3.278 \(\int \frac{(c+d x)^3}{x^5 (a+b x)^2} \, dx\)

Optimal. Leaf size=162 \[ \frac{c^2 (2 b c-3 a d)}{3 a^3 x^3}-\frac{3 c (b c-a d)^2}{2 a^4 x^2}+\frac{(b c-a d)^2 (4 b c-a d)}{a^5 x}+\frac{b (b c-a d)^3}{a^5 (a+b x)}+\frac{b \log (x) (5 b c-2 a d) (b c-a d)^2}{a^6}-\frac{b (5 b c-2 a d) (b c-a d)^2 \log (a+b x)}{a^6}-\frac{c^3}{4 a^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.137446, antiderivative size = 162, 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} \[ \frac{c^2 (2 b c-3 a d)}{3 a^3 x^3}-\frac{3 c (b c-a d)^2}{2 a^4 x^2}+\frac{(b c-a d)^2 (4 b c-a d)}{a^5 x}+\frac{b (b c-a d)^3}{a^5 (a+b x)}+\frac{b \log (x) (5 b c-2 a d) (b c-a d)^2}{a^6}-\frac{b (5 b c-2 a d) (b c-a d)^2 \log (a+b x)}{a^6}-\frac{c^3}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^5*(a + b*x)^2),x]

[Out]

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

Mathematica [A]  time = 0.0830055, size = 155, normalized size = 0.96 \[ -\frac{\frac{4 a^3 c^2 (3 a d-2 b c)}{x^3}+\frac{18 a^2 c (b c-a d)^2}{x^2}+\frac{3 a^4 c^3}{x^4}+\frac{12 a (b c-a d)^2 (a d-4 b c)}{x}+\frac{12 a b (a d-b c)^3}{a+b x}-12 b \log (x) (5 b c-2 a d) (b c-a d)^2+12 b (5 b c-2 a d) (b c-a d)^2 \log (a+b x)}{12 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(x^5*(a + b*x)^2),x]

[Out]

-((3*a^4*c^3)/x^4 + (4*a^3*c^2*(-2*b*c + 3*a*d))/x^3 + (18*a^2*c*(b*c - a*d)^2)/x^2 + (12*a*(b*c - a*d)^2*(-4*
b*c + a*d))/x + (12*a*b*(-(b*c) + a*d)^3)/(a + b*x) - 12*b*(5*b*c - 2*a*d)*(b*c - a*d)^2*Log[x] + 12*b*(5*b*c
- 2*a*d)*(b*c - a*d)^2*Log[a + b*x])/(12*a^6)

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Maple [B]  time = 0.012, size = 320, normalized size = 2. \begin{align*} -{\frac{{c}^{3}}{4\,{a}^{2}{x}^{4}}}-{\frac{{d}^{3}}{{a}^{2}x}}+6\,{\frac{c{d}^{2}b}{{a}^{3}x}}-9\,{\frac{{c}^{2}d{b}^{2}}{{a}^{4}x}}+4\,{\frac{{b}^{3}{c}^{3}}{{a}^{5}x}}-{\frac{{c}^{2}d}{{a}^{2}{x}^{3}}}+{\frac{2\,{c}^{3}b}{3\,{a}^{3}{x}^{3}}}-2\,{\frac{b\ln \left ( x \right ){d}^{3}}{{a}^{3}}}+9\,{\frac{{b}^{2}\ln \left ( x \right ) c{d}^{2}}{{a}^{4}}}-12\,{\frac{{b}^{3}\ln \left ( x \right ){c}^{2}d}{{a}^{5}}}+5\,{\frac{{b}^{4}{c}^{3}\ln \left ( x \right ) }{{a}^{6}}}-{\frac{3\,c{d}^{2}}{2\,{a}^{2}{x}^{2}}}+3\,{\frac{{c}^{2}db}{{a}^{3}{x}^{2}}}-{\frac{3\,{c}^{3}{b}^{2}}{2\,{a}^{4}{x}^{2}}}+2\,{\frac{b\ln \left ( bx+a \right ){d}^{3}}{{a}^{3}}}-9\,{\frac{{b}^{2}\ln \left ( bx+a \right ) c{d}^{2}}{{a}^{4}}}+12\,{\frac{{b}^{3}\ln \left ( bx+a \right ){c}^{2}d}{{a}^{5}}}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ){c}^{3}}{{a}^{6}}}-{\frac{{d}^{3}b}{{a}^{2} \left ( bx+a \right ) }}+3\,{\frac{c{d}^{2}{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-3\,{\frac{{c}^{2}d{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}+{\frac{{c}^{3}{b}^{4}}{{a}^{5} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/x^5/(b*x+a)^2,x)

[Out]

-1/4*c^3/a^2/x^4-1/a^2/x*d^3+6/a^3/x*c*b*d^2-9/a^4/x*b^2*c^2*d+4/a^5/x*b^3*c^3-c^2/a^2/x^3*d+2/3*c^3/a^3/x^3*b
-2*b/a^3*ln(x)*d^3+9*b^2/a^4*ln(x)*c*d^2-12*b^3/a^5*ln(x)*c^2*d+5*b^4/a^6*ln(x)*c^3-3/2*c/a^2/x^2*d^2+3*c^2/a^
3/x^2*b*d-3/2*c^3/a^4/x^2*b^2+2*b/a^3*ln(b*x+a)*d^3-9*b^2/a^4*ln(b*x+a)*c*d^2+12*b^3/a^5*ln(b*x+a)*c^2*d-5*b^4
/a^6*ln(b*x+a)*c^3-b/a^2/(b*x+a)*d^3+3*b^2/a^3/(b*x+a)*c*d^2-3*b^3/a^4/(b*x+a)*c^2*d+b^4/a^5/(b*x+a)*c^3

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Maxima [A]  time = 1.1008, size = 371, normalized size = 2.29 \begin{align*} -\frac{3 \, a^{4} c^{3} - 12 \,{\left (5 \, b^{4} c^{3} - 12 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{4} - 6 \,{\left (5 \, a b^{3} c^{3} - 12 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{3} + 2 \,{\left (5 \, a^{2} b^{2} c^{3} - 12 \, a^{3} b c^{2} d + 9 \, a^{4} c d^{2}\right )} x^{2} -{\left (5 \, a^{3} b c^{3} - 12 \, a^{4} c^{2} d\right )} x}{12 \,{\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} - \frac{{\left (5 \, b^{4} c^{3} - 12 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} \log \left (b x + a\right )}{a^{6}} + \frac{{\left (5 \, b^{4} c^{3} - 12 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} \log \left (x\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^5/(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/12*(3*a^4*c^3 - 12*(5*b^4*c^3 - 12*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 - 2*a^3*b*d^3)*x^4 - 6*(5*a*b^3*c^3 - 12*a
^2*b^2*c^2*d + 9*a^3*b*c*d^2 - 2*a^4*d^3)*x^3 + 2*(5*a^2*b^2*c^3 - 12*a^3*b*c^2*d + 9*a^4*c*d^2)*x^2 - (5*a^3*
b*c^3 - 12*a^4*c^2*d)*x)/(a^5*b*x^5 + a^6*x^4) - (5*b^4*c^3 - 12*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 - 2*a^3*b*d^3)*
log(b*x + a)/a^6 + (5*b^4*c^3 - 12*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 - 2*a^3*b*d^3)*log(x)/a^6

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Fricas [B]  time = 2.35555, size = 788, normalized size = 4.86 \begin{align*} -\frac{3 \, a^{5} c^{3} - 12 \,{\left (5 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 2 \, a^{4} b d^{3}\right )} x^{4} - 6 \,{\left (5 \, a^{2} b^{3} c^{3} - 12 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - 2 \, a^{5} d^{3}\right )} x^{3} + 2 \,{\left (5 \, a^{3} b^{2} c^{3} - 12 \, a^{4} b c^{2} d + 9 \, a^{5} c d^{2}\right )} x^{2} -{\left (5 \, a^{4} b c^{3} - 12 \, a^{5} c^{2} d\right )} x + 12 \,{\left ({\left (5 \, b^{5} c^{3} - 12 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b^{2} d^{3}\right )} x^{5} +{\left (5 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 2 \, a^{4} b d^{3}\right )} x^{4}\right )} \log \left (b x + a\right ) - 12 \,{\left ({\left (5 \, b^{5} c^{3} - 12 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b^{2} d^{3}\right )} x^{5} +{\left (5 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 2 \, a^{4} b d^{3}\right )} x^{4}\right )} \log \left (x\right )}{12 \,{\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^5/(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/12*(3*a^5*c^3 - 12*(5*a*b^4*c^3 - 12*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - 2*a^4*b*d^3)*x^4 - 6*(5*a^2*b^3*c^3
- 12*a^3*b^2*c^2*d + 9*a^4*b*c*d^2 - 2*a^5*d^3)*x^3 + 2*(5*a^3*b^2*c^3 - 12*a^4*b*c^2*d + 9*a^5*c*d^2)*x^2 - (
5*a^4*b*c^3 - 12*a^5*c^2*d)*x + 12*((5*b^5*c^3 - 12*a*b^4*c^2*d + 9*a^2*b^3*c*d^2 - 2*a^3*b^2*d^3)*x^5 + (5*a*
b^4*c^3 - 12*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - 2*a^4*b*d^3)*x^4)*log(b*x + a) - 12*((5*b^5*c^3 - 12*a*b^4*c^2*
d + 9*a^2*b^3*c*d^2 - 2*a^3*b^2*d^3)*x^5 + (5*a*b^4*c^3 - 12*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - 2*a^4*b*d^3)*x^
4)*log(x))/(a^6*b*x^5 + a^7*x^4)

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Sympy [B]  time = 2.48642, size = 466, normalized size = 2.88 \begin{align*} - \frac{3 a^{4} c^{3} + x^{4} \left (24 a^{3} b d^{3} - 108 a^{2} b^{2} c d^{2} + 144 a b^{3} c^{2} d - 60 b^{4} c^{3}\right ) + x^{3} \left (12 a^{4} d^{3} - 54 a^{3} b c d^{2} + 72 a^{2} b^{2} c^{2} d - 30 a b^{3} c^{3}\right ) + x^{2} \left (18 a^{4} c d^{2} - 24 a^{3} b c^{2} d + 10 a^{2} b^{2} c^{3}\right ) + x \left (12 a^{4} c^{2} d - 5 a^{3} b c^{3}\right )}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} - \frac{b \left (a d - b c\right )^{2} \left (2 a d - 5 b c\right ) \log{\left (x + \frac{2 a^{4} b d^{3} - 9 a^{3} b^{2} c d^{2} + 12 a^{2} b^{3} c^{2} d - 5 a b^{4} c^{3} - a b \left (a d - b c\right )^{2} \left (2 a d - 5 b c\right )}{4 a^{3} b^{2} d^{3} - 18 a^{2} b^{3} c d^{2} + 24 a b^{4} c^{2} d - 10 b^{5} c^{3}} \right )}}{a^{6}} + \frac{b \left (a d - b c\right )^{2} \left (2 a d - 5 b c\right ) \log{\left (x + \frac{2 a^{4} b d^{3} - 9 a^{3} b^{2} c d^{2} + 12 a^{2} b^{3} c^{2} d - 5 a b^{4} c^{3} + a b \left (a d - b c\right )^{2} \left (2 a d - 5 b c\right )}{4 a^{3} b^{2} d^{3} - 18 a^{2} b^{3} c d^{2} + 24 a b^{4} c^{2} d - 10 b^{5} c^{3}} \right )}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/x**5/(b*x+a)**2,x)

[Out]

-(3*a**4*c**3 + x**4*(24*a**3*b*d**3 - 108*a**2*b**2*c*d**2 + 144*a*b**3*c**2*d - 60*b**4*c**3) + x**3*(12*a**
4*d**3 - 54*a**3*b*c*d**2 + 72*a**2*b**2*c**2*d - 30*a*b**3*c**3) + x**2*(18*a**4*c*d**2 - 24*a**3*b*c**2*d +
10*a**2*b**2*c**3) + x*(12*a**4*c**2*d - 5*a**3*b*c**3))/(12*a**6*x**4 + 12*a**5*b*x**5) - b*(a*d - b*c)**2*(2
*a*d - 5*b*c)*log(x + (2*a**4*b*d**3 - 9*a**3*b**2*c*d**2 + 12*a**2*b**3*c**2*d - 5*a*b**4*c**3 - a*b*(a*d - b
*c)**2*(2*a*d - 5*b*c))/(4*a**3*b**2*d**3 - 18*a**2*b**3*c*d**2 + 24*a*b**4*c**2*d - 10*b**5*c**3))/a**6 + b*(
a*d - b*c)**2*(2*a*d - 5*b*c)*log(x + (2*a**4*b*d**3 - 9*a**3*b**2*c*d**2 + 12*a**2*b**3*c**2*d - 5*a*b**4*c**
3 + a*b*(a*d - b*c)**2*(2*a*d - 5*b*c))/(4*a**3*b**2*d**3 - 18*a**2*b**3*c*d**2 + 24*a*b**4*c**2*d - 10*b**5*c
**3))/a**6

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Giac [B]  time = 1.24402, size = 504, normalized size = 3.11 \begin{align*} \frac{{\left (5 \, b^{5} c^{3} - 12 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b^{2} d^{3}\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{6} b} + \frac{\frac{b^{9} c^{3}}{b x + a} - \frac{3 \, a b^{8} c^{2} d}{b x + a} + \frac{3 \, a^{2} b^{7} c d^{2}}{b x + a} - \frac{a^{3} b^{6} d^{3}}{b x + a}}{a^{5} b^{5}} + \frac{77 \, b^{4} c^{3} - 156 \, a b^{3} c^{2} d + 90 \, a^{2} b^{2} c d^{2} - 12 \, a^{3} b d^{3} - \frac{4 \,{\left (65 \, a b^{5} c^{3} - 129 \, a^{2} b^{4} c^{2} d + 72 \, a^{3} b^{3} c d^{2} - 9 \, a^{4} b^{2} d^{3}\right )}}{{\left (b x + a\right )} b} + \frac{6 \,{\left (50 \, a^{2} b^{6} c^{3} - 96 \, a^{3} b^{5} c^{2} d + 51 \, a^{4} b^{4} c d^{2} - 6 \, a^{5} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac{12 \,{\left (10 \, a^{3} b^{7} c^{3} - 18 \, a^{4} b^{6} c^{2} d + 9 \, a^{5} b^{5} c d^{2} - a^{6} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}}}{12 \, a^{6}{\left (\frac{a}{b x + a} - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^5/(b*x+a)^2,x, algorithm="giac")

[Out]

(5*b^5*c^3 - 12*a*b^4*c^2*d + 9*a^2*b^3*c*d^2 - 2*a^3*b^2*d^3)*log(abs(-a/(b*x + a) + 1))/(a^6*b) + (b^9*c^3/(
b*x + a) - 3*a*b^8*c^2*d/(b*x + a) + 3*a^2*b^7*c*d^2/(b*x + a) - a^3*b^6*d^3/(b*x + a))/(a^5*b^5) + 1/12*(77*b
^4*c^3 - 156*a*b^3*c^2*d + 90*a^2*b^2*c*d^2 - 12*a^3*b*d^3 - 4*(65*a*b^5*c^3 - 129*a^2*b^4*c^2*d + 72*a^3*b^3*
c*d^2 - 9*a^4*b^2*d^3)/((b*x + a)*b) + 6*(50*a^2*b^6*c^3 - 96*a^3*b^5*c^2*d + 51*a^4*b^4*c*d^2 - 6*a^5*b^3*d^3
)/((b*x + a)^2*b^2) - 12*(10*a^3*b^7*c^3 - 18*a^4*b^6*c^2*d + 9*a^5*b^5*c*d^2 - a^6*b^4*d^3)/((b*x + a)^3*b^3)
)/(a^6*(a/(b*x + a) - 1)^4)