \left (x\right )}{a^{4}} - \frac{2 \, a^{2} c + 6 \,{\left (b^{2} c - a b d\right )} x^{2} - 3 \,{\left (a b c - a^{2} d\right )} x}{6 \, a^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/x^4/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.96911, size = 201, normalized size = 2.34 \begin{align*} \frac{6 \,{\left (b^{3} c - a b^{2} d\right )} x^{3} \log \left (b x + a\right ) - 6 \,{\left (b^{3} c - a b^{2} d\right )} x^{3} \log \left (x\right ) - 2 \, a^{3} c - 6 \,{\left (a b^{2} c - a^{2} b d\right )} x^{2} + 3 \,{\left (a^{2} b c - a^{3} d\right )} x}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/x^4/(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.832352, size = 165, normalized size = 1.92 \begin{align*} \frac{- 2 a^{2} c + x^{2} \left (6 a b d - 6 b^{2} c\right ) + x \left (- 3 a^{2} d + 3 a b c\right )}{6 a^{3} x^{3}} + \frac{b^{2} \left (a d - b c\right ) \log{\left (x + \frac{a^{2} b^{2} d - a b^{3} c - a b^{2} \left (a d - b c\right )}{2 a b^{3} d - 2 b^{4} c} \right )}}{a^{4}} - \frac{b^{2} \left (a d - b c\right ) \log{\left (x + \frac{a^{2} b^{2} d - a b^{3} c + a b^{2} \left (a d - b c\right )}{2 a b^{3} d - 2 b^{4} c} \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/x**4/(b*x+a),x)

[Out]

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

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Giac [A]  time = 1.21994, size = 134, normalized size = 1.56 \begin{align*} -\frac{{\left (b^{3} c - a b^{2} d\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (b^{4} c - a b^{3} d\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, a^{3} c + 6 \,{\left (a b^{2} c - a^{2} b d\right )} x^{2} - 3 \,{\left (a^{2} b c - a^{3} d\right )} x}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/x^4/(b*x+a),x, algorithm="giac")

[Out]

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