3.231 \(\int \frac{(c+d x)^3}{x^6 (a+b x)} \, dx\)

Optimal. Leaf size=150 \[ -\frac{c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 a^3 x^3}-\frac{b^2 \log (x) (b c-a d)^3}{a^6}+\frac{b^2 (b c-a d)^3 \log (a+b x)}{a^6}+\frac{c^2 (b c-3 a d)}{4 a^2 x^4}+\frac{(b c-a d)^3}{2 a^4 x^2}-\frac{b (b c-a d)^3}{a^5 x}-\frac{c^3}{5 a x^5} \]

[Out]

-c^3/(5*a*x^5) + (c^2*(b*c - 3*a*d))/(4*a^2*x^4) - (c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2))/(3*a^3*x^3) + (b*c -
a*d)^3/(2*a^4*x^2) - (b*(b*c - a*d)^3)/(a^5*x) - (b^2*(b*c - a*d)^3*Log[x])/a^6 + (b^2*(b*c - a*d)^3*Log[a + b
*x])/a^6

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Rubi [A]  time = 0.0897376, antiderivative size = 150, 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} \[ -\frac{c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 a^3 x^3}-\frac{b^2 \log (x) (b c-a d)^3}{a^6}+\frac{b^2 (b c-a d)^3 \log (a+b x)}{a^6}+\frac{c^2 (b c-3 a d)}{4 a^2 x^4}+\frac{(b c-a d)^3}{2 a^4 x^2}-\frac{b (b c-a d)^3}{a^5 x}-\frac{c^3}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^6*(a + b*x)),x]

[Out]

-c^3/(5*a*x^5) + (c^2*(b*c - 3*a*d))/(4*a^2*x^4) - (c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2))/(3*a^3*x^3) + (b*c -
a*d)^3/(2*a^4*x^2) - (b*(b*c - a*d)^3)/(a^5*x) - (b^2*(b*c - a*d)^3*Log[x])/a^6 + (b^2*(b*c - a*d)^3*Log[a + b
*x])/a^6

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)^3}{x^6 (a+b x)} \, dx &=\int \left (\frac{c^3}{a x^6}+\frac{c^2 (-b c+3 a d)}{a^2 x^5}+\frac{c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^4}+\frac{(-b c+a d)^3}{a^4 x^3}-\frac{b (-b c+a d)^3}{a^5 x^2}+\frac{b^2 (-b c+a d)^3}{a^6 x}-\frac{b^3 (-b c+a d)^3}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac{c^3}{5 a x^5}+\frac{c^2 (b c-3 a d)}{4 a^2 x^4}-\frac{c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^3}+\frac{(b c-a d)^3}{2 a^4 x^2}-\frac{b (b c-a d)^3}{a^5 x}-\frac{b^2 (b c-a d)^3 \log (x)}{a^6}+\frac{b^2 (b c-a d)^3 \log (a+b x)}{a^6}\\ \end{align*}

Mathematica [A]  time = 0.0744288, size = 188, normalized size = 1.25 \[ \frac{-10 a^3 b^2 c x^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )+30 a^2 b^3 c^2 x^3 (c+6 d x)+15 a^4 b x \left (4 c^2 d x+c^3+6 c d^2 x^2+4 d^3 x^3\right )-3 a^5 \left (15 c^2 d x+4 c^3+20 c d^2 x^2+10 d^3 x^3\right )-60 a b^4 c^3 x^4-60 b^2 x^5 \log (x) (b c-a d)^3+60 b^2 x^5 (b c-a d)^3 \log (a+b x)}{60 a^6 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(x^6*(a + b*x)),x]

[Out]

(-60*a*b^4*c^3*x^4 + 30*a^2*b^3*c^2*x^3*(c + 6*d*x) - 10*a^3*b^2*c*x^2*(2*c^2 + 9*c*d*x + 18*d^2*x^2) + 15*a^4
*b*x*(c^3 + 4*c^2*d*x + 6*c*d^2*x^2 + 4*d^3*x^3) - 3*a^5*(4*c^3 + 15*c^2*d*x + 20*c*d^2*x^2 + 10*d^3*x^3) - 60
*b^2*(b*c - a*d)^3*x^5*Log[x] + 60*b^2*(b*c - a*d)^3*x^5*Log[a + b*x])/(60*a^6*x^5)

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Maple [B]  time = 0.01, size = 305, normalized size = 2. \begin{align*} -{\frac{{c}^{3}}{5\,a{x}^{5}}}-{\frac{{d}^{3}}{2\,a{x}^{2}}}+{\frac{3\,bc{d}^{2}}{2\,{a}^{2}{x}^{2}}}-{\frac{3\,{b}^{2}{c}^{2}d}{2\,{a}^{3}{x}^{2}}}+{\frac{{b}^{3}{c}^{3}}{2\,{a}^{4}{x}^{2}}}-{\frac{c{d}^{2}}{a{x}^{3}}}+{\frac{{c}^{2}bd}{{a}^{2}{x}^{3}}}-{\frac{{c}^{3}{b}^{2}}{3\,{a}^{3}{x}^{3}}}-{\frac{3\,{c}^{2}d}{4\,a{x}^{4}}}+{\frac{{c}^{3}b}{4\,{a}^{2}{x}^{4}}}+{\frac{{b}^{2}\ln \left ( x \right ){d}^{3}}{{a}^{3}}}-3\,{\frac{{b}^{3}\ln \left ( x \right ) c{d}^{2}}{{a}^{4}}}+3\,{\frac{{b}^{4}\ln \left ( x \right ){c}^{2}d}{{a}^{5}}}-{\frac{{b}^{5}\ln \left ( x \right ){c}^{3}}{{a}^{6}}}+{\frac{b{d}^{3}}{{a}^{2}x}}-3\,{\frac{{b}^{2}c{d}^{2}}{{a}^{3}x}}+3\,{\frac{{b}^{3}{c}^{2}d}{{a}^{4}x}}-{\frac{{b}^{4}{c}^{3}}{{a}^{5}x}}-{\frac{{b}^{2}\ln \left ( bx+a \right ){d}^{3}}{{a}^{3}}}+3\,{\frac{{b}^{3}\ln \left ( bx+a \right ) c{d}^{2}}{{a}^{4}}}-3\,{\frac{{b}^{4}\ln \left ( bx+a \right ){c}^{2}d}{{a}^{5}}}+{\frac{{b}^{5}\ln \left ( bx+a \right ){c}^{3}}{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/x^6/(b*x+a),x)

[Out]

-1/5*c^3/a/x^5-1/2/a/x^2*d^3+3/2/a^2/x^2*c*b*d^2-3/2/a^3/x^2*b^2*c^2*d+1/2/a^4/x^2*b^3*c^3-c/a/x^3*d^2+c^2/a^2
/x^3*b*d-1/3*c^3/a^3/x^3*b^2-3/4*c^2/a/x^4*d+1/4*c^3/a^2/x^4*b+1/a^3*b^2*ln(x)*d^3-3/a^4*b^3*ln(x)*c*d^2+3/a^5
*b^4*ln(x)*c^2*d-1/a^6*b^5*ln(x)*c^3+1/a^2*b/x*d^3-3/a^3*b^2/x*c*d^2+3/a^4*b^3/x*c^2*d-1/a^5*b^4/x*c^3-1/a^3*b
^2*ln(b*x+a)*d^3+3/a^4*b^3*ln(b*x+a)*c*d^2-3/a^5*b^4*ln(b*x+a)*c^2*d+1/a^6*b^5*ln(b*x+a)*c^3

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Maxima [A]  time = 1.05762, size = 352, normalized size = 2.35 \begin{align*} \frac{{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (b x + a\right )}{a^{6}} - \frac{{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (x\right )}{a^{6}} - \frac{12 \, a^{4} c^{3} + 60 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} - 30 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 20 \,{\left (a^{2} b^{2} c^{3} - 3 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 15 \,{\left (a^{3} b c^{3} - 3 \, a^{4} c^{2} d\right )} x}{60 \, a^{5} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^6/(b*x+a),x, algorithm="maxima")

[Out]

(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(b*x + a)/a^6 - (b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*
b^3*c*d^2 - a^3*b^2*d^3)*log(x)/a^6 - 1/60*(12*a^4*c^3 + 60*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b
*d^3)*x^4 - 30*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x^3 + 20*(a^2*b^2*c^3 - 3*a^3*b*c^2*d +
3*a^4*c*d^2)*x^2 - 15*(a^3*b*c^3 - 3*a^4*c^2*d)*x)/(a^5*x^5)

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Fricas [A]  time = 2.37232, size = 548, normalized size = 3.65 \begin{align*} -\frac{12 \, a^{5} c^{3} - 60 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} \log \left (b x + a\right ) + 60 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} \log \left (x\right ) + 60 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 30 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 20 \,{\left (a^{3} b^{2} c^{3} - 3 \, a^{4} b c^{2} d + 3 \, a^{5} c d^{2}\right )} x^{2} - 15 \,{\left (a^{4} b c^{3} - 3 \, a^{5} c^{2} d\right )} x}{60 \, a^{6} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^6/(b*x+a),x, algorithm="fricas")

[Out]

-1/60*(12*a^5*c^3 - 60*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5*log(b*x + a) + 60*(b^5*c^
3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5*log(x) + 60*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*
d^2 - a^4*b*d^3)*x^4 - 30*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*x^3 + 20*(a^3*b^2*c^3 - 3*
a^4*b*c^2*d + 3*a^5*c*d^2)*x^2 - 15*(a^4*b*c^3 - 3*a^5*c^2*d)*x)/(a^6*x^5)

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Sympy [B]  time = 2.1519, size = 418, normalized size = 2.79 \begin{align*} \frac{- 12 a^{4} c^{3} + x^{4} \left (60 a^{3} b d^{3} - 180 a^{2} b^{2} c d^{2} + 180 a b^{3} c^{2} d - 60 b^{4} c^{3}\right ) + x^{3} \left (- 30 a^{4} d^{3} + 90 a^{3} b c d^{2} - 90 a^{2} b^{2} c^{2} d + 30 a b^{3} c^{3}\right ) + x^{2} \left (- 60 a^{4} c d^{2} + 60 a^{3} b c^{2} d - 20 a^{2} b^{2} c^{3}\right ) + x \left (- 45 a^{4} c^{2} d + 15 a^{3} b c^{3}\right )}{60 a^{5} x^{5}} + \frac{b^{2} \left (a d - b c\right )^{3} \log{\left (x + \frac{a^{4} b^{2} d^{3} - 3 a^{3} b^{3} c d^{2} + 3 a^{2} b^{4} c^{2} d - a b^{5} c^{3} - a b^{2} \left (a d - b c\right )^{3}}{2 a^{3} b^{3} d^{3} - 6 a^{2} b^{4} c d^{2} + 6 a b^{5} c^{2} d - 2 b^{6} c^{3}} \right )}}{a^{6}} - \frac{b^{2} \left (a d - b c\right )^{3} \log{\left (x + \frac{a^{4} b^{2} d^{3} - 3 a^{3} b^{3} c d^{2} + 3 a^{2} b^{4} c^{2} d - a b^{5} c^{3} + a b^{2} \left (a d - b c\right )^{3}}{2 a^{3} b^{3} d^{3} - 6 a^{2} b^{4} c d^{2} + 6 a b^{5} c^{2} d - 2 b^{6} c^{3}} \right )}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/x**6/(b*x+a),x)

[Out]

(-12*a**4*c**3 + x**4*(60*a**3*b*d**3 - 180*a**2*b**2*c*d**2 + 180*a*b**3*c**2*d - 60*b**4*c**3) + x**3*(-30*a
**4*d**3 + 90*a**3*b*c*d**2 - 90*a**2*b**2*c**2*d + 30*a*b**3*c**3) + x**2*(-60*a**4*c*d**2 + 60*a**3*b*c**2*d
- 20*a**2*b**2*c**3) + x*(-45*a**4*c**2*d + 15*a**3*b*c**3))/(60*a**5*x**5) + b**2*(a*d - b*c)**3*log(x + (a*
*4*b**2*d**3 - 3*a**3*b**3*c*d**2 + 3*a**2*b**4*c**2*d - a*b**5*c**3 - a*b**2*(a*d - b*c)**3)/(2*a**3*b**3*d**
3 - 6*a**2*b**4*c*d**2 + 6*a*b**5*c**2*d - 2*b**6*c**3))/a**6 - b**2*(a*d - b*c)**3*log(x + (a**4*b**2*d**3 -
3*a**3*b**3*c*d**2 + 3*a**2*b**4*c**2*d - a*b**5*c**3 + a*b**2*(a*d - b*c)**3)/(2*a**3*b**3*d**3 - 6*a**2*b**4
*c*d**2 + 6*a*b**5*c**2*d - 2*b**6*c**3))/a**6

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Giac [A]  time = 1.19287, size = 366, normalized size = 2.44 \begin{align*} -\frac{{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac{12 \, a^{5} c^{3} + 60 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 30 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 20 \,{\left (a^{3} b^{2} c^{3} - 3 \, a^{4} b c^{2} d + 3 \, a^{5} c d^{2}\right )} x^{2} - 15 \,{\left (a^{4} b c^{3} - 3 \, a^{5} c^{2} d\right )} x}{60 \, a^{6} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^6/(b*x+a),x, algorithm="giac")

[Out]

-(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(abs(x))/a^6 + (b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*
b^4*c*d^2 - a^3*b^3*d^3)*log(abs(b*x + a))/(a^6*b) - 1/60*(12*a^5*c^3 + 60*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^
3*b^2*c*d^2 - a^4*b*d^3)*x^4 - 30*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*x^3 + 20*(a^3*b^2*
c^3 - 3*a^4*b*c^2*d + 3*a^5*c*d^2)*x^2 - 15*(a^4*b*c^3 - 3*a^5*c^2*d)*x)/(a^6*x^5)